The Probability of Large Queue Lengths and ... - Semantic Scholar

Advances in Applied Probability, 1995

The Probability of Large Queue Lengths and Waiting Times in a Heterogeneous Multiserver Queue Part I: Tight Limits Wojciech Szpankowski2 Department of Computer Science Purdue University West Lafayette, IN 47907 USA

John. S. Sadowsky1 Department of Electrical Engineering Arizona State University Tempe, AZ 85287-5706 USA

Abstract We consider a multiserver queuing process speci ed by i.i.d. interarrival time, batch size and service time sequences. In the case that dierent servers have dierent service time distributions we say the system is heterogeneous . In this paper we establish conditions for the queuing process to be characterized as a geometrically Harris recurrent Markov chain, and we characterize the stationary probabilities of large queue lengths and waiting times. The queue length is asymptotically geometric and the waiting time is asymptotically exponential. Our analysis is a generalization of the well known characterization of the GI/G/1 queue obtained using classical probabilistic techniques of exponential change of measure and renewal theory.

American Mathematical Society subject classi cations: Primary 60K25; Secondary 60F10, 60J05, 60K15, 60K20. Key Words: Queueing theory, Harris recurrent Markov chains, Foster-Lyapunov theory, Markov additive processes, renewal theory. J. S. Sadowsky is supported by the National Science Foundation (NCR-9003007). W. Szpankowski is supported by the National Science Foundation (NCR-9206315 and CCR-9201078), AFOSR (90-0107), and by the National Library of Medicine (R01 LM05118). 1

2

1. Introduction and Summary We consider a multiserver queuing process with batch arrivals that is determined by an i.i.d. interarrival time sequence, an i.i.d. batch size sequence, and an i.i.d. service time sequence for each of c servers, 1  c < 1. This model occurs in numerous applications, including manufacturing processes, multi-processor computer networks, telecommunications networks and various types of service counters. Dierent servers may have dierent service time distributions, in which case we say the system is heterogeneous . For example, a bank may have some highly ecient tellers and some less experienced tellers who will generally take longer to provide their services. Another example of a heterogeneous system is a distributed computer network with dissimilar processors. The results presented here extend the classical GI/G/1 analysis that is generally credited to Feller (1971, Ch. XII). We brie y review the GI/G/1 case in order to place the new results in the proper context. Let X be a random variable with distribution FX (
). Let X () def = log(E[eX ]) be its cumulant function , and put DX def = f: X () < 1g. As a mapping R ! (?1; +1], X (
) is convex (so DX is an interval) and l.s.c. (lower semicontinuous). X (
) is analytic on DXo , X (0) = 0, and when DXo 6= ; we have 0X (0) = E[X ] and 00X (0) = var[X ] where the derivatives are limits of interior derivatives in the case that 0 is an endpoint of DX . For any  2 DX , de ne the twisted distribution





= exp x ? X () FX (dx): FX() (dx) def

(1)

The twisted distribution's cumulant function is X() ( ) = X ( +  ) ? X (), hence, 0X () = E() [X ] and 00X () = var() [X ]. When fXk g is an i.i.d. sequence with L(Xk ) = FX (
), we call the i.i.d. sequence distribution with L(Xk ) = FX() (
) the -conjugate distribution. The GI/G/1 queuing process is determined by the i.i.d. interarrival time and service time sequences fAk g and fBk g. De ne the arrival rate  def = E[A]?1 and the service rate  def = E[B ]?1 . Then it is well known that the waiting time sequence fWk g is a Markov chain, and it is positive Harris recurrent, with stationary distribution (
), if and only if  < . De ne () def = A (?) + def def B (),  = supf: ()  0g and ! = exp(?B ()). One nds that 0 (0) = ( ? )=(), hence, by convexity,  > 0 implies 0 (0) < 0, which in turn implies the stability condition  < . The classical result states that there exists constants 0 < cW ; cQ < 1 such that

P ( Wk  w )  cW e?w and 1

P ( Q k  n )  c Q ! n

(2)

if and only if 0 <  < 1, () = 0 and 0 () < 1. We refer to (2) as the tight limits . The rst step of the classical proof is to relate the queuing problem to a level crossing problem via Lindley's identity. Let fXk g be an i.i.d. sequence with distribution L(Xk ) = L(B ? A), and fk = Pk=1 X and the stopping time Kw observe that X (
) = (
). De ne the random walk W fk  wg. Then P (Wk  w) = P0 (Kw < 1). (See Feller (1971, p. 198).) Next, = inf fk  1: W () = 0 and convexity gives E() [Xk ] = 0 () > 0, which implies P () (Kw < 1) = 1. Thus, by change of measure to the -conjugate distribution we obtain

h  f  i ?w P (Wk  w) = P0 (Kw < 1) = E() exp ? ( W e : Kw ? w )

Finally, since E() [Xk ] = 0 () < 1 (by assumption), it is a well known consequence of the renewal fKw ? w has a limiting distribution, and hence, E() [exp(?(W fKw ? theorem that the \overshoot" W w)) ] ! cW with 0 < cW < 1. If any of the conditions 0 <  < 1, () = 0 and 0 () < 1 fail, revisiting the above arguments yields ew P (Wk  w) ! 0. Now consider the heterogeneous multiserver { batch arrival model. fAk g will denote the interP arrival time sequence. The kth batch of jobs arrives at time Tk = k=1 A . Nk is the number of jobs in the kth batch. There are c servers, which we index i = 1; ::; c. Bi;j is the service time of the j th job processed by server i. We assume that the c + 2 sequences fAk g, fNk g, and fBi;j : j = 1; 2; ::g, i = 1; ::; c, are independent and i.i.d. Thus, the model is speci ed by the interarrival time distribution FA (
), the batch size distribution FN (
), and the c service time distributions Fi (
), i = 1; ::; c. The corresponding cumulant functions are denoted A (
), N (
) and i (
). A, N and Bi will denote generic random variables having respective distributions FA(
), FN (
), and Fi (
). We globally assume the following conditions without reference. Basic Assumptions: FA ((0; 1)) = 1 and E[A] < 1; FN (Z+ ) = 1 and E[N ] < 1; and Fi ([0; 1)) = 1 and 0 < E[Bi ] < 1 for each i = 1; ::; c.

Note that we may have Fi (f0g) > 0, which simply means that some jobs require no processing. We P = E[Bi ]?1 . de ne the arrival rate  def = E[N ]=E[A] and the service rate  def = ci=1 i where i def In most (but not all) of our analysis we will assume that the Fi (
)s are spread out , that is, some nite convolution power has a nontrivial component that is absolutely continuous with respect to Lebesgue measure. This mild nonsingularity condition is used primarily for the renewal theory. It will be evident that analogous results will also hold for systems having interarrival time and service times restricted to a common lattice, but we will not consider the arithmetic case here. 2

Our analysis is a generalization of the classical GI/G/1 analysis outlined above. We construct a function (
) from A (
), N (
) and the i (
)s, and we fully characterize (
) as a convex, l.s.c., piecewise analytic function with (0) = 0 and 0 (0) = (E[A]=)( ? ). The construction also identi es the c +2 twisting parameters that determine a -conjugate queuing process distribution. Tight limits are obtained by application of Markov additive process renewal theory to the -conjugate process distribution. The main dierence is that we do not employ a Lindley identity. Instead our results are based on the large deviations behavior of a single busy cycle. In Part I we obtain the following results. Assuming bounded service times, Theorem 2.1 states that  > 0 implies geometric Harris recurrence. The  = 1 case is eliminated by Theorem 2.2, which states that waiting time and queue length processes are actually bounded in this case. The proofs of Theorems 2.1 and 2.2 are both based on construction of Foster-Lyapunov drift conditions that depend on the (
) function construction. We thus establish a connection between the FosterLyapunov theory and quantities usually associated with large deviations analysis. The nal result, Theorem 2.3, establishes the tight limits (2) under a set of sucient conditions that naturally generalize Feller's conditions 0 <  < 1, () = 0 and 0 () < 1. Most of these conditions are also shown to be necessary as well as sucient. Part II completes the analysis. We give a new proof of irreducibility based on spread out Fi (
)s, and we then give a \state dependent" drift criteria proof of positive Harris recurrence. Finally, we complete the stationary distribution characterization. The logarithmic limits w1 log(P (Wk  w)) ! ? and n1 log(P (Qk  n)) ! log(!) are shown to hold assuming no additional hypothesis. In the process of proving the above results, we will also establish tight and logarithmic limits for the stationary busy period maximum waiting time W and maximum queue length Q, thus generalizing the GI/G/1 results of Iglehart (1989). These limits are useful in the asymptotic analysis of Wkmax = maxk W and Qmax k = maxk Q. See Sadowsky and Szpankowski (1992). The tight limits (2) have previously been obtained for the GI/PH/c queue (with single job arrivals) by Takahashi (1981), and Neuts and Takahashi (1981). PH stands for \phase type" distribution (i.e., a class of continuous distributions having rational moment generating functions). The \method-of-phases" is an \analytic" approach based on the matrix geometric form of the stationary queue length distribution and identi cation of ! as the associated Perron-Frobenious eigenvalue. It has been suggested that the general result might be follow from the PH result and a certain continuity property of queuing process distributions, although this argument would involve 3

an interchange of limits. The results of Part II, however, indicate that this interchange of limits would actually yield the wrong value for ! when () < 0! In comparison, the method-of-phases is an \analytic" approach that is distinctly dierent approach than the \probabilistic" methods (change in distribution, renewal theory, etc.) used here. Our results are more general (spread out vs. PH service times, weak sucient conditions, etc.). We suggest that there is also a practical advantage. The constants cW and cQ are quite dicult to evaluate, either by our methods or by the method-of-phases. However, the conjugate queuing process distribution developed here can be used in an ecient Monte Carlo algorithm for directly estimating P (Wk  w) and P (Qk  n). See Bucklew et. al. (1990) and Sadowsky (1991, 1993). Finally, we note that the results are quite dierent when c = 1. See Aldous, et. al. (1992). Part I is organized as follows. Section 2 presents the construction of the function (
) and then states the main results as outlined above. Section 3 investigates the Markov structure of the queuing process and proves Theorems 2.1 and 2.2. Section 4 proves the tight waiting time limit and Section 5 presents the queue length arguments that together prove Theorem 2.3. The required Markov additive process renewal theory is developed in the appendices.

2. Preliminaries and Presentation of Results The purpose of this section is to formally state our main results. This is done in Subsection 2.3. We must rst construct and characterize the function () and associated quantities (Subsection 2.1), and formally describe the workload and queue length processes (Subsection 2.2). First set some basic notation. Z+ and Z+ will denote, respectively, the nonnegative and positive integers. For a vector x 2 Rc , we write _(x) def = mini=1;::;c xi and ^(x) def = maxi=1;::;c xi . For P x 2 R, [x]+ = maxf0; xg. For , 2 Rc, 
def = ci=1 i i . De ne a def = ess inf A (= inffa: = ess sup A, n def = ess inf N , n def = ess sup N , bi def = = ess inf Bi and bi def FA((?1; a]) > 0g), a def ess sup Bi . Also de ne b def = ^(b1 ; ::; bc) and b def = _(b1 ; ::; bc).

2.1: Construction of the Function () and the Twisting Parameters For a function f : (?1; 1] ! (?1; 1], put f ?1 (y) def = supfx: f (x) < yg for each y 2 (?1; 1] (where sup ; = ?1). We will say that f (
) is a nice increasing function if f (x) # ?1 as x # ?1, f (x) " +1 as x " +1, and f (
) is continuous and strictly increasing on (?1; x] where x = supfx: 4

f (x) < 1g. If f (
) is a nice increasing function, then f ?1(
) > ?1, f ?1 (y) # ?1 as y # ?1, f ?1 (
) is strictly increasing on (?1; f ] where f = f (x), f ?1(f (x)) = x for all x  x, and f ?1 (y) = x for all y  f . Moreover, if g: (?1; 1] ! (?1; 1] satis es the properties of an inverse of a nice increasing function (as just listed), then g?1 (
) is a nice increasing function.

Extend the domain of the service time cumulants by putting i (1) = 1. Then each i (
) is a = supf: i () < 1g. Put convex nice increasing function, and is analytic on (?1; i ) where i def i def = i (i ). We may assume, with no loss of generality, that the servers are indexed so that c  c?1 


 1 . For each  2 (?1; 1], de ne B () def = where H() def = f 2 (?1; 1]c : B ( B ),

inf() i=1 max  ( ) ;::;c i i 2H

(3)

Pc  = g. Also de ne  def def B = supf: B () < 1g, B = i=1 i i (
) def = ?i 1 (B (
))

(4)

and i def = ?B1 (i ) for each i = 1; ::; c. As a vector write () = (1 (); ::; c ()).

Lemma 2.1: Assume Fi([0; 1)) = 1, Fi(f0g) < 1 and c 


 1. Then P (i) B (
) is a convex nice increasing function, B (0) = 0,  B = ci=1 i , and B = ^(1 ; ::; c ) (= 1 ).

(ii) 0   c 


  1 =  B , each i (
) is strictly increasing and continuous on (?1; i ], i () = i for all    i , and (0) = 0. Moreover, for each  such that B () < 1 we have () 2 H(), () is the unique solution the minimax problem of de nition (3), and

8 < B () for   i i (i ()) = : : i < B () for i <    B

(5)

(iii) For each i = 1; ::; c such that i+1 < i , where  c+1 = ?1 for the case i = c, B (
) and each  (
) with   i are analytic on the interval ( i+1 ;  i ). Moreover, 0B () = 1=b() where

b() def =

Xi =1

0 ( ())?1 5

for  2 ( i+1 ;  i ):

(6)

Proof: Put B () def = ^(1 (1 ); ::; c (c )) and H () def = H() \ (?1; 1 ] 


 (?1; c]. Then B : (?1; 1]c ! (?1; 1] is convex and l.s.c., and de nition (3) is equivalent to B () P = inf H () B () where inf ; = 1. For  > ci=1 i , we have H () = ; and B () = 1. P For  < ci=1 i , there exists at least one  2 H () such that i < i for each i, and hence, P B () < 1. Thus,  B = ci=1 i . We also have H ( B ) = f(1 ; ::; c )g, and hence, B def = B ( B ) = ^(1 ; ::; c ). P Put e(
) def = ci=1 ?i 1 (
). i < 1 implies i < 1, ?i 1 (
) is continuous and strictly increasing on (?1; i ], and ?i 1 () = i for   i . Thus, it follows that e(
) is continuous and strictly increasing on (?1; B ] (because B = ^(1 ; ::; c )), and e() =  B for all   B (because B = Pc  ). So, e (
) has all of the properties of a nice increasing function inverse, and hence, e?1 (
) i i=1 i is a nice increasing function with supf: e?1 () < 1g =  B . We claim that B (
) = e?1 (
). If this claim holds, then the properties of B (
) and (
) listed in parts (i) and (ii) follow immediately. We have already shown that B () = e?1 () = 1 for all  > B , and that B (B ) = e?1 ( B ) = B . Fix any  < B and put  = e?1 () < B and i = ?i 1 (). By the de nition of e(
),  2 H (), i (i )   with equality whenever i < i , and i < i for at least one i because  < B = Pc  . Thus,  ( ) =    (), so it is enough to show that  ( ) >  for any 2 H() B i=1 i B B    such that 6=  . For at least one i we must have i > i . If i = i , then B ( )  i ( i ) = 1 > . If i < i , then since i (
) is strictly increasing on (?1; i ], we have B ( )  i ( i ) > i (i ) = . Thus, B () =  as claimed, and moreover, i () = ?i 1 (B ()) = ?i 1 () = i , and hence, we have also shown that () is the unique solution to the minimax problem in de nition (3). That (0) = 0 and B (0) = 0 follows from i(0) = 0 for each i. Fix 1; 2 2 (?1; B ] and  2 (0; 1). By the convexity of B (
) we have B (1 + (1 ? )2 )  B ((1 ) + (1 ? )(2 ))   B (1 ) + (1 ? ) B (2 ). Thus, B (
) is also convex. At this point, we have all of parts (i) and (ii). c 


 1 implies  c 


  1 . Suppose i+1 < i for some i. For  2 ( i+1 ;  i ) we have c c X Xi ?1 X  =  () =  (B ()) +  =1

=1

=i+1

and the inverses in the rst sum on the right above are analytic and strictly increasing on ( i+1 ;  i ). Thus, B (
) is analytic on ( i+1 ;  i ) by the analytic implicit function theorem. 0B () = b()?1 follows by dierentiating the above display. This proves (iii).

6

For each  2 R, de ne





() def = A (?) + N B () ;

(7)

and put D def = f: () < 1g. By Lemma 2.1, (
) is a convex and l.s.c. mapping R ! (?1; 1], and (0) = 0. Moreover, (
) is analytic on each interval ( i+1 ;  i ) \ Do with derivative 0 )   b 0 () = Ab((?  (  ) ?  (  ) (8) ) where b() is as de ned in (6) and 0 B ()) () def = N0((? : (9) A )

P

In Section 4 we will see that () and () def = ci=1 0i (i ())?1 are, respectively, the arrival and service rates for the -conjugate queuing process distribution. In particular, (0) = E[N ]=E[A] P =  = the arrival rate, and (0) = ci=1 E[Bi ]?1 =  = the service rate. As in the GI/G/1 case, we would like to say that the sign of 0 () determines -conjugate stability condition () < or > (). But this is not quite true in general. It is b(), not (), that appears in the dierentiation formula (8), and we may have b() < () when  >  c due to the missing terms in (6). However, if   c , then b() = (), and hence, the sign of 0 () does determine the -conjugate stability in this case. Notice that this is an issue only for heterogeneous systems. In the homogeneous case, i = B for each i = 1; ::; c.

2.2: The Workload and Queue Length Markov Chains If no new jobs are added to the queue after time t, let t + Wi (t) denote the time instant at which server i becomes idle. Wi (t) is called the ith server's workload at time t, and W(t) = (W1 (t); ::; Wc (t)) is the workload process . The waiting time of a single job added to the system at P time t is W (t) = _(W(t)). At time Tk = k=1 A a batch of Nk jobs arrives at the queue. We write Wk? = W(Tk? ) and Wk+ = W(Tk+ ) to denote the workload vector just before and just after the kth batch is added to the system. Let Q(t) denote the number of jobs in queue at time t, not counting the jobs currently being processed. Let Ri (t) denote the service time remaining for the job being processed by server i at time t. R(t) = (R1 (t); ::; Rc (t)) is the residual service time vector , and we call (R(t); Q(t)) the queue length process . As above, write Qk = Q(T  ) and Rk = R(T  ). 7

We will initialize the queuing process by taking the 0th batch of jobs to be inserted into the system at time t = 0. Pw (
) (P (
)) will denote the workload process distribution initialized with W0? = w (P (W0? 2
) =  (
)). Similar notation is used for the queue length process. It turns out that fWk g and f(Rk ; Qk )g are Markov chains on the respective state spaces [0; 1)c and [0; 1)c  Z+ . This is well known, but we will examine the Markov structure in detail in Section 3. We require some de nitions from Markov chain theory. See Nummelin (1984), or Meyn and Tweedie (1992,1993). Let fXk g be a Markov chain taking values in a locally compact topological space S with Borel eld S . For a measure (
) de ne S def = fB 2 S : (B ) > 0g. fXk g is said to be irreducible if there exists a positive measure (
), called an irreducibility measure , such that Px(Xk 2 B ) > 0 for some k = k(x; B ) < 1 for all x 2 S and B 2 S. An irreducible chain is Harris recurrent if for all x 2 S and B 2 S we have Px (Xk 2 B i:o:) = 1. Harris recurrent chains have a unique (up to a scale factor) - nite invariant measure (
). Moreover, (
) is a \maximal" irreducibility measure in the sense that    for any other irreducibility measure (
). If (
) is nite, then the chain is positive Harris recurrent . In this case we always take (
) to be normalized to a probability distribution and we call (
) the stationary distribution . A set C 2 S is a petite set if for some  2 (0; 1) there exists a nontrivial substochastic measure k  (
) such that P1 k=1  Px (Xk 2
)   (
) for all x 2 C . If for some k0 < 1 we have Px (Xk 2
)   (
) for all x 2 C , then C is a small set . Clearly, small sets are petite. Also, these notions are equivalent when fXk g is aperiodic. See Meyn and Tweedie (1993). As will be noted in Part II, our chains are generally not aperiodic. If fXk g is Harris recurrent and there exists a petite set C 2 S such that supx2C Ex [KC ] < 1 where KC = inffk  1: Xk 2 C g, then fXk g is positive Harris recurrent. If the stronger condition supx2C Ex[rKC ] < 1 for some r > 1 holds, then fXk g is said to be geometrically Harris recurrent. This implies geometric convergence to stationary behavior. We again refer the reader to the references for further exposition. Hereafter, we will use (
) to denote the stationary distribution of fWk? g, and b (
) will denote the stationary distribution of f(R?k ; Q?k )g, whenever these chains are positive Harris recurrent. Perhaps the most common way to show that a queuing process is irreducible is by idle period regeneration . The following lemma presents a simple sucient condition. 0

8

Lemma 2.2: If a > n ^(b1 ; ::; bc), then fWk? g is irreducible with irreducibility measure 0(
), and

all bounded sets are small sets.

Sketch of Proof: Consider only the case a = 1. Let C be any bounded set, and put b = ^(b1; ::; bc) and v = supw2C ^(w) + n b . Then Pw (W1? 2
)  0 (
) for all w 2 C where = FA((v + n; 1)) FN (fng) [ mini Fi([0; b + ]) ]n > 0. The case a < 1 requires more than one step to drive Wk? to 0 when v is large.

For the single server queue,  <  and the independence of A, B and N implies E[NB ? A] < 0, which in turn implies ess inf fNB ? Ag = n b ? a < 0. Thus, for c = 1 only,  <  actually implies irreducibility. For c > 1, irreducibility is a somewhat more dicult issue. When a < b we may still have  < , but full idle period regenerations never occur. In Part II, we will give a new proof of irreducibility for the case that the service time distributions are spread out, and it is also shown that compact sets are petite for that case. Fortunately, between Lemma 2.2 and the results of Part II most systems of practical interest are covered. Outside of the scope of these results it appears to be dicult prove irreducibility and identify the petite sets.

2.3: Statement of Results Our rst result gives a sucient condition for geometric Harris recurrence. Recall that  def = supf: ()  0g. As discussed above, the irreducibility and petite set hypothesis may be veri ed using either Lemma 2.2 or results from Part II.

Theorem 2.1: Assume that fWk? g is irreducible, all compact sets are petite, b < 1 and  > 0. Then both fWk? g and f(R?k ; Q?k )g are geometrically Harris recurrent Markov chains. We next characterize the  = 1 case. To see what to expect, consider the GI/G/1 case. When  = 1 we have () = log( E[ exp([B ? A]) ] )  0 for all  > 0. This implies B ? A  0 a.s., and since B and A are independent, ess sup [B ? A] = b ? a  0. Thus, the queue is either a deterministic  =  system (in which case (
)  0), or the system eventually empties and Wk?  0 after that point in time. We say that \the queue is bounded" if there is a constant w < 1 and a stopping time K such 9

that Pw (K < 1) = 1 for all w 2 [0; 1)c , and Wk+  w for all k  K . In other words, fw 2 [0; 1)c : _(w)  wg is an absorbing set. (Notice that this does not imply that each components Wi;k+ is bounded.) Likewise, if the queue has a bounded waiting time as above, then there also exists a constant q < 1 such that Q+k  q for all k  K .

Theorem 2.2: Assume  < . Then the following are equivalent: (i)  = 1.

P (ii) There exists a vector e 2 [0; 1]c such that ci=1 ei = 1 and ei bi n  a for each i = 1; ::; c. (iii) The queue is bounded. We remark that the hypothesis  <  in Theorem 2.2 is needed only to eliminate the deterministic  =  system, for which (
)  0. We require a few more de nitions. De ne K1 def = inffk  1: Wk? = 0g, W def = maxfWk+ : 0  k  K1 g = the maximum waiting time during the rst busy period, and Q def = maxfQ+k : k < K1 g = the maximum queue length during the rst busy period. A distribution  (
) on [0; 1)c is an admissible initial distribution for W0? if E [ exp(()
W0? ) ] < 1 and P (W  w) > 0 for all w < 1. Likewise, b(
) on [0; 1)c  Z+ is an admissible initial distribution for (R?0 ; Q?0 ) i h  < 1 and Pb(Q  n) > 0 for all n < 1. Obviously, any if Eb exp ()
R?0 + B () Q?0 distribution with compact support satis es the exponential moment condition. Thus, by Theorem 2.2 admissible initial distributions exist if and only if  < 1. Finally, de ne ! def = exp( ?B () ).

Theorem 2.3: Assume that fWk? g and f(R?k ; Q?k )g are positive Harris recurrent, and that the Fi (
)s are spread out. Let  (
) and b(
) be any admissible initial distributions. (i) (Sucient Conditions) Assume 0 <  < 1,   c , () = 0, 0i (i ()) < 1 for each i, and 0N (B ()) < 1. Then there exists constants 0 < cW ; cW ; cQ ; cQ < 1 such that

P ( W  w )  cW e?w and and

P ( Wk+  w )  cW e?w and as w ! 1 and n ! 1. 10

Pb( Q  n )  cQ !n;

(10)

Pb ( Q+k  n )  cQ !n

(11)

(ii) (Necessary Conditions) If any of the conditions 0 <  < 1,    c and () = 0 fail, then

ew P ( W  w ) ! 0 and

!?n Pb( Q  n ) ! 0;

(12)

!?n Pb ( Q+k  n ) ! 0:

(13)

and likewise,

ew P ( Wk+  w ) ! 0 and

We believe that (12) and (13) also hold if either 0N (B ()) = 1 or 0i (i ()) = 1 for some i. If so, then all of the conditions listed under part (i) of Theorem 2.3 are necessary as well as sucient. The proof would require an excursion into degenerate Markov additive process renewal theory. We will discuss this only brie y at the end of Appendix B. The above results leave us with two unanswered questions. First, can positive Harris recurrence be obtained under weaker conditions than those of Theorem 2.1? In particular, can we eliminate b < 1? Second, when the tight limits fail, do the parameters  and ! still identify the correct logarithmic behavior, that is, do the quantities in (12) and (13) vanish less than exponentially fast? These are precisely the issues addressed in Part II.

3. The Markov Structure

Subsections 3.1 and 3.2 consider the Markov structure of fWk? g and f(R?k ; Q?k )g, respectively. Subsections 3.3 and 3.4 prove, respectively, of Theorems 2.1 and 2.2. Both proofs are based on the construction of a \drift criteria." The section is completed by a presentation of the cycle structure that forms the basis for the analysis of Sections 4 and 5.

3.1: The Workload Process The evolution of W(t) is best described in terms of c ctitious single server queues, one for each

server. FIFO servicing priority is equivalent to \inserting" jobs into the parallel queues system one-by-one by assigning them to the queue with the current minimum waiting time. Put Mk def = Pk N = the total number of jobs that arrive in batches  = 0; ::; k. Let m = 1; 2; ::: index the =0  jobs in the order that they are inserted into the system, so the jobs in the kth batch are indexed m = Mk?1 + 1; ::; Mk (with M?1 = 0 for k = 0). Let Y(m) denote the workload vector just prior to and just after the insertion of the mth job, so Wk? = Y? (Mk?1 + 1) and Wk+ = Y+(Mk ). Let 11

Ji (m) denote the index of the last job inserted in the ith queue just after the mth job is inserted into the system. The process is initialized at time t = 0 with workload Y? (1) = W0? , and Ji (0) = 0 for i = 1; ::; c. We have the following recursive set of equations. First determine I (m) such that YI?(m) = _(Y?(m)). We assume that there is a tie breaking rule that determines a unique value I (m). For brevity, we consider only deterministic tie breaking rules, however, by augmentation

of the sample space in the usual way it is a minor matter to allow randomized tie breaking rules. Next, we have 8 < Ji (m ? 1) + 1 for i = I (m) Ji (m) = : ; (14) Ji(m ? 1) for i 6= I (m)

8 ? < Y (m) + Bi;Ji(m) for i = I (m) Yi+(m) = : i ; (15) Yi? (m) for i 6= I (m) and 8 + < [ Y (m) ? Ak+1 ]+ when Mk = m for some k Yi? (m + 1) = : i + : (16) Yi (m) otherwise Since Wk? = Y?(Mk?1 +1) and Wk+ = Y+ (Mk ), it follows from the above recursive structure that fWk? g, fWk+ g and f(Wk? ; Wk+ )g are Markov chains. In the GI/G/1 case, Wk? = [Wk??1 + Bk?1 ? Ak ]+ = a re ected random walk. Much of the fk? = W0? +Pk=1(B?1 ?A ), which behavior of Wk? can be deduced by studying the random walk W is \coupled" to the GI/G/1 waiting time process during the rst busy period. It is a trivial matter to construct a \tilde process" for our multiserver queuing model. Simply neglect the re ections in (16). Initialize with Ye ? (1) = W0? and Jei (0) = 0 for i = 1; ::; c. For m = 1; 2; :::, recursively de ne Ie(m), Jei (m) and Ye (m) as in formulas (14) { (16) with the only modi cation being that f k+ def f k? def = Ye + (Mk ) and = Ye ?(Mk?1 + 1), W Yei? (Mk + 1) = Yei+ (Mk ) ? Ak+1 in (16). De ne W f k ). Next de ne the random walks fk def = _(W W j X ? eSi(j ) def = Wi;0 + Bi;j0 j 0 =1

(17)

for i = 1; ::; c. Then, since there are no re ections in the \tilde process," for Mk?1 + 1  m  Mk we have

Yei?(m) = Sei (Jei (m ? 1)) ? Tk and Yei+ (m) = Sei(Jei (m)) ? Tk : 12

(18)

Thus, the index Ie(m) is determined by SeIe(m) (JeIe(m) (m ? 1)) = mini=1;::;c Sei (Jei (m ? 1)), or equivalently, SeIe(1) (JeIe(1) (0))  SeIe(2) (JeIe(2) (1))  SeIe(3) (JeIe(3) (2)) 




with each Sei (j ) occurring exactly once. (The tie breaking rule determines the ordering when there are equalities.) So we see that Ie(m) and the Jei (m)s are completely determined by the random f k g are thus completely determined by the walks Sei (j ), i = 1; ::; c. The unre ected processes fW c + 2 random walks Tk , Mk and Sei (j ), i = 1; ::; c.

f k = Wk fk  Wk for all k 2 Z+. On the event fW0+ > 0g we have W Lemma 3.1: We have W fk?  0g. for all k < K1 def = inffk  1: Wk? = 0g = inffk  1: W f k = Wk for all k < K1 is obvious. When I (m) = i, Ji (m) = j and Proof: The fact that W P Mk?1 < m  Mk , put Ui;j def = Tk ? supft  Tk : Wi (t) > 0g  0. Put Si (j ) def = Wi;?0 + jj 0 =1 [Bi;j 0 + Ui;j0 ]. In this manner we have adjusted the random walk de nition (17) to account for the re ections in (16) that were neglected to de ne the \tilde process." Of course, Si (j ) is not a random walk, but for Mk +1  m  Mk+1 we have Yi+ (m) = Si (Ji (m)) ? Tk , and the original queuing process satis es the ordering SI (1) (JI (1) (0))  SI (2) (JI (2) (1))  : : : with each Si (j ) occurring exactly once. Since Ui;j  0, we have Sei(j )  Si(j ) for all i; j , and hence, SeIe(m) (JeIe(m) (m ? 1))  SI (m) (JI (m) (m ? 1)) and fk  Wk. (Notice that this proof does not imply Ie(m) = I (m), Jei (m) = Ji(m), it follows that W fi;k  Wi;k for xed i and k  K1 .) or even W fi;k ! (= ? 1) E[A] a.s. Moreover, if fWk? g is Harris recurrent and  < , Lemma 3.2: k?1 W then (fw : _(w) = 0g) = P (W0? = 0) > 0. Sketch of Proof: By the law of large numbers, Sei (j )  j=i a.s. We also have Sei (Jei (m))  Se1(Je1 (m)) for each i, and hence, Jei (m)  (i =1 )Je1 (m) a.s. Since Pci=1 Jei (m) = m, we have Je1 (m)  (1 =) m. Likewise, Jei(m)  (i=) m for each i, and hence, Sei (Jei (m))  m=. Since fi;k ! (= ? 1) E[A] follows from (18). Mk  E[N ] k and Tk  E[A] k, k?1 W fk?  0 for some If  < , then by the above result we have Pw (Wk? = 0 for some k < 1) = Pw (W k < 1) = 1 for all w 2 [0; 1)c . If fWk? g is Harris recurrent, then this implies P (W0? = 0) > 0.

As a trivial application of Lemmas 3.1 and 3.2, observe that the system is unstable when  > . 13

fk  w i:o:) = 0 for all w < 1. That is, Pw (Wk  w i:o:)  Pw (W

3.2: The Queue Length Process The processes f(Rk ; Qk )g are also Markov chains. We will not formally specify the Markov structure of these processes, as we have done for the workload processes, but we do formulate the associated \tilde process." First, observe that queue length and workload processes are connected. We have W (t) = 0 if and only if Q(t) = 0 and _(R(t)) = 0. De ne the sequence of stopping times K0 = 0 and

K` = inf fk > K`?1 : Wk? = 0g = inf fk > K`?1 : Q?k = 0 and _ (R?k ) = 0g

(19)

for ` 2 Z+ . (Note that the ` = 1 case agrees with the previous de nition of K1 .) Then, whenever K` < 1, we have (R?K` ; Q?K` ) = (WK? ` ; 0). Let Sei (j ) be the random walk as de ned in (17) with Wi;?0 replaced by Ri;?0 . (On the event fQ?0 = 0g we have W0? = R?0 , hence, on this event there is no dierence.) De ne Jbi+(t) = inffj  0: Sei (j ) > tg, and (20) Rei+ (t) = Sei (Jbi+ (t)) ? t: In renewal theory, Re i+(t) is called a forward process . As a vector, write Re + (t) = (Re1+ (t); ::; Re c+ (t)), and put Re +k = Re (Tk+ ) and

Qe +k = Q?0 + Mk ?

c X Jbi+ (Tk ): i=1

(21)

We state the following is obvious lemma without proof.

Lemma 3.3: We have the alternative expression K1 = inffk  1: Qe ?k < 0g, (Rk ; Qk ) = (Re k ; Qe k ) for all k < K1 , and Qe +k  E[A] ( ? ) k a.s. The next lemma indicates the close coupling between the chains fWk? g and f(R?k ; Q?k )g.

Lemma 3.4: Assume  < . Then K` < 1 a.s. for each ` 2 Z+, and moreover, fWk? g is Harris recurrent (positive Harris recurrent) if and only if f(R?k ; Q?k )g is Harris recurrent (positive Harris

recurrent).

14

Proof: That K` < 1 a.s. for each ` 2 Z+ is obvious. Suppose fWk? g is Harris recurrent and put E def = fw 2 [0; 1): _(w) = 0g, so K1 = inffk: Wk? 2 E g. >From Lemma 3.2,  <  implies Pw (K1 < 1) = 1 for all w 2 [0; 1), which implies (E ) > 0 and Pw (Wk? 2 E i.o.) = 1 by Harris recurrence, and (
) def = (
\ E ) serves as an irreducibility measure for fWk? g. Since (R?K` ; Q?K` ) = (WK? ` ; 0), 0 (
) serves as an irreducibility measure for f(R?k ; Q?k )g. By Theorem 2.1 of Nummelin (1984) there exists a small set C  E for fWk? g with (C ) > 0. Speci cally, for some k0 2 Z+ we have Pw (Wk? 2
)  1C (w)  (
) where  (
) is a nontrivial substochastic measure, and we may assume that  (
) is concentrated on E . Then Cf0g is a small set for f(R?k ; Q?k )g with 0 (Cf0g) = (C ) > 0, and P(r;q) ((R?k ; Q?k ) 2
)  1Cf0g (r; q)  0 (
). Moreover, P(r;0) ((R?k ; Q?k ) 2 C f0g i.o.) = Pr (Wk? 2 C i.o.) = 1, which implies f(R?k ; Q?k )g is Harris recurrent. A similar argument proves the converse. Now suppose fWk? g is positive Harris recurrent. Applying the small set minorizations of the previous paragraph, from Theorem 5.2 of Nummelin (1984), after appropriate scaling, we nd that b (B f0g) = (B ) for all Borel sets B  E . Put KC = inffk 2 Z+ : Wk? 2 C g = inffk 2 Z+: (R?k ; Q?k ) 2 C f0gg. Then, using Corollary 5.3(a) of Nummelin (1984), we evaluate 0

0

b([0; 1)c  Z

0

+)

= =

Z

ZCf0g C

E(r;q) [KC ] b ( d(r; q) )

Ew [KC ] (dw) = ([0; 1)c ) < 1

where the niteness follows by the assumed positive recurrence of fWk? g. A similar argument proves the converse.

3.3: Proof of Theorem 2.1 Here we establish geometric recurrence under the assumptions that fWk? g is irreducible, compact sets are petite, b < 1 and  > 0. For xed  2 [0; 1) and w 2 [0; 1)c de ne v() (w) def = exp( ()
w ), and Z 1 (?) h() (w) def =  FA ([a; 1)) e(a?w) da; (22) w

where FA(?) (
) is the twisted distribution as de ned in (1). The next lemma provides the basis for a \geometric drift condition."

15

Lemma 3.5: For any   0 such that () < 1, we have   Ew [ v() (W1? ) ]  e() 1 + eN (B ()) h() (_(w)) v() (w)

(23)

for all w 2 [0; 1)c . Proof: Put B  (m) def = BI (m);JI m (m) = the service time of the mth job inserted into the system. Then we have v() (Y+ (m)) = exp( I (m) ()B  (m) ) v() (Y? (m)), and hence, (



h

E v() (Y+ (m)) Y? (m)

)

i



h

i

= E exp( I (m) ()B  (m) ) Y? (m) v() (Y? (m))





= exp I (m) (I (m) ()) v() (Y? (m))

 eB () v() (Y?(m)) because i (i ())  B () by (5). By iterative conditioning it follows that Ew [ v() (W0+ ) j N0 = n ]  enB () v() (w), and hence, Ew [ v() (W0+ ) ]  eN (B ()) v() (w). Next, for w 2 [0; 1)c de ne

h() (w)

=

c X i=1

i ()

Z1 wi

F (?) ([a; 1)) A

exp

c X =1

 ()[a ? w ]+ 



!

da:

Since Wi;?1 = [Wi;+0 ? A1 ]+ = Wi;+0 ? A1 + [A1 ? Wi;+0 ]+ , using the twisting formula (1) we obtain

h i E v() (W1? ) j W0+ = w

# ! = E exp ?A1 + i ()[A1 ? wi ]+ W0+ = w v() (w) i=1 ! Z1 c X +  ( ?  ) A i()[a ? wi ] FA(?) (da) v() (w) exp = e 0 "

c X

i=1

h i = eA (?) 1 + h() (w) v() (w)

where the last equality follows from integration-by-parts. Combining this with the bound of the rst paragraph and using () def = A (?) + N (B ()) yields

h i  Ew [ v() (W1? ) ]  e() v() (w) + Ew v() (W0+ )h() (W0+ ) :

>From (22), observe that h() (
) is nonincreasing, and h() (0) = e?A (?) ? 1 (which is evaluated by integration-by-parts and using (1)). Next observe that h() (w)  h() (_(w); ::; _(w)) = h() (_(w)). 16

Thus, in the last display, h() (W0+ )  h() (_(W0+ ))  h() (_(W0? )) = h() (_(w)). >From the rst paragraph, Ew [ v() (W0+ ) ]  eN (B ()) v() (w). Thus, the last display above is reduced to (23). Put A = fw 2 [0; 1)c : ^(w) ? _(w)  b g. A is an absorbing set; that is, Pw (W1? 2 A) = 1 for all w 2 A, and Pw (KA < 1) = 1 for all w 2 [0; 1)c where KA = inffk: Wk? 2 Ag. (Formally, this is a consequence of the proof of Lemma B.3 in Appendix B.) Thus, we may restrict fWk? g to the absorbing set A. Fix  > 0 such that () < 0. Since h() (w) # 0, we may x w < 1 suciently large so that

h

= e() 1 + eN (B ()) h() (w)

i

< 1:

Put C = fw 2 A: _(w)  wg. Then C  [0; w + b ]c is compact, and hence, petite by assumption. >From (23) we have Ew [ v() (W1? ) ]  v() (w)

for all w 2 AnC:

Also, v() (
) is clearly bounded on C . Geometric Harris recurrence of fWk? g now follows by Theorem 6.2 in Meyn and Tweedie (1992). By Lemma 3.4, f(R?k ; Q?k )g is positive Harris recurrent. To get geometric recurrence, we use the following notion. Let fXk g be a generic positive Harris recurrent Markov chain, and put KB = inffk  1: Xk 2 B g. A set D 2 S is geometrically regular if for all B 2 S there is an r > 1 such that supx2D Ex [ rKB ] < 1. Clearly, any subset of a geometrically regular set is geometrically regular. By Theorem 6.2 of Meyn and Tweedie (1992), the set C above is geometrically regular for fWk? g, hence, so is Ce = fw: _(w) = 0 and ^(w)  bg  C (if we take w > b). We also have Pw (Wk? 2 Ce i:o:) = 1 by Lemma 3.2. Put Cb = C  f0g. Since (R?K` ; Q?K` ) = (WK?` ; 0), we have P(r;q) ((R?k ; Q?k ) 2 Cb i:o:) = 1, which implies Cb 2 Sb , and by geometric regularity of Ce we have

h i h i sup E(r;q) rKCb = sup Ew rKCe < 1:

(r;q)2Cb

e

w 2C

Thus, f(R?k ; Q?k )g is also geometrically Harris recurrent.

17

3.4: Proof of Theorem 2.2 P De ne E def = fe 2 [0; 1]c : ci=1 ei = 1g. Assuming  < , Theorem 2.2 states that the following are equivalent: (i)  = 1; (ii) there exists a vector e 2 E such that ei bi n  a for each i; and (iii) the queue is bounded. For e 2 [0; 1] and  2 R de ne

9131 0 2 0 8 N < = X b i (; e) = A (?) + N (i (e)) = log@ E4 exp@  :e Bi;j ? A ; A 5 A ; def

P

j =1

which is the cumulant function of the random variable e Nj=1 Bi;j ? A. For  2 R and e 2 E de ne b (; e) def = maxi=1;::;c b i (; ei ), and note that b (; e) is jointly l.s.c. in (; e). Also observe that () = inf e2E b (; e). De ne  (e) def = supf  0: b (; e)  0g for each e 2 E . Then  = supf: ()  0g = supe2E supf: b (; e)  0g = supe2E (e). Moreover, for all  2 [0; 1), fe:  (e)  g = fe: b (; e)  0g = a closed set because b (;
) is l.s.c., and hence,  (
) is u.s.c. Observe that the following statements are equivalent: (a) (e) = 1; (b) b i (; ei )  0 for all  > 0 and each i = 1; ::; c; (c) ei PNj=1 Bi;j ? A  0 a.s. for each i; and (d) ei ess sup PNj=1 Bi;j = ei n bi  a for each i. Statement (ii) of Theorem 2.2 says that (d) holds for some e 2 E , but we may equivalently work with (a), (b) or (c). Assume (ii), equivalently,  (e ) = 1 for some e 2 E . Then   (e ) = 1. Thus, (ii) ) (i). Suppose that (ii) fails, equivalently,  (e) < 1 for all e 2 E . Since  (
) is u.s.c., we have  = supe2E (e) < 1, and hence, (i) fails. Thus, (i) , (ii). P Next we show (ii) ) (iii). Assume (ii) and de ne v(w) def = ci=1 ei wi . For any i such that +  v (W+ )=e , and hence, it is sucient to show that v (W+ ) is ei > 0, we have Wk+  Wi;k k k i bounded. First, as in the proof of Theorem 2.1, we may restrict attention to the absorbing set A def = fw 2 [0; 1)c : wi ? _(w)  bi for each ig. Next, write 1

v(W+ ) k

= v(Wk? ) +

P

c X

JiX (Mk )

i=1 j =Ji(Mk?1 +1)

ei Bi;j :

By applying ei Bi;j  ei bi  a=n and ci=1 [Ji (Mk?1 + 1) ? Ji (Mk )] = Nk  n, the last display is reduced to the bound v(Wk+ )  v(Wk? ) + a. Put C = fw 2 A: _(w)  ag. When Wk+ 2 AnC we ? = [W + ? Ak+1 ]+  W + ? a, and hence, v(W? )  v(W+ ) ? a. Thus, combining have Wi;k k k+1 i;k i;k +1 18

these bounds yields whenever Wk+ 2 AnC:

v(Wk++1 )  v(Wk+ )

P

P

Observe that   n=a = ci=1 ei n=a  ci=1 b?i 1  . Since we assume  < , one of these inequalities must be strict, and this implies that there exists ;  > 0 and an integer 1  m < 1   ?  such that P v(Wk++m )  v(Wk+ ) ?  jWk+ = w >  for all w 2 AnC . (For example, Wi;k +1 def + wi ? (a + ) with probability  = FA ([a + ; 1)).) Put K = inffk: Wk 2 C g, so we have P (K < 1jW0+ = w) = 1 for all w 62 C . Whenever Wk+ 2 C , we have v(Wk+ )  a + Pci=1 ei bi and P v(Wk++1 )  v def = 2a + ci=1 ei bi . Thus, v(Wk+ )  v for all k  K ; that is, the queue is bounded. To complete the proof we show that (iii) ) (ii) using a contrapositive argument. Assume (ii) fails. If either a = 0 or n = 1, then obviously (iii) fails (the queue is not bounded). So assume a > 0 and n < 1. Put e~i = a=(bi n), so Pci=1 e~i < 1 by the assumption that (ii) fails. If Pci=1 e~i = 0, P P then each bi = 1, in which case (iii) fails. So, suppose ci=1 e~i > 0. Put ei = e~i = c=1 e~ , so e 2 E and bi > a=(ei n) for each i since (ii) fails. Fix  > 0 suciently small so that bi > (a + )=(ei n). Then with positive probability we can have an arbitrarily long run of Ak s  a +  and Nk s = n. For ei > 0, we can have an arbitrarily long run of Bi;j s  (a + )=(ei n). For ei = 0 the Bi;j 's can be arbitrarily large for an arbitrarily long run. During such a run, Wk is lower bounded by the waiting time of a deterministic queue with Ak  a + , Nk  n, and Bi;j  (a ? )=(ei n) when ei > 0. (We omit the obvious details.) The deterministic queue is unstable, and hence, Wk can obtain arbitrarily large values, that is, (iii) fails. Thus, by contrapositive, (iii) ) (ii), and this completes the proof.

3.5: Cycles We call the time intervals [TK` ; TK` ) cycles . This diers from the usual de nition found in the literature that is based on idle period regeneration. In our de nition, WK?` = _(WK? ` ) = 0 does not imply the vector equality WK? ` = 0. P Put L+w def = kK=0?1 1[w;1)(Wk+ ) = the total number of batches that arrive during the rst cycle P = kK=0?1 1[n;1) (Q+k ) is the total number of batches during the and produce Wk+  w. Likewise, L+n def rst cycle that produce queue length Q+k  n. The following lemma is an immediate consequence of positive Harris recurrence and Lemma 3.2. +1

1

1

19

Lemma 3.6: Assume  < , and that fWk? g and f(R?k ; Q?k )g are positive Harris recurrent. Then P ( Wk+  w ) = EE [[LKw ]]  1 +

(24)

where  (
) = L (WK? ` ), and

E [ L+ ] Pb ( Q+k  n ) = Eb [Kn ] b 1 where b(
) = Lb (R?K` ; Q?K` ) =   0 (
).

(25)

Remark: We consider only the stationary distribution of the waiting times just after a batch of jobs arrives. However, we could just as easily work with other the stationary distributions. To get the stationary waiting time distribution sampled just before a batch of jobs arrives, replace L+w P by L?w = kK=0?1 1[w;1)(Wk? ). To get the stationary waiting time distribution for arbitrary time, R replace L+w by Lw = 0TK 1[w;1)(W (t)) dt = total time during the rst cycle that W (t)  w, and replace E [K1 ] by E [TK ]. Similar statements hold for queue length. We also remark that (24) and (25) are our main departure from the classical GI/G/1 analysis that relates the queuing problem to a level crossing problem via Lindley's identity. These are not a simple level crossing probabilities, even in the GI/G/1 case. Nonetheless, random walk theory still plays a key role. 1

1

1

4. Workload and Waiting Time Analysis In this section we prove the waiting time limits of Theorem 2.3. De ne Fk def = ( W0? ; A1 ; ::; Ak ; N0 ; ::; Nk ; Bi;j ; j = 1; ::; Jei(Mk ); i = 1; ::; c ) where Jei (m) is as de ned in Section 3.1. Let K be any Fk -stopping time. Fix (; ; ) 2 R(c+2) such that i (i ) < 1 for each i, N ( ) < 1 and A (?) < 1, and let FA(?) (
), FN( ) (
) and Fi(i ) (
) denote the exponentially twisted distributions as in (1). We de ne the (; ; ; K )-conjugate distribution P(;;;K ) (
) as follows. First, the distribution is de ned on F?1 = (W0? ) as P(;;;K )(W0? 2
) =  (
). Given the de nition on Fk we extend the de nition to Fk+1 by de ning the conditional law of Ak+1 , Nk+1 and Bi;j , j = Jei(Mk ) + 1; ::; Jei (Mk+1 ). On the event fK > kg 2 Fk , this conditional law is just the queuing process governed by FA(?) (
), FN( ) (
), Fi(i ) (
), i = 1; ::; c. On the event fK  kg 2 Fk , the process evolves with the original untwisted distributions FA (
), FN (
) and Fi (
). 20

The restrictions of P(;;;K ) (
) and P (
) to the event fK < 1g are equivalent measures, and on this event the likelihood ratio is

dP

  

dP ( ;;;K )

= exp  TK + K A (?) ?  MK + (K + 1) N ( ) +

c h X i=1

 e i! e e ? ?i Si(Ji (MK )) ? Wi;0 + Ji (MK ) i(i ) : (26)

The above likelihood ratio formula is signi cantly simpli ed by taking  = () and  = B (). >From Lemma 2.1, i (i ())  B () with equality if and only if   c . Thus, since Pc Je (M ) = M , we have K i=1 i K

c c X X Jei (MK ) [ i (i ()) ? B () ]  0: (27) Jei (MK ) i (i()) = i=1 i=1 fi;k+ = Yei+(Mk ) = Sei(Jei (Mk )) ? Tk . Thus, using Pci=1 i() =  we obtain Next, W c   X  TK ? i() Sei(Jei (MK )) ? Wi;?0 i=1 c X + ? W ? ) = ? ( )
(W fi;K f K+ ? W0?): = ? i () (W i;0

? B () MK +

i=1

The remaining terms are

K A (?) + (K + 1) N (B ()) = K () + N (B ()): Finally, observe that i (i ()) < 1 for each i, N (B ()) < 1 and A (?) < 1, if and only if  2 D def = f: () < 1g. We hereafter refer to the ((); B (); ; K )-conjugate distribution as simply the (; K )-conjugate distribution, and we modify the notation accordingly. By the above arguments, we have the following change of distribution formula.

Lemma 4.1: For any  2 D, the restrictions of P(;K ) (
) and P (
) to the event fK < 1g are equivalent, and on this event we have

dP

dP(;K )



f K+ ? W0? ) + K () + N (B ()) = exp ? ()
(W

! c X + Jei (MK ) [ i (i()) ? B () ] i=1

and the last term vanishes when   c . 21

(28)

We will also have cause to consider the distribution P() (
) having L(Ak ) = FA(?) (
) and L(Nk ) = FN(B ()) (
) for all k, and L(Bi;j ) = Fi(i ()) (
) for all i; j . We will call this simply the -conjugate distribution, and we may think of it as the (; K )-conjugate distribution with K  1. Notice that P(;K ) (
) = P() (
) on FK . Recall () def = 0N (B ())=0A (?) as de ned in (9). Since E(?) [A] = 0A (?) and E(B ()) [N ] = 0N (B ()), we now observe that () is indeed the -conjugate arrival rate. Likewise, since P = ci=1 0i (i ())?1 is the -conjugate service rate. E(i ()) [Bi ] = 0i (i ()), we now see that () def Now, assume that 0 <  < 1,   c and () = 0. By convexity we have 0 () > 0, and hence, from the dierentiation formula (8) we have () > b() = (). (See the remarks at the end of Section 2.1 regarding b()  ().) Thus, by Lemma 3.2, the -conjugate workload process is unstable. For w > 0 de ne fk+  wg; (29) Kw+ def = inf fk : W

f k for k < K1 (by Lemma 3.1), the random which is an Fk -stopping time. Since Wk = W variables K1 , W and L+w that were originally de ned in terms of f(Wk? ; Wk+ )g can also be def k+)g. We have K1 = inffk  1: W f k? ; W fk?  0g, W = maxk0

So, it is sucient to show that supt>0 E(0;0) [v() (R(t))] < 1. For I  f1; ::; cg put EI def = fr: ri > 0 for i 2 I and ri = 0 for i 62 Ig and vI() (r) def = P (  ) (  ) v() (r) 1EI (r). Then, v() (r) = I vI (r), so we may consider E(0;0) [vI (R(t))] separately for each index set I . If i = 0 for each i 2 I , then vI() (
)  1. So, putting I () def = mini >0;i2I i (), we only need to consider subsets I such that I () > 0. = Let Sei (j ) and Si(j ) be as de ned in Section 3.1. To abbreviate the notation, write Si (m) def def (  ) Si(Ji (m)), S (m) = (S1(m); ::; Sc (m)), Sei (m) = Sei (Jei (m)), etc. For u 2 [0; 1)c , put vbI (u) Q = i2I E[ei ()(Bi ?ui ) ; Bi > ui ] whenever mini2I ui  maxi62I ui  0, otherwise put vbI() (u) = 0. Then, whenever I is a strict subset of f1; ::; cg, we have E(0;0) [vI() (R(t))] =

1 X

m=0

E(0;0) [ vI() (t ? S1 (m); ::; t ? Sc (m)) ; MK (t) ? #I = m ]

"X 1

 E(0;0)

m=0

vbI() (t ? S1 (m); ::; t ? Sc(m))

#

where #I denotes the number of indices in I . For the case I = f1; ::; cg, simply replace MK (t) ? #I = m by MK (t) ? c  m in the rst equality above. The upper bound is the same. We obtain a 24

further upper bound by applying the crude bound





vbI() (u)  exp #I B () ? I () _ (u) 1[0;1) (_(u)): De ne U (s) def = E(0;0) [ number of points S (m) 2 [0; s]c ]. Then the bound is

Zt h i E(0;0) vI(;) (R(t))  e#I B () e?I ()(t?s) dU (s): 0

Integration-by-parts, and then the change of variable  = t ? s yields

Zt 0

e?I ()(t?s) dU (s) = e?I ()t U (t) + I ()

Zt 0

[U (t) ? U (t ? )] e?I () d:

Put s (
) = L(0;0) (R(s); Q(s); TK (s)+1 ? s). Since Sei (j )  Si (j ) (see the proof of Lemma 3.1), we have

U (t) ? U (t ? ) = E(0;0) [ number of points S (m) 2 [0; t]c n[0; t ? ]c ] = Et? [ number of points S (m) 2 (0; ]c ]

 Et? [ number of points Se (m) 2 (0; ]c ]  E(0;0) [ number of points Se (m) 2 [0; ]c ] = Ue ():

def

Since the Sei (j )s are conditionally independent random walks (given R?0 = 0), we have Ue ()  . Thus, sup

Zt

t0 0

e?I ()(t?s) dU (s)  sup e?I ()t Ue (t) + I ()

which yields the desired result.

t0

Z1 0

Ue () e?I () d < 1;

Lemma 4.4: Assume  <  and fWk? g is positive Harris recurrent. Fix  2 D such that () > (). Then P() (W = 1) > 0, and P (W  w) > 0 for all w < 1. Proof: Applying Lemmas 3.1 and 3.2 to the -conjugate process, we conclude that there are = only nitely many cycles with Wk+ drifting o to +1 during the last cycle a.s. Put `() (
) def P() (WK?` 2
; K` < 1) = a substochastic measure. Then 1 X `=0

P ()( K 

` 0 for some ` < 1. Clearly `() (
)   (
) = P (WK+` 2
). Thus, ` P() (W = 1) > 0 for some ` < 1 implies P() (W = 1) > 0. The restrictions of P() (
) and P (
) ` to the event fKw+ < 1g are equivalent, so we have P (W  w) > 0 for all w < 1. ( )

( )

Corollary 4.1: Assume  < , fWk? g is positive Harris recurrent, and () > () for some  > 0. Then (
) = L (WK? ` ) is admissible and (33) holds. Finally, we turn our attention to the functions gw() (
;
) de ned by (30).

Lemma 4.5: Assume 0 <  < 1. Then as w " 1 we have the pointwise limit gw() (
;
) " g() (
;
) = a bounded measurable function.

P

w ?1 1 fk+). By translation of the fw def fk?  ?wg and Le w def = Kke=0 Proof: De ne K = inffk: W [0;1) (W f+0 = ; W0+ = w + z] = E[ Le w j
f+0 = ; W0+ = z]. Since Kfw " 1 initial condition we have E[ L+w j
P 1 (W + as w " 1, we have Le w " Le 1 = 1 k=0 [0;1) fk ) a.s. The expectation converges by monotone convergence. Thus, gw() (
;
) " g() (
;
) pointwise, where

h



i

f+0 = ; W0+ = z exp( ?z ? ()
 ): g() (; z) = E Le 1


(34)

f+  0g with inf ; = 1, and observe that fLe 1  kg  fKk < Fix  2 (0; ). Put Kk = inff  k: W 1g. We now use a modi ed version of the process distributions and the likelihood ratio formula (28). The modi cation is that we initialize with W0+ instead of W0? . The resulting modi cation of (28) is that W0? is replaced by W0+ , and the term N (B ()) is deleted. To obtain an upper + W fK+  0, we fi;K bound we may delete the terms Jei (MKk )[i (i ()) ? B ()]  0, and, since W k k + f can also delete the term ?()
WKk  0. Fix  2 (0; ), so () < 0. Then f+0 = ; W0+ = z ) P ( Le 1  k j


f+0 = ; W0+ = z )  P ( Kk < 1 j
f+ h   i +=z  E(;Kk) exp ()
W0+ + Kk () Kk < 1
=  ; W 0 0    exp z + ()
 + () k ;

where the last bound follows because Kk  k and () < 0, and Wi;+0 = i + z . Summing over k, 26

we have

h



f+0 = ; W0+ = z E Le 1


i

+ ()
 ) :  exp ( z 1 ? e()

Since  <  implies i ()  i (), we have () ? ())
 )  1 g() (; z)  exp( ? ( ? )1z ?? e(( < 1 ) 1 ? e() for all  2 [0; 1)c and z  0.

(35)

Corollary 4.2: Assume fWk? g is positive Harris recurrent, the Fi (
)s are spread out, and the conditions listed in part (i) of Lemma 2.3 hold. Let  (
) be any admissible initial distribution. Then

i h f+ + i h w E [ L+ ] = eN (B ()) E() v() (W? ) ; W = 1 E g() (
lim e  1 ; V1 ) w 0  w!1 +

(36)

+ (
) is the limiting distribution in Lemma 4.2, and moreover, the limit is nite and positive.

where

At this point we have established the tight P ( Wk+  w ) limit of Theorem 2.3 part (i). The constant cW is just the right side of (36) divided by E [K1 ]. To get the limit for P ( W  w ), simply replace L+w by 1fW wg in the preceding arguments. Part (ii) of Theorem 2.3 addresses the necessity of the conditions 0 <  < 1,    c and () = 0. If any of these conditions fail, the claim is that ew P ( Wk+  w ) ! 0, etc. We only give a quick sketch of the proof. The case  = 0 is trivial. The case  = 1 follows from Theorem 2.2. So assume 0 <  < 1, in which case, by Lemma 4.5, gw() (
;
) is a uniformly bounded sequence of functions. If () < 0, then the additional likelihood ratio term ()Kw+ must be included in the expectation in expression (31). Since Kw+ " 1 as w " 1, the expectation will vanish, by dominated convergence. Likewise, if  > c , then by Lemma 2.1 we have i (i ()) < B () for at least one i, and the expectation in (31) will vanish due to the additional likelihood ratio terms Ji (MKw )[i (i ()) ? B ()] # ?1 a.s. We believe that the conditions 0N (B ()) < 1 and 0i (i ()) < 1 for each i are also necessary, but this is more dicult to prove. Assuming the other conditions hold, one must reconsider the fk+1 ?+ f+k ; W fk+)g. If 0N (B ()) = 1, then E [W renewal limit for the Markov additive process f(
f+k g is presumably null Harris fk+] = 1. If 0i (i()) < 1 for any i, then the Markov chain f
W recurrent. In either case, the limit of Lemma 4.2 should be degenerate. Appendix B provides some further discussion of the a degenerate case. +

27

5. Queue Length Analysis In this section we consider the proof of the queue length limits of Theorem 2.3. The arguments are almost identical to the workload analysis of the previous section, so we only give a quick sketch. De ne Fbk def = (R?0 , Q?0 , A1 ; ::; Ak , N0 ; ::; Nk , Bi;j : j  Jbi+ (Tk ), i = 1; ::; c) where Jbi+ (t) is as de ned in Section 3.2. Let K be an Fbk -stopping time. De ne the (; K )-conjugate distribution P(;K ) (
) exactly as in Section 4, except that we use Fbk in place of Fk . The dierence is that in Section 4 service times are assigned to jobs at the instant that they enter the system, while in this section service times are assigned to jobs only when they exit the queue and processing begins. Thus, in Section 4 the service times for jobs in queue at time TK+ are samples from the twisted distributions Fi(i ()) (
), while here they are ultimately sampled from the untwisted distributions Fi (
). The restrictions of P(;K ) (
) and P (
) to the event fK < 1g are equivalent, and the likelihood ratio is

dP

dP(;K )

= exp  TK + K A (?) ? B () MK + (K + 1) N (B ()) +

c h X i=1

i! + ? + e b b ?i() (Si (Ji (TK )) ? Ri;0) + Ji (TK ) i (i()) : P

By (20), (21) and other de nitions in Section 3.2 we have MK = Qe +K ? Q?0 + ci=1 Jbi+ (TK ) and + . Using these relationships along with Pc  ( ) =  and  ( ( )) = Sei(Jbi+ (TK )) = TK + Rei;K i i i=1 i B () for   c , the above likelihood ratio is reduced and we obtain the following analog of Lemma 4.1.

Lemma 5.1: Assume  2 D \ (?1; c]. The restrictions of P(;K ) (
) and P (
) to the event fK < 1g are equivalent, and on this event we have dP = exp ?  () (Qe + ? Q? ) ? ()
(Re ? R? ) K B 0 0 K dP(;K )  + K () + N (B ()) :

(37)

De ne Kn+ def = inffk  1: Qe +k  ng and observe that fQ  ng = fKn+ < K1 g. Also, the P = random variable L+n in Lemma 3.6 can be written as L+n = Kk=0?1 1[n;1) (Qe +k ). De ne vb() (r; q) def 1

28

exp( B () q + ()
r ). Then, Lemma 5.1 yields

h



E [ L+n ] = eN (B ()) E(;Kn ) L+n exp ? B () Qe +Kn ? ()
Re Kn +

+

+

 i

vb() (R?0 ; Q?0 ) ; Q  n : De ne



h

i

gbn() (r; q) def = E L+n Re 0 = r; Qe +0 = n + q exp( ? B () q ? ()
r )

(38)

where the expectation with respect to the untwisted probability law. Then by conditioning on FbKn we obtain

+

h

i

E [ L+n ] = eN (B ()) E() gbn() (Re Kn ; Qe +Kn ? n) vb() (R?0 ; Q?0 ) ; Q  n !n +

+

(39)

where ! def = exp(?B ()). Formula (39) is obviously the queue length analog of the waiting time expression (31). As in Section 4, we characterize the limiting behavior of (39) using renewal theory. Lemma 5.2 is proved in Appendices A and C.

Lemma 5.2: Assume the Fi (
)s are spread out, 0 <  < 1, () > (), 0i(i ()) < 1 for each i = 1; ::; c and 0N (B ()) < 1. Then there exists a unique probability distribution + (
) on [0; 1)c  Z+ such that the following holds. Let g: [0; 1)c  Z+ ! [0; 1) be bounded and measurable, and let gn (
;
) be uniformly bounded measurable functions such that gn (
;
) ! g(
;
) pointwise. Let  (
) be a distribution and let v(
)  0 be a measurable function on [0; 1)c  Z+ such that E [ v(R?0 ; Q?0 ) ] < 1. Then i ()h g (R + ? n) v (R? ; Q? ) ; Q  n + ;Q e e lim E n 0 0 Kn Kn n!1  h i h i = E g(Re +1 ; V1+ ) E() v(R?0 ; Q?0 ) ; Q = 1 : (40) +

+

+

where (Re +1 ; V1+ ) have joint distribution

+ (
).

Veri cation of the conditions of Lemma 5.2 follow almost exactly as in Section 4. Lemmas 4.3 and 4.4 are directly applicable to the queue length analysis. Thus, we nd that b(
) is admissible. Following the proof of Lemma 4.5, it is straightforward to show that gn() (
;
) " g() (
;
) = a bounded measurable function. 29

Appendix A: Markov Additive Processes Renewal Theory Let ? denote either R or Z, and put ?+ def = ?\[0; 1). Let (S; S ) be a measurable state space. Let f(Xk ; Zk )g be a Markov chain taking values on S? such that P ( (Xk+1 ; Zk+1) 2
j (X ; Z);   k) P = P ( (Xk+1 ; Zk+1 ) 2
j Xk ). Then f(Xk ; Yk )g, where Yk = Y0 + k=1 Z , is called a Markov additive process . We will work with the following conditions: (i) There is a function f : ? ! [0; 1] such that f (z ) is nonincreasing for z  0, f (z ) = 0 for z < 0, R Px( jZ1 j  z )  f (z) for all (x; z) 2 S  ?+ and cZ def = 01 f (z ) dz < 1. (Notice that Ex [ jZ1 j ]  cZ for all x 2 S.) (ii) fXk g is positive Harris recurrent with stationary distribution (
), and E [Z1 ] > 0. (iii) There is a positive integer k0 < 1, a > 0, a probability measure  (
) on S  ?, and a set C 2 S such that (41) P(x;0) ( (Xk ; Yk ) 2
)  1C (x)  (
): 0

0

(iv) In the case ? = R,  (dx  dz ) = e(dx) (b ? a)?1 1[a;b] (z ) dz for some a < b. Condition (iii) is the basis for a regeneration scheme for the skeletons f(X +k ; Y +k ):  = 0; 1; 2:::g, where 0 = 0; ::; k0 ? 1 is xed. Brie y, the probability space is augmented with a sequence of 0{1 valued \coin toss" random variables  constructed as follows. If X +k 62 C , then  = 0. If X +k 2 C , then  = 1 with probability , otherwise  = 0. The event f = 1g is the regeneration event. The regeneration times for the 0 th skeleton are K0 (0 ) = 0 and K` (0 ) = inff > K`?1 (0 ):  = 1g for `  1. The following Lemma is proved in Ney and Nummelin (1984) and Nummelin (1984). 0

0

0

0

0

0

0

0

Lemma A.1: Assume n  fXk g is Harris recurrent and  (iii) holds. Then for xed 0o 2 f0; ::; k0 ? 1g, the blocks X +k ; [Y +k ? Y +K` ( )k ] :  = K` (0 ) + 1; ::; K`+1 (0 ) , for `  0, are independent. Moreover, for `  1 the blocks are i.i.d. with distribution determined by L(X +(K`( )+1)k , [Y +(K`( )+1)k ? Y +K`( )k ] ) =  (
). 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

The basic ideas of the proof of Lemma A.2 below are the same as the proof of Theorem 4.1 of Athreya, et. al. (1978). The much stronger hypotheses (conditions (i) { (iv)) used here allows us 30

to relax the continuity requirement on the function f (
;
). Also, for our application we must work with k0 > 1, which was not worked out by Athreya et. al.

Lemma A.2: Assume conditions (i) { (iv). Let f : S  ? ! [?1; 1] be a (bounded) measurable function such that jf (x; z )j  f (z ) for all (x; z ) 2 S  ?+ . Then we have the limit # "X R 1 E [ f (X; z) ] dz 1 ?1  lim E f ( X ; y ? Y ) = : (42) k k ( x ;y ) y!1 E [Z ] 0

0



k=0

1

P

1 Z where Z = Y Proof: Begin by writing Y +K`( )k = Y +K ( )k + `m?=1 m m  +Km ( )k ? Y +Km ( )k . By lemma A.1, fZ mg is an i.i.d. sequence with marginal distribution L(Z m ) = i hP L (YK (0)k ), which does not depend on (x0 ; y0; 0 ). De ne F0(y) def = E K=1(0) f (Xk ; y ? Yk ) . Then using Lemma A.1 we have 0

0

0

0

0

0

1

0

0

0

+1

0

0

0

1

1

2 E(x ;y )4 0

0

0

0

K`+1 X(0 ) =K`(0 )+1

3 f (X +k ; y ? Y +k ) 5 0

0

0

0

"

= E(x ;y ) F0 y ? Y +K ( )k ? 0

0

0

0

1

0

0

`X ?1 m=1

Zm

!#

"

= E F y?

`?1 X m=1

Z m ;x0 ; y0 ; 0

!#

where F (y; x0 ; y0 ; 0 ) = E(x ;y ) [F0 (y ? Y +K ( )k )]. Thus,

"X 1

E(x ;y ) 0

0

=0

0

0

0

f (X +k ; y ? Y +k ) 0

0

0

1

#

0

0

0

3 2 K ( ) "X 1  # X = E(x ;y )4 F y ? Y ` ; x0 ; y 0 ; 0 f (X +k ; y ? Y +k ) 5 + E 1

0

P

0

0

=0

0

0

0

0

`=1

(43)

where Y ` = `m=1 Z m . Since jf (x; z )j  f (z )  1 and Ex [K1 (0 )] < 1 (by positive Harris recurrence), the rst term on the right side of (43) will vanish as y ! 1 by dominated convergence. Thus, the limit (42) will follow by application of the i.i.d. key renewal theorem to the second term in (43), and then adding up the k0 limits for 0 = 0; ::; k0 ? 1. We only need to verify the hypothesis of the i.i.d. key renewal theorem. By positive Harris recurrence we have E[ jZ 1 j ] = E [ jYK (0)k j ]  cZ k0 E [K1 (0)] < 1, and 0

1

31

0

E[Z 1 ] = E [YK (0)k ] = E [Z1 ] k0 E [K1 (0)] > 0 because E [Z1 ] > 0. Next, we have 1

Z1

?1

0

2 K (0) Z 3 1 X jF0 (y)j dy  E4 jf (Xk ; y ? Yk )j dy 5  cz E [ K1 (0) ] < 1; 1

?1

=1

0

0

and hence, F (
; x0 ; y0 ; 0 ) is also integrable. We only need to show direct Riemann integrability for the case ? = R. By condition (iv), and using the Markov additive process de ning property,

3 2 k X E4 f (x ; y ? [z + Yk ]) Xk = x ; Yk ? Yk = z ; K1 (0) = k1 5 =1 ZbX k = f (x; y ? [z + z]) b dz ?a 1

0

0

0

0

1

a =1

which is a continuous function of y. Taking the expectation, a standard dominated convergence argument yields that F0 (
) is also continuous, and hence, directly Riemann integrable. Likewise, so is F (
; x0 ; y0 ; 0 ).

Lemma A.3: Assume conditions (i) { (iv) and de ne Ky def = inf fk : Yk  yg. Then there exists a probability distribution (
) on S  ?+ such that for all (x0 ; y0 ) 2 S  R we have the limit h ?
i = E [ g(X; Z ) ] lim E g X ;Y ? y (44) y!1 (x0 ;y0 )

Ky Ky

whenever g: S  ?+ ! [0; 1) is a bounded measurable function. Sketch of Proof: Put Ky0 = 0 and Kyi+1 = inffk > Kyi : Yk?1 < y; Yk  yg with inf ; = 1. In particular, Ky = Ky1 . Also, de ne I = supfi : Kyi < 1g. (We have I < 1 a.s.) Without loss of generality, suppose sup(x;y) = g(x; y) = 1. i h For z > 0, put fa (x; z ) = E(x;y?z) g(X1 ; Y1 ? y); Ky1 = 1 (which does not depend on y  0), and for z  0 put fa (x; z ) = 0. Then, fa (x; z )  Px (Z1  z )  f (z ), and a straightforward computation yields

"X 1

E(x ;y ) 0

0

k=0

fa(Xk ; y ? Yk )

h 

#

"X I 

= E(x ;y ) 0



0

i=1

g XKyi ; YKyi ? y

i

#

:

For z > 0, put fb(x; z ) = E(x;y?z) g XKy ; YKy ? y ; Ky1 = 1; Ky2 < 1 (which does not depend on the value of y  0), and for z  0 put fb (x; z ) = 0. Then, fb(x; z )  Px (Z1  z )  f (z ), and a 2

2

32

straightforward computation yields

"X 1

E(x ;y ) 0

0

k=0

fb (Xk ; Yk ? y)

#

"X I 

= E(x ;y ) 0

Thus, putting f (x; z ) = fa (x; z ) ? fb (x; z ), we have

"X 1

E(x ;y ) 0

0

k=0

f (Xk ; Yk ? y)

#

0

i=2

g XKyi ; YKyi ? y ; I  2 :

 ?

= E(x ;y ) g XKy ; YKy ? y 0

#



0




and the limit (44) follows by application of Lemma A.2. Lemma A.4 addresses the degenerate case. For g(
;
)  0, we have E(x ;y ) [g(XKy ; YKy ? y)]  P g(X ; Y ? y)]. Thus, the proof of Lemma A.4 is obtained by reconsidering the proof E(x ;y ) [ 1 k k k=1 of Lemma A.2 with E[Z ` ] = 1. For brevity, we omit this proof. 0

0

0

0

Lemma A.4: Assume conditions (iii) and (iv), fXk g is Harris recurrent, R [ ?Ex[Z1 ] ]+ (dx) < 1 g(x; z ) dz (dx) < 1. Then 1, R [ Ex [Z1] ]+ (dx) = 1, and g(
;
)  0 is such that RS R?1 E(x ;y ) [g(XKy ; YKy ? y)] ! 0 as y " 1. 0

0

Appendix B: The Workload Markov Additive Process In this appendix we prove Lemma 4.2 as an application of Lemma A.3. Notice that since there f+k =
f?k+1, hence, we may work with either are no re ections in the \tilde process," we have
f+k g. f?k g or f
f


Lemma B.1: Assume that Fi (
) is spread out. Fix any w 2 (0; 1). Then there exists a > 0 and a ji < 1 such that for each j  ji there are constants ai (j ) < bi (j ) and i (j ) > 0 such that   (45) Pw Sei(j ) 2 ds ; Sei(j ) ? Sei(j ? 1) >  i(j ) 1[ai (j);bi(j)](s) ds for all w 2 [0; w ]c . Moreover, bi (j ) ? ai (j ) = O(j ). Proof: Let F j (
) denote the j th convolution power of F (
). If F (
) is spread out, there exists

> 0, j y < 1 and a < b such that F j y (ds)  1[a;b] (s) ds. For each j  j y we have ay(j ) < by(j ) and (j ) > 0 such that F j (ds)  (j ) 1[ay (j );by(j )] (s) ds. Moreover, by convolution of uniform densities, we can x the ay (j )s and by(j )s so that for each j0 = 0; ::; j y ? 1 we have ay (j0 + mj y ) 

33

ay(j0 ) + (m + 1)ay(j y ) and by(j0 + mj0 )  by(j0 ) + (m ? 1)by(j y ) for all m  1. Take ji < 1 suciently large so that byi (j ? 1) ? ayi (j ? 1) > w + . The lemma follows by applying the above minorizations to Fi(j ?1)  Fi (
) where Fi (
) = Fi (
\ ( ; 1)).

Lemma B.2: Assume that each Fi (
) is spread out. Fix any  2 (0; 1). There exist a positive integer k0 < 1, a > 0 and a probability measure  (
) on [0; 1)c  R such that f  f? f ? f    1 c ()  (
) (46) P (
0 =  ; W0 = 0 k ;W ) 2

0

k0

[0; ]

and  (
) satis es condition (iv) in Appendix A. Proof: Fix > 0 such that Fi (( ; 1)) > 0 for each i. Let the ai (j )s and bi (j )s be as constructed in the proof of Lemma B.1. Then there exists an m0 < 1 such that for each m  m0 there is an P = ^(a1 (j1 ); ::; ac (jc )) < integer vector (j1 ; ::; jc ) such that ji  ji for each i, ci=1 ji = m, and a(m) def def b(m) = _(b1 (j1 ); ::; bc (jc)). By taking m0 suciently large, we also have b(m) ? a(m)  . Fix k0 < 1 so that P (Mk ?1 = m ) > 0 for some m  m0 . Let U be a random vector that is uniformly distributed on the cube [a(m ); a(m ) + ]c and is independent of Tk . Then (46) holds with  (
) = L((U1 ? _(U); ::; Uc ? _(U); _(U) ? Tk ) and = P (Mk ?1 = m ) c mini i (ji ). 0

0

0

0

Lemma B.3: Assume that E[Bi] < 1 for each i. Then for all suciently large  < 1 there exists

an  > 0 such

h



fc )
f0 =  E ^(


whenever  62 [0;  ]c and _( ) = 0.

i

 ^() ? 

(47)

Proof: De ne Sei (m) def = Sei(Jei (m)) and Se  (m) def = (Se1 (m); ::; Sec (m)). Then
e ?i;k = Sei (Mk?1 ) ? _(Se i (Mk?1 )). Also, put B (m) = Bi;j with (i; j ) = (Ie(m); JeIe(m) (m)). Let Se  (0) =  with _( ) = 0. Then we claim that _(Se  (c))  minm=1;::;c B  (m). Let [i] , i = 1; ::; c, denote the components of the vector  ordered so that [1]  [2]  :::  [c]. After job m = 1 is inserted there are two possibilities; either _(Se  (1)) = [1] + B (1), or _(Se  (1)) = [2] . In the rst case, the claim holds. So proceed assuming latter case. After job m = 2 is inserted, either _(Se (2)) = [i] + B (i) for i = 1 or i = 2, or _(Se (2)) = [3]. Again, the claim holds in the rst case, so proceed assuming the second, and so on. After c ? 1 jobs have been inserted we have either

34

_(Sei(c ? 1)) = [i] + B (i) for some i  c ? 1 or _(Se  (c ? 1)) = [c]. In the latter case, insertion of the cth job yields _(Se  (c)) = mini=1;::;c[[i] + B  (i)]  mini=1;::;c B  (i). Now, write

h



f?c )
f?0 =  E ^(


i



h

i

f?c ) ; ^(Se (Mc?1 )) = ^() Se  (0) =  = E ^(
h f?  (Mc?1 )) > ^( ) Se  (0) =  i : e + E ^(
) ; ^ ( S c

(48)

We upper bound the two terms above separately. f?c ) = ^(Se  (Mc?1)) ? To upper bound the rst term on the right side of (48), observe that ^(
_(Se (Mc?1 )), and _(Se (Mc?1 ))  _(Se  (c))  minm=1;::;c B (m) because Mc?1  c. Thus, on f?c )  ^() ? minm=1;::;c B (m). De ne G(b) def the event f^(Se  (Mc?1 )) = ^( )g, we have ^(
= def mini=1;::;c Fi ([b; 1)) and G(b) = maxi=1;::;c Fi ([b; 1)) for b 2 [0; 1). By a standard construction, we may construct i.i.d. sequences fB  (m)g and fB  (m)g such that B  (m)  B  (m)  B  (m) nPMc?  o B (m) < ^()  a.s., P (B  (m)  b) = G(b) and P (B  (m)  b) = G(b). Then m =1 o nPMc?  e  (Mc?1)) = ^()g. Thus, for  62 [0;  ]c we have m=1 B (m) < ^( )  f^(S 1

1





 (m) ; ^(Se  (Mc?1 )) = ^( ) Se  (0) =  E mmin B =1;::;c



2 3 MX Z1 c?  (m) <  5 "  (m) ;  E4 mmin B B G(b)c db =1;::;c 1

m=1

0

as  " 1, and hence, when  62 [0;  ]c with suciently large  < 1, we may upper bound the rst term in (48) as

Z f?c ) ; ^(Se (Mc?1 )) = ^() j Se (0) =  ]  ^() ? 1 1 Gc(b) db: E[ ^(
2 0

(49)

Next consider the second term in (48). Since ^( ) = [c] = i for some i 2 f1; ::; cg, we have f^(Se  (Mc?1)) > ^()g = fSei (Mc?1 ) > [c] for some i 6= i g. The insertion of the mth job can increase maxi6=i Sei (m) by at most B  (m). Thus, for  62 [0;  ]c we have f^(Se  (Mc?1 )) > ^( )g  c? B  (m)   g. When ^(S e  (m)) > ^(Se  (m ? 1)) we have ^(Se (m)) ? _(Se  (m))  B (m). fPMm=1 f?c )  maxm=1;::;Mc? B (m), and the second Thus, on the event f^(Se  (Mc?1 )) > ^()g we have ^(
1

1

35

term in (48) is upper bounded by



h

f?c ) ; ^(Se  (Mc?1)) > ^() Se (0) =  E ^ (


i

2 3 MX c?    B (m)   Se (0) =  5 B (m) ;  E4 m=1max ;::;Mc? m=1 2 3 MX c?  E4 max B (m) ; B  (m)   5 1

1

1

m=1;::;Mc?1

(50)

m=1

for all  62 [0;  ]c . (Notice that the last bound does not depend on the initial condition .) By dominated convergence the bound (50) will vanish as  " 1. Lemma B.3 follows by applying (49) and (50) to (48) with suciently large  < 1.

fk g is a positive Lemma B.4: Assume E[Bi] < 1 and Fi (
) is spread out for each i. Then f
Harris recurrent Markov chain, and for any  2 (0; 1), [0;  ]c is a small set. ?

fk 2 [0;  ]c Proof: By Lemma B.2, to establish Harris recurrence it is sucient to show that Pw (
f?k 2 [0;  ]cg, Applying Lemma B.3, x  < 1 so that (47) holds for some  > 0. Put K = inffk:
f?k 2 [0;  ]cg. For  62 [0; ]c, (47) and Theorem 4.2 in Meyn and Tweedie and K  = inffk = c:
f?0 = ]  E[K  j
f?0 = ]  c ^ ()=, and this proves Harris recurrence with (1992) gives E[K j
small set [0;  ]c and irreducibility measure  (
 R). Using the same arguments as in the proof of Lemma B.3, for all  2 [0;  ]c we have h



f?0 =  f?c )
E ^(


i

   (m)   + E m=1max B < 1: ;::;Mc? 1

f?c:  = 1; 2; :::g is positive Harris recurrent. By By Proposition 5.10 in Nummelin (1984), f
f?k g is positive Harris recurrent. irreducibility, f


Lemma B.5: Assume E[N ] < 1, E[Bi ] < 1 and Fi (
) is spread out for each i, and  > . Then fk ; W fk)g satis es conditions (i) through (iv) in Appendix A. the Markov additive process f(
fk??1, and let ~ (
) denote the stationary distribution of the positive fk? ? W Proof: Put Zk? = W f?k g. Clearly, jZ1?j  Z = A1 + Pci=1 PNj=1 Bi;j . Put f (z) = P (Z  z), Harris recurrent chain f
P fk?]  E~ [Z1?] k  so cZ = E[A] + E[N ] ci=1 E[Bi ] < 1. By and Lemma 3.2 we have Ew [W 0

36

(= ? 1) E[A] k, hence, E~ [Z1? ] = (= ? 1) E[A] > 0. Lemmas B.2, B.3 and B.4 verify the other conditions. Proof of Lemma 4.2: First, recall that fKw+ < K1 g = fW  wg. For v < w,

h

fK+ ? w) ; W  v f+Kw ; W Ew g (
w

i

? jjgjj Pw (v  W < w) h f+ f + i ? w ) ; ; W  Ew g (
W  w Kw K w h f+ f + i  Ew g (
Kw ; WKw ? w) ; W  v + jjgjj Pw (v  W < w)

+

+

+

+

+

+

where jjgjj = sup g( ; z ). Successive conditioning on FKv , the dominated convergence theorem and Lemma A.3 yield +

h

f+Kw ; W fK+ ? w) ; W  v g (
lim E w w!1 w +

i

+

h

h

f+ f+ f + = Ew wlim !1 E g(
Kw ; WKw ? w) j WKv +

+

+

i

; W v

i

f+1; V1+ ) ] Pw (W  v): = E [ g (
Letting w " 1, the last two displays yields

h

f+Kw ; WK+ ? w) ; Kw+ < K1 g (
lim inf E w w!1 w +

i

+

f+1; V1+) ] Pw (W  v) ? jjgjj Pw (v  W < 1)  E [ g (
Let v " 1, so Pw (W  v) # Pw (W = 1) and Pw (v  W < 1) # 0. A similar argument upper bounds the lim sup. Replacing g(
;
) by gw (
;
) follows by dominated convergence. To obtain Lemma 4.2, simply apply the above arguments to the -conjugate distribution. Finally, we comment on the case that the necessary conditions 0 <  < 1,   c and () = 0 hold, but either 0N (B ()) = 1, or 0i (i ()) = 1 for at least one i. In these cases we argue that the degenerate renewal limit of Lemma A.4 is applicable, ultimately yielding the same conclusion as part (ii) of Theorem 2.3. In the case E(B ()) [N ] = 0N (B ()) = 1, which clearly yields E() [Z1 ] = f+k g is null Harris recurrent 1 for all . If E(i ()) [Bi ] = 0i (i()) = 1 for some i, then evidently f
37

RR

with - nite invariant measure (
). It must also be shown that g() (; z) dz (d ) < 1. This task is reduced by applying the bound (35) developed in the proof of Lemma 4.5. >From (35), R we easily obtain g() ( ; z ) dz  C e?
 where C < 1, = () ? () for some  2 (0; ). Moreover, note that since 0 <    c, we have i > 0 for each i. Thus, it would be sucient to R show that e?
 (d ) < 1.

Appendix C: The Queue Length Markov Additive Process In this Appendix we provide only the key elements of the proof of Lemma 5.2. The full proof follows essentially the same arguments as in Appendix B.

Lemma C.1: Assume that each Fi (
) is spread out and x r 2 (0; 1). Then there exist a positive integer k0 < 1, a > 0 and a probability measure  (
) on [0; 1)c  Z such that   P (Re ?k ; Qe ?k ) 2
j Re ?0 = r; Qe ?0 = 0  1[0;r]c (r)  (
): (51) 0

0

Proof: Let the ai (j )s and bi (j )s and be as constructed in the proof of Lemma B.1 (using r in place of ). Fix some  > 0, and for each k < 1, nd a t0 (k) > 0 such that P (Tk 2 [t0 (k); t0 (k) + ]) > 0. There exists a k0 < 1 such that for all k  k0 there is a vector j(k) such that ji (k)  ji (from Lemma B.1) and [t0 (k); t0 (k) + ]  [ai (ji (k)); bi (ji (k)) ? ]. Then (51) holds with ! c X ? c  ( dr  fqg ) = 1[0; ]c (r) dr P Mk ?1 ? ji(k0 ) = q 0

and

= c

mini i (ji ).

i=1

Lemma C.2: For suciently large r < 1, there exists an  > 0 such E[ ^(Re ?1 ) j Re ?0 = r ]  ^(r) ? 

(52)

whenever r 62 [0; r ]c .

e ?0 = rg, either ^(Re ?1 ) = ^(r) ? A1, or for some i 2 f1; ::; cg we have ri  A1 and Proof: Given fR ^(Re ?1 )  Bi;j where j  Jbi (A1 ). Put B(r) = maxi=1;::;c maxjJbi(A ) Bi;j on the event fRe ?0 = rg. 1

38

Then E[ ^(Re ?1 ) j Re ?0 = r ]  ^(r) ? E[ A1 ; A1  ^(r) ]

+ E[ B (r) ; B (r)  ^(r) ? A1 ]:

(53)

For r 62 [0; r ]c , we have fA1  rg  fA1  ^(r)g, and hence, E[A1 ; A1  ^(r)]  E[A1 ; A1  r] " E[A1 ] as r " 1. Next, we have B (r)  B = maxi maxj =1;::;N Bi;j , and hence, for r 62 [0; r ]c the last term in the above display is upper bounded by E[B ; B  r ? A1 ] # 0 as r " 1. 0

Lemma C.3: Assume each Fi (
) is spread out. fRe ?k g is a positive Harris recurrent Markov chain, and for any r > 0 the set [0; r]c is a small set.

Proof: Lemma C.3 follows form Lemmas C.1 and C.2. The proof is a direct parallel to the proof of Lemma B.4.

Lemma C.4: Assume each Fi(
) is spread out and  > . Then the Markov additive process f(Re ?k ; Qe ?k )g satis es conditions (i) through (iii) in Appendix A. e?   P Proof: Put f (z ) = P N0 + ci=1 Jbi (A1 )  z R 0 = 0 and apply Lemmas C.1, C.2 and C.3. Lemma 5.2 follows from Lemma A.3, the hypothesis having been veri ed by Lemmas C.1 { C.4. The proof follows exactly the same lines as the proof of Lemma 4.2 given in Appendix B.

Acknowledgment The authors would like to express their sincere thanks to Sean Meyn (University of Illinois at Urbana-Champaign), Richard Tweedie (Colorado State University) and Steve Lalley (Purdue University). Their valuable advice help proved indispensable in the course of this work.

39

References [1] Aldous, D., Hofri, M. and Szpankowski W. (1992), Maximum size of a dynamic data structure: hashing with lazy deletion revisited, SIAM J. Computing , 21, 713-732. [2] Arjas, E., Nummelin, E. and Tweedie, R. L. (1978), Uniform limit theorems for non-singular renewal and Markov renewal processes, J. Appl. Prob. 15, 112-125. [3] Athreya, K. B., McDonald, D. and Ney, P. (1978), Limit theorems for simi-Markov processes and renewal theory for Markov chains, Ann. Probab. 6, 788-797. [4] Bucklew, J. A., Ney, P. and Sadowsky, J. S. (1990), Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains, J. Appl. Prob. 27, 44-59. [5] Feller, W. (1971), An Introduction to Probability and its Applications, Vol. II, John Wiley & Sons, New York. [6] Iglehart, D. L. (1989), Extreme values in the GI/G/1 queue, Ann. Math. Statis. 43, 627-635. [7] Meyn, S. P. and Tweedie, R. L. (1992), Stability of Markovian processes I: criteria for discrete time chains, Adv. Appl. Probab. 24, 542-574. [8] Meyn, S. P. and Tweedie, R. L. (1993), Markov Chains and Stochastic Stability , SpringerVerlag, New York. [9] Neuts, M. and Takahashi, Y. (1981), Asymptotic behavior of the stationary distribution in the GI=PH=c queue with heterogeneous servers, Z. Wahr. 57, 441-452. [10] Ney, P. and Nummelin, E. (1984), Some limit theorems for Markov additive processes, Proc. of Symp. on Semi-Markov Models, Brussels. [11] Nummelin, E. (1984), General Irreducible Markov Chains and Non-negative Operators, Cambridge University Press, Cambridge. [12] Sadowsky, J. S. (1991), Large deviations theory and ecient simulation of excessive backlogs in a GI/GI/m queue, IEEE Trans. Auto. Control AC-36, 1383-1394. [13] Sadowsky, J. S. (1993), On the optimality and stability of exponential twisting in Monte Carlo estimation, IEEE Trans. Inform. Theory IT-39, 119-128. 40

[14] Sadowsky, J. S. and Szpankowski, W. (1992), Maximum queue length and waiting time revisited: G/G/c queue, Probab. in Eng. & Inform. Sci., 6, 157-170. [15] Takahashi, Y. (1981), Asymptotic exponentiality of the tail of the waiting-time distribution in a PH=PH=c queue, Adv. Appl. Probab. 25, 391-403.

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