The Annals of Applied Probability 1998, Vol. 8, No. 1, 98 ] 111
THE LINEAR GEODESIC PROPERTY IS NOT GENERALLY PRESERVED BY A FIFO QUEUE BY A. J. GANESH 1
AND
NEIL O’CONNELL
University of London and BRIMS If a FIFO queue is fed by several input streams that jointly satisfy a sample path large deviation principle ŽLDP. with ‘‘linear geodesics,’’ then the cumulative departures Žup to a large time. also satisfy the LDP with a rate function which depends in a relatively simple way on the rate function corresponding to the inputs: this was demonstrated in a recent paper by the second author. It suggests the possibility of an iterative scheme which would allow one to determine the large deviation behavior of more complicated networks. To do this, however, one would require that the linear geodesic property be preserved: in this paper we demonstrate that in general it is not preserved. This is true even in the case of a single input stream.
1. Introduction and preliminaries. There has been considerable recent interest in the large deviations behavior of queueing systems. This started with the observation that for a single server queue, the tails of the queue length distribution can be characterized in terms of the large deviations behavior of the arrivals and service processes. ŽThis is actually a classical result, originally due to Cramer ´ in the iid case; for more general statements in the context of queueing systems, see w 2, 7, 8, 9x .. Since then there have been many attempts to extend the theory to more complicated networks. A starting point in this quest is to consider the effect of interactions in a shared buffer, which is served according to a FIFO Žfirst-in]first-out. policy. More precisely, if the arrival streams are assumed to jointly satisfy a large deviation principle ŽLDP., then what can be said about the joint large deviation behavior of the corresponding departure streams? A partial answer to this question was presented in w 7x , where the notion of decoupling of effective bandwidths was introduced. There it is shown that there is a region over which the large deviation rate functions for the cumulative departures and arrivals agree, and bounds are given outside that region. Chang and Zajic w 4x consider the case of a single arrival stream and stochastic service rate. In w 10x , a full description of the rate function for the cumulative departures is given in general, under the hypothesis that the arrival processes jointly satisfy a sample path LDP with ‘‘linear geodesics.’’ w Roughly
Received July 1996; revised May 1997. 1 Supported in part by a fellowship from BP and the Royal Society of Edinburgh. AMS 1991 subject classifications. 60F10, 60K25, 60G17. Key words and phrases. Sample path large deviations, queueing networks, multiclass queues.
98
THE LINEAR GEODESIC PROPERTY
99
speaking, this means that the most likely path to an extreme value is a straight line. It is usually assumed, or is a consequence of the stochastic model of the arrival process, that the sample paths of the arrival process satisfy a LDP with rate function of the form I Ž f . s HLU Ž f˙Ž t .. dt. If LU is convex, then the sample path f that minimizes I Ž f . subject to the boundary conditions f Ž0. s a and f Ž1. s b is described by the straight line joining Ž0, a. and Ž1, b ..x This begs the question, which we will address in this paper: do the departures also satisfy this hypothesis? If so, then one could treat quite complicated networks by successive iteration of the single-buffer results in w 10x . We know of one example where this is the case, namely if the inputs and service are independent Poisson processes and the queue is stable: then the outputs, in equilibrium, are also independent Poisson processes with the same rates as the corresponding inputs. However, we find that it is not generally the case, even when there is jut one input stream. The remainder of this section is devoted to giving some background and a formal description of the problem. Counterexamples to the above suggestion are given in the next section. But first we discuss some special cases when the departure process does have linear geodesics. Suppose there is a single input stream. If the service process is deterministic, then the departure process has linear geodesics. So, a recursive analysis of networks of such queues is possible, as in w 3x . Even if the service process is stochastic, we show that, conditional on the departure rate from a queue exceeding its mean, the departure process has linear geodesics. We are typically interested in the probability of queue lengths exceeding some large threshold, and in well-designed networks this requires departure rates exceeding their mean. Therefore, we have linear geodesics in the region of interest, and so the study of networks of queues using a recursive approach is again feasible. Such an approach has been taken in w 1x , in the context of quite general arrival and service processes and a single class of customers. We show in this paper that this approach cannot be extended easily to networks with more than one traffic class. In fact, even if the service process is deterministic and there are only two traffic types, the departure process need not have linear geodesics. ŽThis is true even if we condition on the aggregate departure rate exceeding its mean.. We now give a formal description of the problem. Consider a discrete time queue with d arrival streams X s Ž X 1 , . . . , X d . sharing an infinite buffer according to a FIFO policy with stochastic service rate C. The number of arrivals of each type in time slot k is denoted by X k , while the maximum number of customers of any type that can be served in this time slot is denoted by Ck . We will begin by assuming that the queue is empty at time slot 0. Define
Ž 1.
n
An s
Ý
ks1
n
Xk ,
Bn s
Ý
Ck .
ks1
Let A n s Ý djs1 A nj denote the total number of arrivals, and Dn the total number of departures up to time n. Assuming that the queue is work-con-
100
A. J. GANESH AND N. O’CONNELL
serving, we have
Ž 2.
Dn s
inf Ž A k y Bk . q Bn .
0FkFn
The amount of work, Dn s Ž Dn1 , . . . , Dnd ., serviced from each input stream by time n, is defined as follows. Set
Ž 3.
Tn s sup k F n: A k F Dn 4 ,
Ž 4.
Dn s A T n q Ž Dn y AT n . X T nq1rXT nq1 .
Recall that X k s Ž X k1 , . . . , X kd . denotes the amount of work arriving from the different streams in time slot k. The expression Tn denotes the last time slot such that all arrivals up to it have been served by time n Žsome of the arrivals in time slot Tn q 1 may also have been served.. In words, work is serviced in the order received and simultaneous arrivals from different sources are thoroughly mixed in the queue. Define S nŽ t . s ŽA w n t x rn, Bw n t x rn., R nŽ t . s Dw n t x rn. For each positive integer k, let L k denote the subspace of paths in L`Žw 0, 1x k . with nondecreasing components, and by A k ; L k the set of those paths with absolutely continuous components starting at zero. The following hypotheses are employed in w 10x . ŽH1. For all g g R, sup k Ew exp g Ž X k q Ck .x - `. ŽH2. For each l g R dq 1 , the limit
Ž 5.
L Ž l . s lim
nª`
1 n
log E exp Ž lS n Ž 1 . .
exists as an extended real number and is finite in a neighborhood of the origin. The sequence S n satisfies the large deviation principle ŽLDP. in L dq 1 with good rate function I given by
¡ L Ž f˙ . ds, ¢`,
~H0
1
Ž 6.
IŽ f . s
U
if f g A dq1 , otherwise,
where LU is the convex conjugate of L. ŽH3. The arrival and service processes are asymptotically independent in the sense that
Ž 7.
LU Ž x, c . s LUa Ž x . q LUb Ž c . .
We refer to the hypothesis ŽH2. as the ‘‘linear geodesic property.’’ It follows from ŽH2., the convexity of LU and Jensen’s inequality, that the optimal path from point to point is a straight line. Such a LDP has been shown to hold quite generally by Dembo and Zajic w 6x : roughly speaking, it holds provided the sequence is, in some sense, stationary and mixing. Under the above hypotheses, it was shown in w 10x that the sequence R n satisfies the LDP in
101
THE LINEAR GEODESIC PROPERTY
L d with good rate function given by Id Ž c . s inf I Ž f . : D Ž f . s c 4 ,
Ž 8. where D: C
Ž 9.
dq 1
ªC
d
is defined by
AŽ f . s Ž f 1 , . . . , f d . ,
Ž 10.
DŽ f . Ž t . s
Ž 11. Ž 12.
T Ž f . Ž t . s inf r : A Ž f . Ž r . s D Ž f . Ž t . 4 ,
inf
0F n F1
A Ž f . Ž n t . y f dq 1 Ž n t . q f dq1 Ž t . ,
D Ž f . s A Ž f . (T Ž f . .
Here Ž9. follows from Ž2., Ž10. from Ž3. and Ž11. from Ž4.. AŽ f . denotes the Žscaled. joint arrival process, while AŽ f . s f 1 q ??? qf d denotes the aggregate arrival process. For the scaled processes, T Ž f .Ž t . denotes the last time, arrivals up to which depart by time t. Since the queue was assumed to be empty at time 0 and the service discipline is FIFO, the departures in all the streams up to time t, denoted DŽ f .Ž t ., are precisely the arrivals in all the streams up to time T Ž f .Ž t .. The term DŽ f . describes the scaled departure process corresponding to the scaled arrival and service processes described by f . By expressing the object of interest, the scaled departure process, as a continuous function, D, of the arrival and service processes, Ž12. sets the stage for applying the contraction principle. The contraction principle then yields the LDP in Ž8. for the sample paths of the departure process. Using the contraction principle once more, we obtain a LDP for the departure rate: Dnrn satisfies the LDP in R d with good rate function
Ž 13.
LUd Ž z . s inf Id Ž c . : c Ž 0 . s 0, c Ž 1 . s z 4 ,
for Id as in Ž8.. From this, it was derived in w 10x that
½
LUd Ž z . s inf bLUa Ž xrb . q s LUa
Ž 14.
ž
zyx
s
qbLUb Ž c . q Ž 1 y b . LUb
ž
/
zyx 1yb
/
:
5
b , s g w 0, 1 x , c g R, b q s F 1, x F b c . The last result has the interpretation that the most likely path of the arrival and service processes which results in the departure process having mean rate z on the interval w 0, n x is as follows. The arrival process has rate xrb on the interval w 0, b n x and rate Žz y x.rs on w b n, Ž b q s . n x . The service rate during w 0, b n x is c, which is greater than the aggregate arrival rate during this period. So the queue is empty during w 0, b n x . The queue is nonempty throughout w b n, n x , during which period the service rate, at Ž z y x .rŽ1 y b ., is no larger than the total arrival rate, which is Ž z y x .rs . Therefore, the aggregate departure rate is equal to the aggregate arrival rate, xrb , during the first phase, w 0, b n x , when the queue is empty, and equal to
102
A. J. GANESH AND N. O’CONNELL
the service rate, Ž z y x .rŽ1 y b ., during the second phase, w b n, n x , when the queue is never empty. The rigorous statement underlying this intuition, proved in w 10x , is the following:
Ž 15.
LUd Ž z . s Id Ž c . ,
where c Ž t ., 0 F t F 1 is specified by,
Ž 16.
c Ž 0 . s 0,
c˙ Ž t . s
¡x ,
~
0 - t - b,
b zyx
¢1 y b ,
b - t - 1.
Here, b , x are those achieving the infimum in Ž14., and Id is as defined in Ž8.. The result in Ž14. applies to a queue started empty. A similar but more involved expression was derived for a queue in equilibrium. Under the above hypotheses, we can derive an expression for the asymptotics of the queue length distribution. The problem of extending this derivation to an arbitrary queue in a feed-forward queueing network remains open. The arrival process into any queue in such a network is an aggregate of the departure process from its predecessors Žor splittings thereof. and possibly of an external arrival process. Therefore, the result above suggests that we approach this problem using the LDP for the departure process. This would work if the departure process also satisfied hypotheses ŽH1. ] ŽH3.. In the next section we give examples to show that there are situations where the departure process fails to satisfy ŽH2., both for the queue initially empty and for the system started in equilibrium. 2. Counterexamples. 2.1. Single customer class. Consider a queue with a stochastic server and a single class of customers Žso d s 1.. Then Ž14. simplifies to
½
LUd Ž z . s inf bLUa Ž xrb . q s LUa
Ž 17.
ž
zyx
s
qbLUb Ž c . q Ž 1 y b . LUb
ž
/ zyx
1yb
/
:
5
b , s g w 0, 1 x , c g R, b q s F 1, x F b c , where now x and z are scalars. Let EX s LX aŽ0. denote the mean number of arrivals, and EC s LX b Ž0. the mean number of services in each time slot. The following properties of LUa , LUb are well known; see w 5x for instance. LUa Žrespectively, LUb . is nonnegative and zero only at EX Žrespectively, EC .. Both LUa and LUb are convex, and finite on a nonempty interval, in the interior of which they are analytic. We assume
103
THE LINEAR GEODESIC PROPERTY
that the queue is stable, namely EX - EC. Suppose that for some a g w 0, EX x ,
Ž 18.
LUa Ž a . s LUb Ž a .
and also that these functions are finite in a neighborhood of a . Without loss of generality, we can take a to be the largest number in w 0, EX x for which Ž18. holds. Then, since LUaŽ EX . s 0 - LUb Ž EX ., we have LUaŽ x . - LUb Ž x . for all x g Ž a , EX x , and consequently that Ž LUa . X Ž a . - Ž LUb . X Ž a .. It follows from this that
Ž 19.
' « ) 0: LUb Ž x . - LUa Ž x . - q` ; x g w a y « , a x ,
and also that
Ž 20. ' 0 - x 1 - a - x 2 - EX :
x1 q x 2 2
sa,
LUb Ž x 1 . q LUa Ž x 2 . 2
- LUa Ž a . .
We shall show that in this case, the most likely departure path having mean rate a is not linear. Let f Ž t . s Ž f 1 Ž t ., f 2 Ž t .., be defined on w 0, 1x by f Ž0. s 0 and
Ž 21.
f˙1 Ž t . s
½
x2 ,
0 - t - 12 ,
EX ,
1 2
f˙ 2 Ž t . s
- t - 1,
½
EC,
0 - t - 12 ,
x1 ,
1 2
- t - 1,
where x 1 , x 2 are as in Ž20.. Then, since EX - EC, we have from Ž9. ] Ž12. that
Ž 22.
DŽ f . Ž t . s
½
x2 t , 1 2
x2 q Ž t y
1 2
. x1 ,
0 F t F 12 , 1 2
F t F 1,
and in particular that DŽ f .Ž1. s a . Therefore, by Ž8. and Ž13., LUd Ž a . F I Ž f .
Ž 23.
s
H0
LUa Ž f˙1 Ž s . . ds q
s
H0
LUa Ž x 2 . ds q
1
1r2
H1r2 L
q s
1 2
1
U b
H0
1
LUb Ž f˙ 2 Ž s . . ds
H1r2 L 1
U a
Ž EX . ds q H
1r2
0
LUb Ž EC . ds
Ž x 1 . ds
LUa Ž x 2 . q LUb Ž x 1 .
- LUa Ž a . . The first equality above follows from Ž6. and Ž7., the second from the definition of f in Ž21., and the last from the fact that LUaŽ EX . s LUb Ž EC . s 0; see w 5x , for example. The last inequality above holds because of Ž20.. Notice that the departure process DŽ f . in Ž22., corresponding to the arrival process f 1 and service process f 2 , is not linear but has different slopes x 2 and x 1 in two different periods of equal length.
104
A. J. GANESH AND N. O’CONNELL
Next, let c Ž t . be linear on w 0, 1x with c Ž0. s 0 and c Ž1. s a , so that c˙ Ž t . s a for all t g Ž0, 1.. Consider any f g A 2 such that c s DŽ f .. That is, Ž f 1 , f 2 . is any pair of arrival and service processes Žexcluding those whose rate function is q`., corresponding to which c is the departure process. Then, by Ž9. ] Ž12.,
Ž 24.
c Ž t . s inf
0FsFt
f1 Ž s. y f 2 Ž s. q f 2 Ž t . ,
from which it is clear that c Ž t . F f 1 Ž t .. In particular, f 1 Ž1. G a . If f 1 Ž1. s a , then, by Ž6., IŽ f . G
Ž 25.
H0
1
G LUa
LUa Ž f˙1 Ž s . . ds
žH ˙ Ž . / 1
f 1 s ds
0
s LUa Ž a . .
The first inequality is due to the nonnegativity of LUa , LUb ; the second holds because of Jensen’s inequality and the convexity of LUa , while the equality is because f 1 Ž1. was assumed to be a . If f 1 Ž1. ) a , define
Ž 26.
t s sup t g w 0, 1 x : f 1 Ž t . F a t 4
and note that t - 1. Hence, by the continuity of f , f 1 Žt . s at . LEMMA 1. Suppose that c Ž t . s a t for all t g w 0, 1x , where c is defined by Ž24., and that f g A 2 . Then, with t given by Ž26.,
f 2 Ž t . y f 2 Žt . s a Ž t y t .
; t g w t , 1x .
PROOF. As noted above, f 1 Žt . s at s c Žt ., the latter equality holding by hypothesis regarding c . From this, we see that the infimum in Ž24. corresponding to t s t is achieved at s s t . Consequently, Ž24. implies that
Ž 27.
c Ž t . s inf
tFsFt
f1 Ž s. y f 2 Ž s. q f 2 Ž t .
; tGt.
If the infimum above is achieved at t for all t g w t , 1x , then, for all t in this interval, f 2 Ž t . y f 2 Žt . s c Ž t . y c Žt . s a Ž t y t ., and so the lemma is established. Otherwise, because f is absolutely continuous, one of the following must hold:
Ž 28.
' « ) 0: f˙1 Ž s . y f˙ 2 Ž s . - 0 ; s g Ž t , t q « .
or
Ž 29.
T J inf s ) t : f 1 Ž s . y f 2 Ž s . - f 1 Ž t . y f 2 Ž t . 4 g Ž t , 1 . .
In the former case, the infimum in Ž27. corresponding to t s t q « is achieved at s s t q « , and so
Ž 30.
c Ž t q « . y c Ž t . s f 1 Ž t q « . y f 1 Ž t . ) a« ,
THE LINEAR GEODESIC PROPERTY
105
where the inequality follows from the definition of t in Ž26.. In the latter case, we see from the continuity of f that the infimum in Ž27. corresponding to t s T is achieved at s s T. So c ŽT . s f 1 ŽT . and
Ž 31.
c ŽT . y c Žt . s f1ŽT . y f1Žt . ) a ŽT y t . , where the inequality follows from Ž26.. Now, both Ž30. and Ž31. contradict the hypothesis that c Ž t . s a t for all t g w 0, 1x . Therefore, neither Ž28. nor Ž29. can hold, implying that the infimum in Ž27. must be achieved at t for all t g w t , 1x . This completes the proof of the lemma. I From Lemma 1 and Ž26., we obtain, using Ž6., that IŽ f . G
Ž 32.
1 H0 L Ž f˙ Ž s . . ds q Ht L Ž f˙
t
U a
U b
1
2
Ž s . . ds
G t LUa Ž a . q Ž 1 y t . LUb Ž a . s LUa Ž a . .
The first inequality is due to the nonnegativity of LUa and LUb , and the second is due to their convexity and Jensen’s inequality w note that f 1 Žt . s at , while f 2 Ž1. y f 2 Žt . s a Ž1 y t . by Lemma 1x . The equality follows from the definition of a in Ž18.. Let c be given by c Ž t . s a t, t g w 0, 1x . Since either Ž25. or Ž32. applies to any f g A 2 for which D f s c , observe from Ž8. that Id Ž c . G LUaŽ a .. Therefore, by Ž23., c does not achieve the infimum in Ž13. corresponding to z s a . In other words, the departure process with constant rate a is not the most likely to achieve an average departure rate a ; this is achieved by a process with a nonlinear path. This implies that Id cannot be expressed in the form Id Ž j . s
H0
1
LU Ž j˙. ds
for any convex function LU and so the departure process does not satisfy hypothesis ŽH2.. The above conclusion applies to a queue started empty. We now consider a queue in stationarity. Let Q0 denote the queue length at time 0. It is shown in w 10x that the scaled queue lengths Q0rn satisfy a LDP in R with rate function L, which is explicitly computed. For our purposes, it is enough to note that LŽ0. s 0 and that LŽ q . G 0 for all q ) 0. Suppose the scaled initial queue length is q, and that the scaled process of arrivals and services is described by f s Ž f 1 , f 2 .. Then, the scaled departure process up to time t is given by
Ž 33.
D Ž q, f . Ž t . s f 2 Ž t . n
inf
0F n F1
q q f1Ž n t . y f 2 Ž n t . q f 2 Ž t . ,
where x n y denotes min x, y4 . Notice that since f Ž0. s 0, we recover Ž10. for the departure process from an empty queue by substituting q s 0. We shall show that for any q ) 0 and f g L 2 such that DŽ q, f Ž t .. s a t for all t g w 0, 1x , we have LŽ q . q I Ž f . G LU Ž a .. This will enable us to conclude that the departure process does not have linear geodesics, even in equilibrium.
106
A. J. GANESH AND N. O’CONNELL
Let c be linear on w 0, 1x with c Ž t . s a t for all t g w 0, 1x . Fix q ) 0 and let f g C 2 be such that DŽ q, f . s c . Then, by Ž33., either f 2 Ž t . s a t for all t g w 0, 1x , or q q f 1 Ž s . y f 2 Ž s . - 0 for some s g w 0, 1x . In the former case, we have by Ž6., Ž7. and the nonnegativity of the LU that
Ž 34.
IŽ f . G
H0
1
LUb Ž f˙ 2 Ž s . . ds s LUb Ž a . .
In the latter case, we have by the continuity of f that
t J inf s g w 0, 1 x : q q f 1 Ž s . y f 2 Ž s . - 0 4 g 0, 1 . . It follows that DŽ q, f Ž t .. s f 2 Ž t . for all t g w 0, t x , whereas, for t g w t , 1x , D Ž q, f . Ž t . y D Ž q, f . Ž t . s f 2 Ž t . y f 2 Ž t . q inf
tFsFt
q q f1 Ž s. y f 2 Ž s. ,
because q q f 1 Ž s . y f 2 Ž s . takes its minimum value on w 0, t x at t , and this value is zero. Hence, we can rewrite the above as
Ž 35.
D Ž q, f . Ž t . y D Ž q, f . Ž t . s inf
tFsFt
Ž f1 Ž s. y f1 Žt . . y Ž f 2 Ž s. y f 2 Žt . .
q f 2 Ž t . y f 2 Žt . .
Define f˜Ž t . s f Ž t . y f Žt ., t g w t , 1x . Then we have from above that
Ž 36. c Ž t . s
½
f2Ž t. , inf t F s F t
f˜1 Ž s . y f˜ 2 Ž s . q f˜ 2 Ž t . q f 2 Ž t . ,
if t g w 0, t x , if t g Ž t , 1 x .
Comparing this with Ž24., we see that the departure process on w t , 1x is identical to that from an empty queue with arrival and service processes given by f˜. This is not surprising because the queue does, in fact, become empty at time t by definition of t . Since f˜, restricted to w t , 1x , is merely a shifted version of f on this interval, I Ž f . s I Ž f˜. for f restricted to this interval. Therefore, IŽ f . G
H0 L Ž f˙ t
U b
2
Ž s . . ds q I Ž f˜ . .
Now, since c Ž t . s a t, f˙ 2 Ž s . s a for all s g w 0, t x . Also, by the same derivation as leads to Ž25. and Ž32., we have I Ž f˜. G Ž1 y t . LUaŽ a .. Finally, since LUaŽ a . s LUb Ž a . by definition of a , and LŽ q . G 0 for all q, we get L Ž q . q I Ž f . G LUa Ž a . . This holds for all initial queue lengths q G 0 and arrival and service processes f that result in a linear departure process c Ž t . s a t. Note that Ž23. continues to hold for departures in equilibrium because it was derived for departures from an empty queue, and we have LŽ0. s 0; see w 10x . Therefore,
Ž 37.
LUd Ž a . - inf L Ž q . q I Ž f . : D Ž q, f . Ž t . s a t ; t g w 0, 1 x 4 ,
which implies that, conditional on a mean departure rate of a , the most likely path is not linear. Thus, even in equilibrium, the departure process does not necessarily have linear geodesics.
107
THE LINEAR GEODESIC PROPERTY
We end this subsection with some comments about the scope and implications of the above results. A careful look at the proof shows that the result relied on a being less than EX and on the rate functions of the arrival and service processes intersecting at a . If the service process is deterministic, the latter cannot happen, and in this case it can be shown that the departure process has linear geodesics. This makes it possible to analyze networks of deterministic server queues, as in w 3x . Likewise, if we consider only a ) EX, then too it can be shown that the departure process conditioned on having mean rate a is linear. Since we are typically interested in the problem of queue lengths exceeding some large threshold, and since in well-designed networks this requires departure rates exceeding their mean, we are usually only interested in the rate function of departures for a ) EX. Since we have linear geodesics in this region, the study of networks of queues using a recursive approach is again feasible. Such an approach has been taken in w 1x . We shall next show that neither of these features comes to our rescue when dealing with multiclass queues. In this case, the joint departure process can have nonlinear geodesics even if the server is deterministic, and even if we consider departures whose aggregate rate exceeds their mean. 2.2. Two customer classes. Consider a queue multiplexing two customer classes and served deterministically at rate c. Suppose that customers from the first class arrive deterministically at rate a, while those of the second have a stochastic arrival process satisfying hypotheses ŽH1. ] ŽH3. with the rate function LU2 . We assume that the mean aggregate arrival rate is strictly less than the service rate, c. Note that the two arrival streams are trivially independent, as are the arrival and service processes. We denote the large deviations rate function of the first arrival process by LU1 and that of the service process by LUb . So
Ž 38.
LU1 Ž x . s
½
0, q`,
if x s a, else,
LUb Ž x . s
½
0, q`,
if x s c, else.
For some « ) 0 and b - c q « y a, let z s Ž a y « , b . and consider the departure process conditioned to have mean rate z. We shall show that this departure process does not have linear geodesics. Let c g A 2 be linear with c Ž0. s 0 and c Ž1. s z, so c˙ s z. We show that there is no f g A 3 with I Ž f . - q` such that DŽ f . s c . In other words, there is no process of arrivals and services whose rate function is finite, corresponding to which c is the departure process. Suppose otherwise. Let f g A 3 have I Ž f . - q`, so that
Ž 39.
f 1 Ž t . s at ,
f 3 Ž t . s ct ,
and suppose that
Ž 40.
D Ž f . Ž t . s c Ž t . s Ž Ž a y « . t , bt . ,
where « ) 0 and a q b y « - c. Observe from Ž9. ] Ž12. that
Ž 41.
DŽ f . Ž t . s Ž f1 Ž T Ž f . Ž t . . , f 2 Ž T Ž f . Ž t . . . .
108
A. J. GANESH AND N. O’CONNELL
Therefore, by Ž39. and Ž40., T Ž f .Ž t . s Ž a y « . tra, and so f 2 ŽŽ a y « . tra. s bt. In addition, by Ž11., D Ž f . Ž t . s f 1 Ž T Ž f . Ž t . . q f 2 Ž T Ž f . Ž t . . s Ž a q b y « . t. But by Ž10.,
Ž 42.
D Ž f . Ž t . s inf
0FsFt
f1 Ž s. q f 2 Ž s. y f 3 Ž s. q f 3 Ž t . ,
and so, by Ž39. and the fact, noted above, that f 2 Ž s . s absrŽ a y « ., we get abs D Ž f . Ž t . s inf as q y cs q ct ay« 0FsFt
¡ct ,
s~
if
¢a y « Ž a q b y « . t , a
a ay«
Ž a q b y « . G c,
else.
Because of our hypothesis that a q b y « - c, we have DŽ f .Ž t . ) Ž a q b y « . t in either case above, contradicting Ž42.. We have thus shown that, if I Ž f . - q`, then DŽ f . s c is impossible for c Ž t . s z t with z s Ž a y « , b .. Therefore, by Ž8., Id Ž c . s q`. We now show that LUd Žz. - q` for z as above. Since « ) 0 was arbitrary, we assume without loss of generality that a y 2 « ) 0 and define a x 1 s a q 2 Ž b y « . y c, x2 s Ž 43. Ž c y a q 2« . . a y 2« Since the only requirement we imposed above was that a q b y « - c, it is clear that b and « can be chosen so that x 1 G 0. Also, x 2 ) 0 since it was assumed that c is larger than a. Let f g A 3 be defined by
Ž 44.
f Ž 0 . s 0,
f˙ Ž t . s
½
Ž a, x 1 , c . , Ž a, x 2 , c . ,
0 - t - 1r2, 1r2 - t - 1.
Since x 1 and x 2 are nonnegative, f has nondecreasing components as required by the definition of A 3. Note that a q x1 s 2Ž a q b y « . y c - c by the hypothesis that a q b y « - c, whereas ac a q x2 s ) c. a y 2« In other words, the aggregate arrival rate a q x 1 is less than the service rate c during w 0, 1r2x whereas, at a q x 2 , it is greater than c during w 1r2, 1x . Therefore, the joint departure process DŽ f . is given by D Ž f . Ž 0 . s 0,
Ž 45.
d dt
¡Ž a, x . , a x c, ¢ž a q x a q x
D Ž f . Ž t . s~
1
2
2
/
c , 2
0 - t - 1r2, 1r2 - t - 1.
THE LINEAR GEODESIC PROPERTY
109
This is intuitively clear from the description of the queue, but can also be formally established using Ž9. ] Ž12.. Hence, we have from Ž43. that D Ž f . Ž 1. s 1
D Ž f . Ž 1. s 2
1 2 1 2
ž ž
aq
ac a q x2
x1 q
/
x2 c a q x2
sa y«,
/
s b.
Therefore, by definition of z, D f Ž1. s z. Furthermore, by Ž44., Ž6. and Ž38., we have LU Ž f . s 12 Ž LU2 Ž x 1 . q LU2 Ž x 2 . . Ž 46. for x 1 , x 2 as in Ž43.. Therefore, LU Ž f . is finite if LU2 Ž x 1 . and LU2 Ž x 2 . are, as is true if, for instance, the second arrival process is Poisson. It now follows from Ž8. and Ž13. that LUd Žz. - q`. But we showed earlier that Id Ž c . s q` for c given by c Ž t . s z t. Therefore, the departure process with linear path does not achieve the infimum in Ž13., implying that the departure process does not satisfy a large deviations principle with action functional that is the integral of a convex rate function. In other words, it is not true that Id Ž j . s
H0
1
LU Ž j Ž s . . ds
for any convex function LU . Consequently, the joint departure process does not satisfy hypothesis ŽH2., and so a recursive approach to estimating asymptotics of the queue lengths in a network does not appear feasible. We now consider the same queueing system in stationarity, rather than started empty. It is shown in w 10x that in stationarity, the scaled queue lengths Q 0rn satisfy a LDP in R 2 with a rate function L that can be computed explicitly. Here Q 0 s Ž Q01 , Q02 . denotes the number of customers of each of the two types in the queue at time zero. It suffices for our purposes to note that LŽq. G 0 for all q G 0, with equality if q s 0. Consider the system starting at time zero with scaled queue length Q 0rn s q. Suppose the arrival and service processes are given by f s Ž f 1 , f 2 , f 3 ., and that c s Ž c 1 , c 2 . is the corresponding departure process. Let a and c be defined as above to be the deterministic rate of the first arrival process and the service process respectively. Let « g Ž0, a. and b ) 0 be such that a q b y « - c. We shall show that if c Ž t . s z t for all t g w 0, 1x , where z s Ž a y « , b ., then LŽq. q I Ž f . s `. Analogous to Ž33., the scaled process of aggregate departures up to time t is given by
Ž 47. D Ž q, f . Ž t . s f 3 Ž t . n inf
0F n F1
q q AŽ f . Ž n t . y f 3 Ž n t . q f 3 Ž t . ,
where q J q 1 q q 2 is the total number in queue at time zero, and AŽ f . J f 1 q f 2 is the aggregate arrival process. Note that setting q s 0 above recovers Ž10. since f Ž0. s 0. Now, if c is to be the departure process, then we must have DŽq, f . s c 1 q c 2 . Since c Ž t . s z t, with z s Ž a y « , b ., the above
110
A. J. GANESH AND N. O’CONNELL
implies that DŽq, f . s Ž a q b y « .rt for all t g w 0, 1x . Recall that if I Ž f . is to be finite, then we must have f 1 Ž t . s at and f 3 Ž t . s ct for all t g w 0, 1x , since the first arrival process and the service process are deterministic with rates a, c, respectively. Therefore, for all such f , Ž47. implies that
Ž a q b y « . t s ct n inf
0FsFt
q q A Ž f . Ž s . y cs q ct.
Since a q b y « - c, the above implies that inf 0 F s F t w q q AŽ f .Ž s . y cs x is strictly negative for all t ) 0. Now AŽ f .Ž s . s f 1 Ž s . q f 2 Ž s ., f Ž0. s 0 and f is continuous if I Ž f . is finite. Therefore, it follows from the above that q s 0, that is, the queue must start empty. Then, by the argument above for departures from an empty queue, there is no process f of arrivals and services such that the departure process is c and I Ž f . is finite. We have also shown that this conclusion does not change if we allow any positive initial queue size q. This completes the proof that Id Ž c . s ` even in equilibrium, where Id denotes the rate function of the departure process. We argued above that LUd Žz. - ` for departures from an empty queue. Since LŽ0. s 0 Žsee w 10x. , this argument applies to departures in equilibrium as well. Thus, the most likely path leading to a mean departure rate z is not linear. This also implies that the rate function Id Ž c . for equilibrium departures cannot be of the form H01LU Ž c˙ .Ž s . ds, for any convex function LU . Therefore, the departure process does not satisfy hypothesis ŽH2., needed to apply the results of w 10x inductively to feed-forward multiclass queueing networks. 3. Conclusion. We considered the problem of characterizing the large deviations behavior of the departure process from a FIFO queue multiplexing several traffic streams. Such a characterization could, in principle, be used iteratively to determine the large deviations behavior of all processes of interest in networks of queues, and thereby to obtain the tail of the queue length and waiting time distributions at each queue in the network. The starting point of our analysis was the general description, in w 10x , of the rate function for the cumulative departures as the solution of a variational problem. It was shown in w 10x that if, in addition, the arrivals satisfy a ‘‘linear geodesics’’ condition, then the variational problem reduces to a finite-dimensional optimization problem. Such a simplification is essential if an iterative approach to analyzing networks of queues is to be practical. This naturally leads to the question of whether the departures also satisfy the ‘‘linear geodesics’’ assumption. We showed in this paper that this is not generally the case, even when there is just one input stream. Nevertheless, in the case of a single input stream, the departures do satisfy the linear geodesics requirement in the regime leading to large queue sizes. So an iterative approach to obtaining the tail of the queue length is possible; see w 1x . However, such is not the case for multiple traffic streams, even when the service rate is deterministic.
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REFERENCES w 1x BERTSIMAS, D., PASCHALIDIS, I. and TSITSIKLIS, J. Ž1994.. On the large deviations behaviour of acyclic networks GrGr1 queues. Preprint. w 2x CHANG, C.-S. Ž1994.. Stability, queue length and delay of deterministic and stochastic queueing networks. IEEE Trans. Automat. Control 39 913]931. w 3x CHANG, C.-S. Ž1995.. Sample path large deviations and intree networks. Queueing Systems Theory Appl. 20 7]36. w 4x CHANG, C.-S. and ZAJIC, T. Ž1995.. Effective bandwidths of departure processes from queues with time varying capacities. IEEE INFOCOM Proceedings. w 5x DEMBO, A. and ZEITOUNI, O. Ž1993.. Large Deviations Techniques and Applications. Jones and Bartlett, Boston. w 6x DEMBO, A. and ZAJIC, T. Ž1995.. Large deviations: from empirical mean and measure to partial sums process. Stochastic Process. Appl. 57 191]224. w 7x DE VECIANA, G., COURCOUBETIS, C. and WALRAND, J. Ž1994.. Decoupling bandwidths for networks: a decomposition approach to resource management. IEEE INFOCOM Proceedings. w 8x DUFFIELD, N. G. and O’CONNELL, N. Ž1995.. Large deviations and overflow probabilities for the general single server queue, with applications. Math. Proc. Cambridge Philos. Soc. 118. w 9x GLYNN, P. W. and WHITT, W. Ž1995.. Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. J. Appl. Probab. To appear. w 10x O’CONNELL, N. Ž1997.. Large deviations for departures from a shared buffer. J. Appl. Probab. To appear. DEPARTMENT OF STATISTICS BIRKBECK COLLEGE UNIVERSITY OF LONDON MALET STREET LONDON WC1E 7HX UNITED KINGDOM E-MAIL:
[email protected]
BRIMS HEWLETT PACKARD LABORATORIES FILTON ROAD STOKE GIFFORD BRISTOL BS 12 6QZ UNITED KINGDOM E-MAIL:
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