A Fluid Queue Driven by a Markovian Queue

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A Fluid Queue Driven by a Markovian Queue Bruno Sericola and Bruno Tuffin

N ˚ 3306 Novembre 1997 ` THEME 1

ISSN 0249-6399

apport de recherche

A Fluid Queue Driven by a Markovian Queue Bruno Sericola  and Bruno Tun y Thème 1  Réseaux et systèmes Projet Model Rapport de recherche n3306  Novembre 1997  15 pages

Abstract: We consider a uid queue receiving its input from the output of a Marko-

vian queue with nite or innite waiting room. The input rate of the uid queue is characterized by a Markov modulated rate process. We derive a new approach for the computation of the stationary buer content. This approach leads to a numerically stable algorithm for which the precision of the result can be given in advance. Key-words: Fluid queue, Markovian queue, Markov process.

(Résumé : tsvp)

{Bruno.Sericola}@irisa.fr Institut Mathématique de Rennes, Université de Rennes 1, Campus universitaire de Beaulieu, 35042 Rennes Cedex, France, {Bruno.Tun}@maths.univ-rennes1.fr  y

Unit´e de recherche INRIA Rennes IRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France) T´el´ephone : 02 99 84 71 00 - International : +33 2 99 84 71 00 T´el´ecopie : 02 99 84 71 71 - International : +33 2 99 84 71 71

Une le d'attente uide pilotée par une le d'attente Markovienne

Résumé : On considère une le d'attente uide dont le ux d'entrée est le ux de

sortie d'une le d'attente Markovienne à capacité nie ou innie. Le processus d'entrée est caractérisé par son taux, lui même modulé par un processus de Markov. Nous obtenons une nouvelle méthode pour le calcul de la distribution stationnaire du contenu du tampon. Cette approche conduit à un algorithme numériquement stable pour lequel la précision peut être donnée à l'avance. Mots-clé : File d'attente uide, le d'attente Markovienne, processus de Markov.

A Fluid Queue Driven by a Markovian Queue

3

1 Introduction In performance evaluation of telecommunication and computer systems, uid queues models with Markov modulated input rates have been widely used in many papers, see among others [3, 6, 9, 1, 8]. The trac arriving to a network queue has already traversed parts of the network and has been modied along its traversal. In such cases, it is the output from a queue which forms the input to the next network element. In the most important part of the litterature on this subject, see for instance [3, 6] and the references therein, the state space of the Markov process that modulates the input rate in the uid queue is supposed to be nite. The case where this state space is innite has been analysed in [9] and [1] for the M=M=1 queue and in [8] for a birth and death process. In this paper, we generalize the problem to a uid queue driven by a Markovian queue. The only requirement needed on the Markov process that modulates the input rate process is that it has a single state such that the input rate is smaller than the output rate of the uid queue and that it has a uniform innitesimal generator, that is, the suppremum of the output rates of the states is bounded. These Markov processes include not only the well-known M=M=1, M=M=K , M=PH=1 and M=PH=K queues with nite or innite waiting room but also the superposition of on-o sources with exponential o periods and phase-type on periods. The method used here to obtain the distribution of the stationary buer content is neither based on spectral analysis nor on the use of Bessel functions as done in [9], [1] and [8], but a direct approach is used which leads to simple recursions. This method is particularly interesting by the fact that it uses only additions and multiplications of positive numbers bounded by one. Thus we obtain a stable algorithm which moreover gives the result with a precision that can be specied in advance. The rest of the paper is organized as follows. In the next section, we present the model and we obtain the solution in terms of recurrence relations whose behavior is studied. In Section 3 we present the algorithm and numerical illustrations are given in Section 4.

2 Model and Solution We describe in this section a uid model with an innite buer for which the input and output rates are controlled by a homogeneous Markov process fXt; t  0g on the RR n3306

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B. Sericola & B. Tun

state space S with innitesimal generator denoted by A and stationary probability distribution denoted by . Let ri be the input rate and ci be the output rate when the Markov process fXt g is in state i. We denote by i the eective input rate of state i, that is i = ri ? ci. We suppose that for every i 2 S we have i 6= 0. It is shown in the Appendix that the case where i = 0 for some i can be reduced to this one. We assume in this paper that the state space S contains only one state withS negative eective input rate. This state is denoted by 0 and thus we have S = f0g S with  < 0 and i > 0 for i 2 S . We suppose that the stability condition is satised, that is +

0

+

X

i2S = X

2

i S

rii cii

< 1;

where  is the trac intensity, so that the limiting behavior exists. We denote by X the stationary state of the Markov process fXtg and by Q the stationary amount of uid in the buer. Let Fj (x) = PrfX = j; Q  xg. We then have the following dierential equations, see for instance [3], for all j 2 S j (x) = j dFdx

X 2

i S

Fi(x)A(i; j );

(1)

with initial condition given by Fj (0) = 0 for every j 2 S . It follows that we have F (0) = PrfQ = 0g = 1 ? . We assume that supf?A(i; i) : i 2 S g is nite and we denote by P the transition probability matrix of the uniformized Markov chain [5] with respect to the uniformization rate  which veries   supf?A(i; i); i 2 S g. The matrix P is then related to A by P = I + A=, where I denotes the identity matrix. The main result of this paper, which is the distribution of the pair (X; Q) is given by the following theorem. +

0

Theorem 2.1 For every j 2 S , we have Fj (x) =

1 X n=0

e? x (

) n! bj (n)

x n 

(2)

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A Fluid Queue Driven by a Markovian Queue

where  = minfiji > 0g and the coecients bj (n) are given by the following recursive expression b0 (0) = 1 ?  and bj (0) = 0 for j 2 S +; and for n  1 and j 2 S ,

X bj (n) = (1 ?  )bj (n ? 1) +  bi(n ? 1)P (i; j ): 2

j

j i S

(3)

Proof. We replace Fj (x) by the expression (2) in equation (1). Thus j

e?x=  

"X 1

n=1

#

n 1 (x=)n 1 X (x=)n? ?x= X (x=) X b (n)A(i; j ); b ( n ) ? b ( n ) = e (n ? 1)! j n! j n! i2S i n n 1

=0

=0

which can be reduced to 1 (x=)n X 1 X )n (b (n + 1) ? b (n)) = X j  (x= j n! j n! i2S bi(n)A(i; j ): n n =0

=0

We then have for every n  0

X j  (bj (n + 1) ? bj (n)) = bi (n)A(i; j ): 2

i S

Using A = (P ? I ), we obtain relation (3). For x = 0, we have Fj (0) = bj (0) for every j 2 S from equation (2), which gives from the initial condition b (0) = 1 ?  and bj (0) = 0 for j 2 S : This completes the proof +

0

We give now some properties of the numbers bj (n) which will be used in the next section in order to develop a precise and stable algorithm to compute the distribution of the buer content.

Proposition 2.2 For every n  0, we have b (n) = 1 ?  + 0

RR n3306

X j S+

2

j bj (n)

?

0

:

(4)

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B. Sericola & B. Tun

Proof. Consider relation (3). By X multiplyingXboth sides by j and by summing over index j , we obtain for n  1, j bj (n) = j bj (n ? 1). It follows that for every j 2S X X j2S n  0 we have j bj (n) = j bj (0) =  (1 ? ), which is equivalent to relation (4). 2

0

2

j S

j S

Lemma 2.3

X 2

j S

j j =  (1 ? ): 0

Proof. XConsider equation (1). By integrating from 0 to 1 and summing over index j , we get j (Fj (1) ? Fj (0)) = 0. Now since Fj (1) = j , F (0) = 1 ?  and Fj (0) = 0 j 2S for j 2 S , we obtain the result. 0

+

Proposition 2.4 For every j 2 S and n  0, we have 0  bj (n)  j . Proof. We proceed by induction. By denition of Fj (x), we have 0  Fj (x)  j for every x  0 and j 2 S . Since Fj (0) = bj (0) for every j 2 S , we have 0  bj (0)  j . Suppose now that we have 0  bj (n ? 1)  j . For j 2 S , we have =j 2 (0; 1), so we easily obtain from relation (3), by using the relation P = , that 0  bj (n)  j . For j = 0, since  < 0, j > 0 and bj (n)  0 for j 2 S , we obtain from relation (4) that b (n)  0 and +

+

0

0

j bj (n) P ?+    1 ?  + j2?S  j j P  = 1 ?  + j2S j j +  ? =  from Lemma 2.3;

b (n) = 1 ?  + 0

P

j S+

2

0

0

0

0

0

and the result follows.

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A Fluid Queue Driven by a Markovian Queue

Proposition 2.5 For every j 2 S , the sequence bj (n) increases and converges to j . Proof. We have, for j 2 S , bj (1)  0 = bj (0) and +

b (1) = 1 ?  +

P

0

j bj (1)  1 ?  = b (0): ?

j S+

2

0

0

Moreover, from relations (3) and (4), we have

X bj (n +1) ? bj (n) = (1 ?  )(bj (n) ? bj (n ? 1))+  (bi(n) ? bi (n ? 1))P (i; j ) for j 2 S

+

2

j

j i S

and

P  (b (n + 1) ? b (n)) j : b (n + 1) ? b (n) = i2S i j ? Since, for j 2 S , we have =j 2 (0; 1) and  < 0 we deduce by induction that for every j 2 S the sequence bj (n) is increasing. It then converges by using Proposition 2.4. For every j 2 S , we denote by lj the limit of bj (n) when n goes to innity. We then have x n 1 X Fj (x) = e? ( n !) bj (n) ?! lj when x ?! 1: n Thus, since Fj (x) = PrfX = j; Q  xg tends to j when x tends to 1, we have lj = j . 0

0

0

+

0

x 

=0

3 Algorithmical Aspects We suppose in this section that the innitesimal generator of the process X has the following block tridiagonal structure.

0 BB AA ;; AA ;; A ; BB A; A; A; B A; : : A = BBB BB : : Ak? ;k B@ Ak;k? Ak;k Ak;k : : 00

01

10

11

12

21

22

23

32

1

1

RR n3306

+1

:

1 CC CC CC CC CC CA

8

B. Sericola & B. Tun

where A ; is the output rate from state 0. Such a structure leads to the innitesimal generators of Markovian queues such as the M/M/1, the M/M/K, the M/PH/1 and the M/PH/K queues with nite or innite waiting room [4]. To compute the probability distribution PrfQ  xg of the buer content we use relations (2), (3) and (4) together with Proposition 2.4 and Proposition 2.5. relation (3) is used only for j 2 S and for j = 0 we use Relation (4). These relations are particularly interesting from a computational point of view. Indeed, the fact that only additions and multiplications of positive and bounded numbers are used in their recurrences is a very important property for what concerns the numerical stability of the computation. Proposition 2.4 and Proposition 2.5 will be used as a criterion to stop the computation in the case where the sequence of the bj (n) is close to its limit j . We denote by ni the dimension of the square matrix Ai;i. Note that n = 1. The transition probability matrix of the uniformized Markov chain has the same block tridiagonal structure that the matrix A. The blocks of matrix P are denoted by Pi;j and we have, since P = I + A=, Pi;i = I + Ai;i= and Pi;j = Ai;j = for i 6= j where I is in this case the identity matrix of dimension ni. We also consider the innite row vector containing the bj (n) for j 2 S . This innite row vector can be rearranged according to the structure of matrix P to be written as (b (n); b (n); b (n); : : :); where b (n) is the scalar b (n) and for j  1, b j (n) is a rowS vector of dimension nj . This consists in rearranging the state space S as S = f0g S S S S


, where for j  1, Sj contains nj states with the same eective input rate equal to j . With this notation, relation (3) can be written, for j  1 and n  1 as 00

+

0

[0]

[0]

[1]

[2]

[ ]

0

1

2

 b j (n) = (1?  )b j (n?1)+  b j? (n ? 1)Pj? ;j + b j (n ? 1)Pj;j + b j (n ? 1)Pj [ ]

[

[ ]

j

j

1]

1

[ ]

[ +1]

+1;j



:

(5) Using this recursion, it can be easily checked that, since b (0) = 0 for j  1, we have b j (n) = 0 for n  0 and j  n + 1: Relation (4) can then be written as [j ]

[ ]

n X

j b j (n)1l [ ]

b (n) = 1 ?  + j ? [0]

=1

0

;

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A Fluid Queue Driven by a Markovian Queue

where 1l is a column vector with all the entries equal to 1, its dimension being given by the context. Denoting by F (x) the probability distribution function of the buer content Q, that is F (x) = PrfQ  xg, we nally get x n 1 X X F (x) = Fj (x) = e? x ( n !) b(n); (7) n j 2S =0

where b(n) = Pj2S bj (n) = Pnj b j (n)1l: From Proposition 2.5 and from the dominated convergence theorem, we obtain that the sequence b(n) is an increasing sequence that converges to 1 when n goes to innity. The computation of F (x) can then be done as follows. For a given error tolerance ", we dene integer N as X ( ) x i n ( ) x N = min n 2 IN e?  i!  1 ? " ; (8) i and we denote by F (N; x) the sum of the N + 1 rst terms of relation (7), that is x n N X F (N; x) = e? x ( n !) b(n): n We then have F (x) = F (N; x) + e(N ); where the rest e(N ) of the series satises x n x n x n 1 N 1 X X X e? x ( n !) = 1 ? e? x ( n !)  ": e(N ) = e? x ( n !) b(n)  n n N n N We also consider integer N 0 dened by N 0 = min fn 2 IN j b(n)  1 ? " g : Since the sequence b(n) is increasing and converges to 1, we have b(n)  1 ? " for every n  N 0 . So we get x n 1 X F (x) = F (N 0; x) + e? x ( n !) b(n) n N0 x n N0 X ) x ( 0 = F (N ; x) + 1 ? e?   ? e0(N 0 ); n! n =0

[ ]

=0

=0

=

+1

=

=0

+1

=

+1

=0

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B. Sericola & B. Tun

where the rest e0(N 0 ) satises

e0 (N 0) =

1 X

x n e? ( n !) (1 ? b(n))  ": x 

n=N 0 +1

The integer N 0 is not known a priori so we will rst compute the integer N and start the computation of F (N; x). This computation will be then stopped in the case where N 0 < N . Note also that the integer N , dened in (8), is an increasing function of x, say N (x). So if the function F (x) has to be evaluated at M points, say x < : : : < xM , we only need to evaluate the values of b(n) for n = 0; 1; : : : ; N (xM ) since these values are independent of the values of x ; : : : ; xM . The pseudocode of the algorithm is given below. input : x <


< xM , " output : PrfQ  x g; : : : ; PrfQ  xM g Compute N from relation (8) with x = xM N0 = N b (0) = 1 ?  n=0 while [ n < N 0 ] do n=n+1 for j = 1 to n do Compute b j (n) from relation (5) endfor Compute b (n) from relation (6) n X b(n) = b j (n)1l j if (b(n)  1 ? ") then N0 = n 1

1

1

1

[0]

[ ]

[0]

[ ]

=0

endif endwhile if (N' = N) then for i = 1 to M do Compute F (N; xi) endfor else N X ( x )n ? for i = 1 to M do Compute 1 ? e + F (N 0; xi) endfor n! n endif 0

xi 

i

=0

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A Fluid Queue Driven by a Markovian Queue

4 Numerical Results We have shown that the algorithm described in the previous section applies to any block tridiagonal innitesimal generator A with a single state having a negative eective input rate. Such a structure for the innitesimal generator includes the following Markovian systems:  The M=M=1 queue with arrival rate and service rate . Take A ; = ? , A ; = and for i  1, Ai;i = , Ai;i? = and so Ai;i = ?( + ).  The M=M=K queue with arrival rate and service rate per server . Take A ; = ? , A ; = and for i  1, Ai;i = , Ai;i? = min(i; K ) and so Ai;i = ?( + min(i; K ) ).  The M=PH=1 queue with arrival rate and (; T ) as phase-type representation of the service time distribution [4]. In this case, we must take A ; = ? , A ; =  , A ; = ?T 1l, and for i  1, Ai;i = I , Ai;i = T ? I , Ai ;i = ?T 1l  The M=PH=K queue with arrival rate and (; T ) as phase-type representation of the service time distribution per server. The blocks Ai;j of its innitesimal generator can be obtained using tensor algebra as done in [7].  All these Markovian queues can also be considered when their capacity is nite since in this case the innitesimal generator A is a nite block tridiagonal matrix.  The superposition of a nite number of independent on-o sources where the o periods are exponentially distributed and the on periods have a phase-type distribution. In order to illustrate our algorithm, we consider the M=M=K queue with arrival rate and service rate per server. The input rate in the uid queue when the M=M=K queue is in state i is then given by ri = min(i; K )r for every i  0, where r is the input rate per server in the uid queue. We suppose that the output rate of the uid queue is constant equal to c, that is ci = c for every i  0 and such that r > c. We then obtain that the eective input rate in the uid queue is given by i = min(i; K )r ? c. We suppose that < K so that the i exists and that  = r

c < 1, which implies that the limiting behavior of the buer contents exists. 00

01

+1

1

00

01

+1

1

00

10

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+1

+1

01

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B. Sericola & B. Tun

Figure 1 shows the complementary cumulative distribution function of the buer content of a uid queue driven by an M=M=K for K = 1 and K = 10. In both cases, the arrival rate is = 0:8, the service rate per server is = 1, the input rate per server in the uid queue is r = 1:2 and the output rate is constant c = 1. In this example we have taken " = 10? . The same function, but for larger values of x, is shown in Figure 2 for = 0:4, = 1, r = 2 and c = 1. In this gure, the vertical axis is in logarithmic scale and we have taken " = 10? . 5

10

1

0.8

0.6

0.4

0.2

0 0

20

40

60

80

100

x

Figure 1: From top to the bottom : PrfQ > xg versus x for the M=M=10 and the M=M=1 queues as input queues with arrival rate = 0:8, service rate = 1 per server, input rate r = 1:2 per server and constant output rate c = 1, which gives  = 0:96.

References [1] I. Adan and J. Resing. Simple analysis of a uid queue driven by an M/M/1 queue. Queueing Systems, 22:171174, 1996.

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A Fluid Queue Driven by a Markovian Queue

1e-05

1e-06

1e-07

1e-08

1e-09

1e-10 100

120

140

160

180

200

x

Figure 2: From top to the bottom : PrfQ > xg versus x for the M=M=10 and the M=M=1 queues as input queues with arrival rate = 0:4, service rate = 1 per server, input rate r = 2 per server and constant output rate c = 1, which gives  = 0:8.

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[2] A. Berman and R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. Academic Press, 1979. [3] D. Mitra. Stochastic theory of a uid model of producers and consumers coupled by a buer. Advances in Applied Probability, 20:646676, 1988. [4] M.F. Neuts. Matrix-Geometric Solutions in Stochastic Models: An Algorithm Approach. Baltimore: The Johns Hopkins University Press, 1981. [5] S.M. Ross. Stochastic Processes. John Wiley and Sons, 1983. [6] T.E. Stern and A.I. Elwalid. Analysis of Separable Markov-Modulated Rate Models for Information-Handling Systems. Advances in Applied Probability, 23:105139, 1991. [7] Y. Takahashi. Asymptotic exponentiality of the tail of the waiting-time distribution in a PH/PH/c queue. Advances in Applied Probability, 13:619630, 1981. [8] E.A. van Doorn and W.R.W. Scheinhardt. A uid queue driven by an innite state birth-death process. In V. Ramaswami and P.E. Wirth, editors, ITC'15, Teletrac Contributions for the Information Age, pages 465475. Elsevier Science B.V., 1997. [9] J. Virtamo and I. Norros. Fluid queue driven by an M/M/1 queue. Queueing Systems, 16:373386, 1994.

Appendix

We consider here that the state space S  of process fXtg, that we suppose irreducible, contains a nite number of zero eective input rates. We write S  = S [ S where S (resp. S ) is the set of states with non-zero (resp. zero) eective input rates. The innitesimal generator A of the process fXtg and the diagonal matrix D of the eective input rates can then be written in the obvious notation as 0

A =

0

ASS ASS0 AS0S AS0S0

!

and D =

! D 0 : 0 0

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A Fluid Queue Driven by a Markovian Queue

In the same way, we denote by FS (x) and FS0 (x) the row vectors containing the Fj (x) for j 2 S and j 2 S respectively. The dierential equations (1) can then be written as 0

(

D = FS (x)ASS + FS0 (x)AS0 S 0 = FS (x)ASS0 + FS0 (x)AS0S0 :

dFS (x) dx

(9)

As A is irreducible, ?AS0 S0 is a non-singular M-matrix [2], so AS0S0 is invertible. Let (i)i2S be the stationary distribution of fXt g. We have:

Proposition where

(

F (x) = ?FS (x)ASS0 A?S0 S0 D = FS (x)A 1

S0 dFS (x) dx

(10)

A = ASS ? ASS0 A?S0S0 AS0S : 1

The results given by Theorem 2.1 can then be used to obtain the solution inPthe following way: the solution G(x) of Section 2 for dG (x)D = G(x)A gives FS (x) = ( i2S i)G(x) dx and then FS0 (x) is given from (10).

Proof. Equations (10) follow immediately from (9). It is well-known that A is an

innitesimal generator and that the stationary probability measure S = (i)i2S of the Markov process with innitesimal generator A is given for every i 2 S by i = P   (x)D = G(x)A a solution G(x) i =( j2S j ). Section 2 then gives for equation dG dx which tends to S as x ! +1. Given that the solution of this equation is unique up to a multiplicative constant, and given that FS (x) tends to S = (i)i2S as x ! +1, we P obtain FS (x) = G(x) j2S j.

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