An RG-factorization approach for a BMAP/M/1 generalized processor-sharing queue

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1 Lingnan University From the SelectedWorks of Prof. LIU Liming January, 2005 An RG-factorization approach for a BMAP/M/1 generalized processor-sharing queue Quan-Lin LI Zhaotong LIAN Liming LIU, Lingnan University, Hong Kong Available at:

2 Stochastic Models, 21: , 2005 Copyright Taylor & Francis, Inc. ISSN: print/ online DOI: /STM AN RG-FACTORIZATION APPROACH FOR A BMAP/M/1 GENERALIZED PROCESSOR-SHARING QUEUE Quan-Lin Li Department of Industrial Engineering, Tsinghua University, Beijing, P.R. China Zhaotong Lian Faculty of Business Administration, University of Macau, Macao, China Liming Liu Department of Industrial Engineering and Engineering Management, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China In this paper, we study a BMAP /M /1 generalized processor-sharing queue. We propose an RG-factorization approach, which can be applied to a wider class of Markovian blockstructured processor-sharing queues. We obtain the expressions for both the distribution of the stationary queue length and the Laplace transform of the sojourn time distribution. From these two expressions, we develop an algorithm to compute the mean and variance of the sojourn time approximately. Keywords Batch Markovian arrival process (BMAP); Generalized processor-sharing; Markov chain of M /G/1 type, Processor-sharing queue, Sojourn time. Mathematics Subject Classification Primary 60K25; Secondary 60J INTRODUCTION Since the pioneering work of Kleinrock [25, processor-sharing queues have attracted considerable interests and have been used widely to study computer and communication systems. Early works on processor-sharing queues are for the study of multiuser mainframe computer systems. These include Kleinrock [25,26, Coffman and Kleinrock [10, and Coffman, Muntz and Trotter [11. Recent works in processor-sharing queues stem from their Received August 2004; Accepted February 2005 Address correspondence to Quan-Lin Li, Department of Industrial Engineering, Tsinghua University, Beijing, China; liquanlin@mail.tsingua.edu.cn

3 508 Li et al. applications to communication networks and web server systems such as modeling congested links with TCP traffic and job scheduler in web servers; see, for example, Núñez-Queija [39, Jelenković and Momčilović [24, and Avrachenkov, Ayesta, and Brown [2. When n customers are present in the system, each customer receives service at a rate 1/n. This service discipline is called the egalitarian processor-sharing and has been extensively studied in the literature. Cohen [12 extended the egalitarian processor-sharing discipline to the generalized processor-sharing discipline: When there are n customers in the system, each customer receives service at a rate f (n), where f (n) is a positive function such that 0 < nf (n) F for n 1 and F > 0. The generalized processor-sharing queues have been studied by some researchers. For the related literature, one can refer to Yashkov [57,58, which also include a detailed discussion of various processorsharing disciplines. Parekh and Gallager [43 first extended the generalized processor-sharing discipline for a single class of customers to multiclass models. This line of research is followed by many authors including Parekh and Gallager [44, Zhang [59, van Uitert and Borst [55, Borst, Mandjes and van Uitert [6, Borst, Boxma, and Jelenković [5 and Ramanan, and Reiman [46. To motivate our work, we discuss briefly results in the literature for various processor-sharing queues. For the M /M /1 processor-sharing queue, Coffman, Muntz and Trotter [11 first derived the Laplace Stieltjes transform of the sojourn time distribution. O Donovan [42 and Fayolle, Mitrani, and Iasnogorodski [15 analyzed the conditional distribution of the sojourn time. Morrison [35 obtained an integral representation for the sojourn time distribution using the Laplace Stieltjes transform. Conditioning on the number of customers found by an arriving customer, Sengupta and Jagerman [52 studied moments of the sojourn time. Núñez- Queija [40 analyzed the M /M /1 processor-sharing queue with server interruptions. Braband [7 considered the waiting time distributions of the M /M /N processor-sharing queue. The author later discussed the waiting time distributions of the closed M /M /N processor-sharing queue, see Braband [8. Guillemin and Boyer [22 used the orthogonal polynomials to discuss the sojourn time distribution. Núñez-Queija [41 studied a Markovian processor-sharing queue with a service rate that varies over time, depending on the number of customers in the system and on the state of a stochastic environment. He used a level-dependent QBD process with finitely-many levels to derive the Laplace Stieltjes transform of the sojourn time distribution. Masuyama and Takine [34 provided a recursive formula to compute the sojourn time distribution in the MAP /M /1 processor-sharing queue. This formula is numerically feasible with a controllable absolute error. For the GI /M /1 processor-sharing queue, the Laplace Stieltjes transform of the sojourn time distribution and its first two moments were first derived by Ramaswami [48. Subsequent works on this model include

4 RG-Factorization Approach for a BMAP/M/1 Queue 509 Cohen [13, Jagerman and Sengupta [23, Knessl [28, Tan, Yang, and Knessl [54, and Yang and Knessl [56. For the GI /G/1 processor-sharing queues, readers may refer to, for example, Sengupta [51, Grishechkin [19, Gromoll, Puha and Williams [21, Gromoll [20, and Puha and Williams [45. Results for processor-sharing queues with bulk arrivals are limited. Kleinrock, Muntz, and Rodemich [27 first analyzed a processor-sharing queue with bulk arrivals. They showed that the derivative of the expected response time conditioned on the job size satisfies an integral equation. Rege and Sengupta [49 studied the M /G/1 processor-sharing queue with bulk arrivals under a general job size distribution. They expressed the response time conditional upon the number of customers present and their residual service requirements, providing an interesting structural insight. Based on the integral equation given in Kleinrock [27, Bansal [3 provided expression for the expected response time of a job as a function of its size when the service time distribution has a rational Laplace transform. Avrachenkov, Ayesta, and Brown [2 used the contraction mapping principle to prove existence and uniqueness of a solution to the integral equation given in Kleinrock [27, and then provided asymptotic analysis as well as some tight bounds for the expected response time conditioned on the job size. Using Crump-Mode-Jagers branching processes, Grishechkin [18 studied the M /G/1 generalized processorsharing queue with batch arrivals. The batch Markovian arrival process (BMAP) is a useful mathematical model for describing bursty traffic in modern communication networks, see, for example, Ramaswami [47, Neuts [36,37, and Lucantoni [32,33. In this paper, we consider a BMAP /M /1 generalized processor-sharing queue. Processor-sharing queues with bulk arrivals are significantly different from the case with single arrivals. So far, the limited results reviewed above are all for conditional expressions of the sojourn time. Thus our work addresses an open problem, i.e., unconditional results for the sojourn time distribution in the BMAP /M /1 generalized processor-sharing queue. Our work also extends the work of Masuyama and Takine [34 in two aspects: Batch arrivals and the generalized processor-sharing discipline. At the same time, we also note that Núñez-Queija [41 used a level-dependent QBD process with finitely-many levels while Masuyama and Takine [34 applied a level-dependent QBD process with infinitely-many levels to discuss Markovian processor-sharing queues. By successfully extending the QBD structure of processor-sharing models to level-dependent Markov chains of M /G/1 type, we propose an effective and unified approach to study the block-structured BMAP /M /1 processor-sharing queue based on the RGfactorization. Since the RG-factorization always exists for any irreducible block-structured Markov chain (see Li and Zhao [31 and Li and Liu [29 ), the RG-factorization approach proposed here can be used to study a wider class of Markovian processor-sharing models.

5 510 Li et al. This paper is divided into six sections. In section 2, we describe the BMAP /M /1 generalized processor-sharing queue. In section 3, we provide an expression for the distribution of the stationary queue length. In section 4, we derive the Laplace transform for the sojourn time distribution and its mean. In section 5, we propose an approximate algorithm for computing the mean sojourn time. Finally, some numerical examples are given in section 6 to illustrate the approximate algorithm. 2. MODEL DESCRIPTION A BMAP /M /1 generalized processor-sharing queue is defined as follows: The arrival process: Arrivals to the queue follow a BMAP with m phases described by coefficient matrices D k, k 0. The matrix D 0 has strictly negative diagonal entries and non-negative off-diagonal entries, and is invertible. For k 1, D k 0, we assume that kd k is finite, and D = D k=0 k is an irreducible infinitesimal generator with De = 0, where e is a column vector of ones. Let be the stationary probability vector of D. Then = ke is the stationary arrival rate. The service times: Service times n for n 1 are i.i.d. and are exponential with a rate. The generalized processor-sharing discipline: When n customers are in the system, each customer receive service at a rate f (n), where f (n) is a positive function such that 0 < C 1 nf (n) C 2 for all n 1, and C 1 and C 2 are two positive constants. The independence: We assume that all the random variables defined above are independent of each other. Remark 1. i) If f (n) = 1/n, then C 1 = C 2 = 1. In this case, the service discipline becomes egalitarian processor-sharing. ii) The condition that 0 < C 1 < + is necessary for guaranteeing the stability of the generalized processor-sharing queue. This will be illustrated in section THE STATIONARY QUEUE LENGTH We first provide a condition under which the BMAP /M /1 generalized processor-sharing queue is stable. Let q(t) and i(t) be the number of customers in the system and the phase of the BMAP input at time t, respectively. Then q(t), i(t); t 0 is a continuous-time level-dependent

6 RG-Factorization Approach for a BMAP/M/1 Queue 511 Markov chain of M /G/1 type whose infinitesimal generator is given by D 0 D 1 D 2 D 3 f (1) I D 0 f (1) I D 1 D 2 Q = 2f (2) I D 0 2f (2) I D 1. (1) 3f (3) I D 0 3f (3) I Let C = inf n 1 nf (n). Then 0 < C < +, since the assumption that 0 < C 1 nf (n) C 2 for all n 1 implies C C 1 > 0. By the boundary condition of the matrix Q, kd k is finite. It is obvious from the main drift condition (see Proposition 4.6 in Asmussen [1 for discrete-time case) that the generalized processor-sharing queue is stable if and only if C >, since nf (n) C > for all n 1. Unlike the egalitarian processor-sharing discipline, we see that the generalized processor-sharing discipline influences stability of the queueing system as pointed out in Yashkov [58. Now, we determine the stationary probability vector of the Markov chain Q. The procedure used here is based on the censoring technique (e.g., see Grassmann and Heyman [16 ) and a RG-factorization. For the matrix Q, we denote by Q k the southeast corner of Q from level k. Let the superscript T denote matrix transpose. For Q k, we denote by ( Q (k) 1,1 T, Q (k) 2,1 T, ) T the first block-column of its fundamental matrix Q k = Q 1 k, which is the minimal non-negative inverse of Q k. We define R (k) j = i=1 D i+j 1 Q (k+1) i,1, k 0, j 1, (2) G (k) = kf (k) Q 1,1, (k) k 1, (3) U k = D 0 kf (k) I + (k + 1)f (k + 1) R (k) 1, k 0. (4) Based on the censoring technique, or a similar argument employed in Lemma 4 of Li and Zhao [30, we have the following Lemma. Lemma 1. For k 1 and j 2, Q (k) j,1 = G (k+j 1) G (k+j 2) G (k+1) ( U 1 k ), U k = 1. [ Q (k) 1,1 Using this lemma, we can obtain Lemma 2 below, which shows some basic relationships among the R-, G- and U -measures given in (2), (3) and (4), respectively.

7 512 Li et al. Lemma 2. i) For k 0 and l 1, R (k) l ii) For k 0, = [ D l + D l+1 G (k+2) + D l+2 G (k+3) G + ( U (k+2) 1 k+1 ). (5) U k = D 0 kf (k) I + D 1 G (k+1) + D 2 G (k+2) G (k+1) + D 3 G (k+3) G (k+2) G (k+1) +. (6) iii) The matrix sequence G (k) is the minimal nonnegative solution to the system of matrix equations kf (k) I +[D 0 kf (k) I G (k) + D 1 G (k+1) G (k) + D 2 G (k+2) G (k+1) G (k) + =0, k 1. (7) Proof. i) It follows from Lemma 1 that Q (k+1) 1,1 = U 1 k+1, j,1 = G (k+j) G (k+j 1) G (k+2) ( U 1 ), j 2. Q (k+1) Thus, (2) becomes R (k) l k+1 =[D l + D l+1 G (k+2) + D l+2 G (k+3) G (k+2) + ( U 1 k+1 ). ii) It follows from (4) that U k = D 0 kf (k) I + D 1 G (k+1) + D 2 G (k+2) G (k+1) + D 3 G (k+3) G (k+2) G (k+1) +. iii) Note that ( U k )( U 1 k ) = I and kf (k) ( U 1 k ) = G (k), we obtain ( U k )G (k) = kf (k), which is equivalent to kf (k) +[D 0 kf (k) I G (k) + D 1 G (k+1) G (k) + D 2 G (k+2) G (k+1) G (k) + =0. A discussion similar to the proof of Theorem in Neuts [36 leads to the conclusion that G (k) is the minimal nonnegative solution to the system of matrix equations in (7). This completes the proof. The following theorem provides the RG-factorization for the continuous-time level-dependent Markov chain of M /G/1 type. The proof is obvious from Lemma 2 and is omitted here. Theorem 3.1. Q = (I R U )U D (I G L ), (8)

8 RG-Factorization Approach for a BMAP/M/1 Queue 513 where (I R U ) = I R (0) 1 R (0) 2 R (0) 3 I R (1) 1 R (1) 2 I R (2) 1 I..., U D = diag(u 0, U 1, U 2, ), I G (1) I (I G L ) = G (2) I. G (3) I The following theorem expresses the stationary probability vector ( 0, 1, 2, ) of the Markov chain Q in terms of the RG-factorization. The proof is similar to the solution procedure in Subsection 5.1 of Li and Zhao [30. Theorem 3.2. The stationary probability vector of Q is given by 0 = z 0, n 1 n = i R n i, (i) n 1, i=0 where z 0 is the stationary probability vector of the infinitesimal generator U 0 of the censored chain of Q to level 0 and the scalar is uniquely determined by k=0 ke = 1. Proof. We now solve the equation Q = 0, or (I R U )U D (I G L ) = 0. (9) Let x = (I R U ), (10)

9 514 Li et al. where x = (x 0, x 1, x 2, ). Then 0 = x 0, n 1 n = x n + i R n i, (i) n 1. i=0 (11) It follows from (9) and (10) that xu D (I G L ) = 0, which is equivalent to { x0 U 0 x 1 U 1 G (1) = 0 x n U n x n+1 U n+1 G (n+1) = 0, n 1. (12) Note that the matrix U 0 is positive recurrent, it is easy to check that the vector ( z 0,0,0, ) is a nonzero non-negative solution to the system of equations (12). Hence, we obtain that x 0 = z 0 and x n = 0 for n 1. Therefore, using (11) we conclude the stated result. Let q k = lim t + P q(t) = k for k 0. It follows from Theorem 3.2 that the distribution q k of the stationary queue length is given by n 1 q 0 =, q n = i R n ie, (i) n 1. i=0 Thus, the mean and variance of the stationary queue length are given by Var[q = E[q = i=0 k 1 k i R k ie, (i) i=0 k 1 { k 1 2 k 2 i R k ie (i) k i R k ie} (i). i=0 4. THE SOJOURN TIME Let W k and I k denote the sojourn time experienced by a customer and the phase of the BMAP input at this customer s arrival epoch, respectively, when there are k customers in the system. We write W k (x, i) = P W k > x, I k = i, W k (x) = (W k (x,1), W k (x,2), W k (x, m)), k 0.

10 RG-Factorization Approach for a BMAP/M/1 Queue 515 Using a standard probabilistic analysis, e.g., Masuyama and Takine [34, the vector sequence W k (x) satisfies the following system of differentialdifference equations and for n 1, d dx W 0(x) = W 0 (x)[d 0 f (1)I + d dx W n(x) = nf (n + 1)W n 1 (x) + W n (x)[d 0 nf (n)i + W l (x)d l, (13) l=1 W n+l (x)d l, l=1 (14) with the initial condition that there exists at least a vector W k0 (0) = 0 for k 0 0. Let W(x) = (W 0 (x), W 1 (x), W 2 (x), ), D 0 f (1) I D 1 D 2 D 3 f (2) I D 0 2f (2) I D 1 D 2 = 2f (3) I D 0 3f (3) I D 1. 3f (4) I D 0 4f (4) I Then the system of equations (13) and (14) can be rewritten as with the initial condition Using (16) and (17), we obtain (15) d W(x) = W(x) (16) dx W(0) = (W 0 (0), W 1 (0), W 2 (0), ) = 0. (17) W(x) = W(0) exp x. We denote by w (s) and w n (s) the Laplace transform of the row vectors W(x) and W n (x), respectively. For example, w (s) = + 0 e sx W(x)dx.

11 516 Li et al. It is easy to verify that + Thus, it follows from (16) that 0 e sx dw (x) = W (0) + sw (s). w (s)( si ) = W(0). For s > 0, we denote by ( si ) 1 max the maximal negative inverse of the matrix si of infinite size with the setting that ( si ) 1 ( si ) 1 max < 0 for an arbitrary inverse ( si ) 1 of si. Then w (s) = W(0)( si ) 1 max. What remains now is find a procedure to efficiently determine the maximal negative inverse ( si ) 1 max. The procedure here is based on the censoring technique and the RG-factorization. Let the matrix sequence G (k) (s) be the minimal non-negative solution to the system of matrix equations (k 1)f (k) I + D 0 [s + kf (k) I G (k) (s) + D 1 G (k+1) (s)g (k) (s) + D 2 G (k+2) (s)g (k+1) (s)g (k) (s) + =0, k 1. For k 0 and l 1, we write U k (s) = D 0 [s + kf (k) I + D 1 G (k+1) (s) + D 2 G (k+2) (s)g (k+1) (s) + D 3 G (k+3) (s)g (k+2) (s)g (k+1) (s) +, R (k) l (s) =[D l + D l+1 G (k+2) (s) + D l+2 G (k+3) (s)g (k+2 (s) + [ U k+1 (s) 1. Similar to the discussion of Theorem 3.1, we obtain Q (s) =[I R U (s)u D (s)[i G L (s), (18) where I R U (s) = I R (0) 1 (s) R (0) 2 (s) R (0) 3 (s) I R (1) 1 (s) R (1) 2 (s) I R (2) 1 (s) I..., U D (s) = diag(u 0 (s), U 1 (s), U 2 (s), U 3 (s), ),

12 RG-Factorization Approach for a BMAP/M/1 Queue 517 I G L (s) = I G (1) (s) I G (2) (s) I G (3) (s) I Let X (l) 1 (s) = R (l) 1 (s), l 0, (19) X k+1(s) (l) = R (l) 1 (s)x (l+1) k (s) + R (l) 2 (s)x (l+2) k 1 (s) + +R (l) k (s)x (l+k) 1 (s), l 0, k 1, (20) Y (l) k (s) = G (l) (s)g (l 1) (s) G (l k+1) (s), l 1, k 1. (21) The proof of the following lemma is obvious according to the definition of matrix inverse such as [I R U (s)[i R U (s) 1 = I. Lemma 3. The matrices I R U (s), U D (s) and I G L (s) are invertible, [I R U (s) 1 = I X (0) 1 (s) X (0) 2 (s) X (0) 3 (s) I X (1) 1 (s) X (1) 2 (s) I X (2) 1 (s) I..., (22) U D (s) 1 = diag ( U 0 (s) 1, U 1 (s) 1, U 2 (s) 1, U 3 (s) 1, ), (23) I Y (1) 1 (s) I [I G L (s) 1 = Y (2) 2 (s) Y (2) 1 (s) I. (24) Y (3) 3 (s) Y (3) 2 (s) Y (3) 1 (s) I The following theorem gives the Laplace transform of the sojourn time distribution and its mean. Theorem 4.1. given by If the initial probability vector of the processor-sharing queue is W(0) = (W 0 (0), W 1 (0), W 2 (0), ) = 0,

13 518 Li et al. then w 0 (s) = [W 0 (0) + W k (0)Y (k) k (s) [ U 0 (s) 1 (25) and for n 1, w n (s) = [W n (0) + + l=0 k=n+1 [ W l (0) + W k (0)Y k n(s) (k) [ U n (s) 1 k=l+1 Furthermore, the mean sojourn time is given by and for n 1, E[W n = E[W 0 = [ W n (0) + + l=0 [ W 0 (0) + k=n+1 [ W l (0) + W k (0)Y k l(s) (k) [ U l (s) 1 X n l(s). (l) (26) W k (0)Y (k) k (0) [ U 0 (0) 1 e W k (0)Y k n(0) (k) [ U n (0) 1 e k=l+1 W k (0)Y k l(0) (k) [ U l (0) 1 X n l(0)e. (l) Proof. It follows from (16) that w (s) = (w 0 (s), w 1 (s), w 2 (s), ) = (W 0 (0), W 1 (0), W 2 (0), )[I G L (s) 1 [ U D (s) 1 [I R U (s) 1. By Lemma 3 and some matrix manipulations, we have (25) and (26). Note that E[W k =wk (0)e for k 0, it is easy to obtain the mean sojourn time. This completes the proof. Using the expression of wk (s), we obtain Var[W k =2 d ds w k (s) s=0 (E[W n ) 2, k 0. In what follows, we discuss two special cases of the sojourn time when the initial probability vector is specified. Besides, we can also analyze the

14 RG-Factorization Approach for a BMAP/M/1 Queue 519 third case: the sojourn time beginning with an embedded point with respect to the BMAP arrival process. Such discussion is standard by means of the method of embedded Markov chains for many queueing systems (see Chapter 5 of Neuts [36 ) and is omitted here. At the same time, it is worthwhile to note that the difference between the actual and virtual sojourn times is due to the initial probability vector. Case 1. Consider a stable processor-sharing queue with the initial probability vector W(0) = ( 0, 1, 2, ), where k is given in Theorem 3.2. We have and for n 1, w n (s) = [ n + w 0 (s) = [ l=0 k=n+1 [ l + k Y (k) k (s) [ U 0 (s) 1 (27) k Y k n(s) (k) [ U n (s) 1 k=l+1 k Y k l(s) (k) [ U l (s) 1 X n l(s). (l) (28) Case 2. queue is The initial probability vector of this stable processor-sharing W n (0) = and W k (0) = 0 for all k = n, where is a probability vector of size m with e = 1. Then w 0 (s) = [ U 0(s) 1, w n (s) = [ U n 1 n(s) 1 + Y n l(s)[ U (n) l (s) 1 X n l(s), (l) for n 1. l=0 5. ALGORITHMS In this section, we provide an approximate algorithm to calculate the stationary probability vector and the mean sojourn time. We present this algorithm in five steps below.

15 520 Li et al. Step one: Modifying level-dependence We need to modify the matrix Q given in (1) to a level-independent infinitesimal generator. Let N be a larger positive integer such that the matrix Q is modified as Q N = where C 1 C 2 C 3 C 4 C (N ) 0 D 0 Nf (N ) I D 1 D 2 Nf (N ) I D 0 Nf (N ) I D 1 Nf (N ) I D 0 Nf (N ) I C (N ) 0 = (0, 0,,0,Nf (N ) I ), C k = (D T N +k 2, DT N +k 3, DT N +k 4,, DT k 1 )T, k 2, (29) and C 1 is given by D 0 D 1 D N 2 D N 1 f (1) I D 0 f (1) I D N 3 D N 2 2f (2) I D N 4 D N f I D 0 f I. Step two: The R-measure The matrix Q N given in (29) is a levelindependent infinitesimal generator of M /G/1 type. Let G be the minimal non-negative solution to the matrix equation Nf (N ) I +[D 0 Nf (N ) I G + D k G k+1 = 0. Using Theorem 1, (18) and (19) in Li and Zhao [30, we define, respectively U = D 0 Nf (N ) I + D k G k

16 and the R-measure RG-Factorization Approach for a BMAP/M/1 Queue 521 R 0,k = R k = i=1 C k+i G i 1 ( U ) 1, k 1, D k+i 1 G i 1 ( U ) 1, k 1. i=1 Step three: The matrix U 0 and the vector x 0 1 in Li and Zhao [30, we define According to Theorem U 0 = C 1 + C k+1 G k 1 ( U ) 1 C (N ) 0. Note that U 0 is the infinitesimal generator of the censored Markov chain of Q N to level 0. Since the Markov chain Q N is positive recurrent, so is the censored chain U 0. Let x 0 be the stationary probability vector of U 0. Then x 0 U 0 = 0 and x 0 e = 1. Step four: The stationary probability vector We denote by ( 0, 1, 2, ) the stationary probability vector of the modified Markov chain Q N, where the size of 0 is Nm and the size of k is m for k 1. Using Subsection 5.1 of Li and Zhao [30, we obtain 0 = cx 0, 1 = 0 R 0,1, k 1 k = 0 R 0,k + i R k i, k 2, i=1 where c = R 0 = x 0 R 0 (I R) 1 e, R 0,k and R = R k. Step five: The stationary queue length distribution Let 0 = ( 0,0, 0,1,, 0,N 1 ). Then the stationary probability vector of the Markov chain Q is approximately given by n 0,n, 0 n N 1, N +l l+1, l 0.

17 522 Li et al. Furthermore, the stationary queue length distribution is given by q n 0,n e, 0 n N 1, q N +l l+1 e, l 0. For a positive recurrent level-dependent QBD process, Neuts and Rao [38 proposed an approximate algorithm to compute the stationary probability vector. With this algorithm, one first needs to modify this QBD process to a corresponding level-independent QBD process; then use the matrix-geometric solution of the modified process to approximate the stationary probability vector. Our algorithm is a generalization, to the M /G/1 type models, of their algorithm. For an asymptotically quasi- Toeplitz 2-dimensional Markov chain, Dudin and Klimenok [14 gave a direct truncation algorithm to deal with a level-dependent Markov chain of M /G/1 type. When the level-dependent Markov chain is modified as the corresponding level-independent case, they used the censoring technique to construct a system of finitely-many linear equations whose solution approximates the stationary probability vector. However, our algorithm is different from that of Dudin and Klimenok [14, although both algorithms use the same censoring technique. One drawback of the direct truncation method is that computational accuracy depends heavily on the truncation size of the censored matrix for some heavy-traffic cases. Note that in this paper the stationary probability vector of a Markov chain of M /G/1 type can be expressed by means of the R-measure, thus our algorithm effectively use all the subvectors in the stationary probability vector, including those corresponding to the censored chain as in Dudin and Klimenok [14. Therefore, our algorithm improves that of Dudin and Klimenok [14. A probabilistic interpretation on this improvement was given in Neuts and Rao [38. On the other hand, we now compare the computational complexity of our algorithm and that of Dudin and Klimenok [14. Let N and N be the truncated number given in Step one of our algorithm and the size of the censored matrix (see (12) in Dudin and Klimenok [14 ), respectively. Using the size of the matrix U 0, it is easy to check that the complexity of our algorithm for computing the vectors k for k 1isO(N 3 m 3 ) while the computational complexity of the algorithm of Dudin and Klimenok [14 is O(N 3 m 3 ). Note that N in general is not more than N under the same precision, thus our algorithm improves the computational complexity of the algorithm of Dudin and Klimenok [14. For the mean sojourn time, we use the following four-step approximate algorithm. Step one: Modifying level-dependence From Section 4, we see that the mean sojourn time only depends on the matrix given in (15). Hence, we need to modify the level-dependent Markov chain of M /G/1 type

18 RG-Factorization Approach for a BMAP/M/1 Queue 523 to its corresponding level-independent Markov chain. Let N be a larger positive integer. Then, the matrix can be modified to N = 1 B 2 B 3 B 4 B (N ) 0 D 0 Nf (N ) I D 1 D 2 (N 1)f (N ) I D 0 Nf (N ) I D 1 B (N 1)f (N ) I D 0 Nf (N ) I where B (N ) 0 = (0, 0,,0,(N 1)f (N ) I ), B k = (DN T +k 3, DT N +k 4, DT,, N +k 5 DT k 1 )T, k 2, and B 1 is given by D 0 f (1) I D 1 D N 3 D N 2 f (2) I D 0 2f (2) I D N 4 D N 3 2f (3) I D N 5 D N (N 2)f (N 1) I D 0 (N 1)f (N 1) I. Step two: The R- and G-measures Let G be the minimal nonnegative solution to the matrix equation (N 1)f (N ) I +[D 0 Nf (N ) I G + Using Theorem 1 in Li and Zhao [30, we define U = D 0 Nf (N ) I + U 0 = B 1 + D k G k, D k 1 G k = 0. k=2 B k+1 G k 1 ( U) 1 B (N ) 0, G 1 = UB (N ) 0 = (0, 0,,0,(N 1)f (N ) U).

19 524 Li et al. By (18) and (19) in Li and Zhao [30, we define a R-measure as R 0,k = R k = i=1 B k+i G i 1 ( U) 1, k 1, D k+i 1 G i 1 ( U) 1, k 1. i=1 Step three: Two matrix sequences We compute matrix sequences below: { X (l) R 0,1, if l = 0, 1 = R 1, if l 1, X (l) k+1 = Y (l) k = k i=1 k i=1 R 0,i X (i) k i+1, if l = 0, R i X (l+i) k i+1, if l 1, { G k 1 G 1, if l = k, G k, if l k + 1. Step four: Calculating the mean sojourn time ( l = 0,,0 }{{}, e T, l zero vectors 0,,0 }{{} N 2 l zero vectors Let ) T, 0 l N 1. Then the mean sojourn time at steady state is approximately given by and for n 1, E[W l = [ 0 + E[W N +n 1 = [ n + k Y (k) k [ U 0 1 l, 0 l N 1, (30) [ l=1 k=n+1 [ l + k Y (k) k n [ U 1 e k Y (k) k [ U 0 1 X (0) n e k=l+1 k Y (k) k l [ U 1 X n le. (l) (31)

20 RG-Factorization Approach for a BMAP/M/1 Queue 525 Given the initial probability vector W n (0) = and W k (0) = 0 for all k = n, (32) we then have E[W l = [ U 0 1 l, 0 l N 1, (33) where = 0,,0 }{{} l zero vectors, }{{} N 2 l zero vectors,, 0,,0 and for n 1, E[W N +n 1 = [ U 1 e + Y (n) n [ U 0 1 X (0) n e n 1 + Y n l[ U (n) 1 X (l) n l e (34) l=1 Remark 2. The GTH algorithm was first presented by Grassmann, Taksar and Heyman [17 in order to compute the stationary probability vector or a first passage time distribution in a finite-state Markov chain. The GTH algorithm has been shown to improve the Gaussian elimination in terms of some sophisticated probabilistic interpretations, e.g., Stewart [53, Seneta [50, and Benzi [4 and references therein. Our algorithm deals with an infinitestate Markov chain similarly with the RG-factorization playing a key role and can be regarded as a generalization of the GTH algorithm. 6. NUMERICAL EXAMPLES We give three simple numerical examples to illustrate how to use the approximate algorithms presented in section 5. We only consider the case with the initial probability vector given in (32), and compute the mean sojourn time using the two expressions (33) and (34). Similarly, we can use the approximate algorithm to compute the steady state mean sojourn time without any difficulty. In the numerical implementations, truncation for those infinite sums is necessary and is easily taken, e.g., see Bright and Taylor [9. The first example is to show how the mean sojourn time depends on the initial probability vector.

21 526 Li et al. Example 1. The positive function in the generalized processor-sharing discipline is taken by two different forms below: f 1 (n) = 1, f n 1 2 (n) = 1 n + n. 2 2n The Markovian arrival process has the matrix descriptor (D 0, D 1 ), where ( ) ( ) D 0 =, D =. 3 0 It is easy to check that the stationary arrival rate of the MAP is equal to 17/7. Let the service rate be 5. Then the processor-sharing queue is stable. Let = (,1 ), which is given in (32). Figure 1, for f 1 (n) and f 2 (n), shows how the mean sojourn time depends on the vector for the parameter [0, 1. We see that E[W 0 and E[W 10 are both monotonically increasing in for [0, 1. The following two examples illustrate how the mean sojourn time depends on the positive function in the generalized processor-sharing discipline Example 2. Let the positive function in the generalized processorsharing discipline be f (n) = 1 n b, 2 b 5. The arrivals to the queue follow a batch Markovian arrival process with the matrix descriptor D k ( ) ( ( ) k 0 2 D 0 =, D 2 4 k = ) k ), k 1. 0 ( 2 3 FIGURE 1 The mean sojourn time depends on.

22 RG-Factorization Approach for a BMAP/M/1 Queue 527 FIGURE 2 The mean sojourn time depends on b. It is easy to check that the stationary arrival rate of the MAP equals 5/3. Let the service rate be 5. Then the processor-sharing queue is stable. Figure 2 for = 1/3 and 2/3 shows how the mean sojourn time depends on the positive function f (n) for the parameter b [2, 5. Both E[W 0 and E[W 10 are monotonically decreasing in b for b [2, 5. Example 3. Let the positive function in the generalized processorsharing discipline be f (n) = 1 a n, 5 a 10, with the same arrival and service processes as that in Example 2. Figure 3 for = 1/3 and 2/3 shows how the mean sojourn time depend on the positive function f (n) for the parameter a [5, 10. We observe that E[W 0 is monotonically decreasing in while E[W 29 is monotonically increasing in a for a [5, 10. FIGURE 3 The mean sojourn time depends on a.

23 528 Li et al. ACKNOWLEDGMENTS The authors thank three referees for their valuable comments and remarks that improved the presentation of the paper. This research is supported in part by the National Natural Science Foundation of China (Grant No ), Hong Kong Research Grant Council (HKUST6133/02E), and a grant from University of Macau (Grant No. RG052/03-04S/LZT/FBA). The first author also acknowledges the Hong Kong University of Science and Technology (HKUST) and University of Macau for providing opportunity and support for his visits. REFERENCES 1. Asmussen, S. Applied Probability and Queues, Second Edition; Springer, Avrachenkov, K.; Ayesta, U.; Brown, P. Batch Arrival M /G/1 Processor-sharing With Application to Multilevel Processor Sharing Scheduling. Technique Report No. 5043; INRIA Sophia Antipolis, Bansal, N. Analysis of the M /G/1 processor-sharing queue with bulk arrivals. Oper. Res. Lett. 2003, 31, Benzi, M. A direct projection method for Markov chains. Linear Algebra and Its Applications 2004, 386, Borst, S.C.; Boxma, O.; Jelenković, P. Reduced-load equivalence and induced burstiness in GPS queues with long-tailed traffic flows. Queueing Systems 2003, 43, Borst, S.C.; Mandjes, M.; van Uitert, M. Generalized processor sharing queues with heterogeneous traffic classes. Adv. Appl. Probab. 2003, 35, Braband, J. Waiting time distributions for M /M /N processor sharing queues. Stochastic Models 1994, 10, Braband, J. Waiting time distributions for closed M /M /N processor sharing queues. Queueing Systems 1995, 19, Bright L.W.; Taylor, P.G. Calculating the equilibrium distribution in level-dependent quasi-birthand-death processes. Stochastic Models 1995, 11, Coffman E.G.; Kleinrock, L. Feedback queueing models for time-shared systems. J. ACM. 1968, 15, Coffman E.G.; Muntz, R.R.; Trotter, H. Waiting time distributions for processor-sharing systems. J. ACM. 1970, 17, Cohen, J.W. The multiple phase service network with generalized processor sharing. Acta Inform. 1979, 12, Cohen, J.W. Letter to the editor: The sojourn time in the GI /M /1 queue with processor sharing [J. Appl. Probab. 1984, 21, by Ramaswami. V. J. Appl. Probab. 1984, 21, Dudin, A.; Klimenok, V. A retrial BMAP /SM /1 system with linear repeated requests. Queueing Systems 2000, 34, Fayolle, G.; Mitrani, I.; Iasnogorodski, R. Sharing a processor among many job classes. J. ACM. 1980, 27, Grassmann, W.K.; Heyman, D.P. Equilibrium distribution of block-structured Markov chains with repeating rows. J. Appl. Prob. 1990, 27, Grassmann, W.K.; Taksar, M.I.; Heyman, D.P. Regenerative analysis and steady state distributions for Markov chains. Opens Res. 1985, 33, Grishechkin, S. On a relationship between processor-sharing queues and Crump-Mode-Jagers branching processes. Adv. Appl. Prob. 1992, 24, Grishechkin, S. GI /G/1 processor sharing queue in heavy traffic. Adv. Appl. Prob. 1994, 26, Gromoll, H.C. Diffusion approximation for a processor sharing queue in heavy traffic. Ann. Appl. Probab. 2004, 14, Gromoll, H.C.; Puha A.L.; Williams, R.J. The fluid limit of a heavily loaded processor sharing queue. Ann. Appl. Probab. 2002, 12,

24 RG-Factorization Approach for a BMAP/M/1 Queue Guillemin, F.; Boyer, J. Analysis of the M /M /1 queue with processor sharing via spectral theory. Queueing Systems 2001, 39, Jagerman, D.L.; Sengupta, B. The GI /M /1 processor-sharing queue and its heavy traffic analysis. Stochastic Models 1991, 7, Jelenković, P.; Momčilović, P. Large deviation analysis of subexponential waiting times in a processor-sharing queue. Math. Oper. Res. 2003, 28, Kleinrock, L. Time-shared systems: a theoretical treatment. J. ACM. 1967, 14, Kleinrock, L. Queueing Systems, Vol. II: Computer Applications; Wiley: New York, Kleinrock, L.; Muntz, R.R.; Rodemich, E. The processor-sharing queueing model for time-shared systems with bulk arrivals. Networks 1971, 1, Knessl, C. Asymptotic approximations for the GI /M /1 queue with processor-sharing service. Stochastic Models 1992, 8, Li, Q.L.; Liu, L.M. An algorithmic approach on sensitivity analysis of perturbed QBD processes. Queueing Systems 2004, 48, Li, Q.L.; Zhao, Y.Q. A constructive method for finding -invariant measures for transition matrices of M /G/1 type. In Matrix Analytic Methods Theory and Applications. Latouche, G., Taylor, P.G., Eds; World Scientific, 2002; Li, Q.L.; Zhao, Y. The RG-factorization in block-structured Markov renewal processes with applications. In Observation, Theory and Modeling of Atmospheric Variability; Xun Zhu, Ed; World Scientific, 2004; Lucantoni, D.M. New results on the single server queue with a batch Markovian arrival process. Stochastic Models 1991, 7, Lucantoni, D.M. The BMAP /G/1 queue: A tutorial. In Models and Techniques for Performance Evaluation of Computer and Communication Systems, Donatiello, L., Nelson, R., Eds.; Springer- Verlag: New York, 1993; Masuyama, H.; Takine, T. Sojourn time distribution in a MAP /M /1 processor-sharing queue. Oper. Res. Lett. 2003, 31, Morrison, J.A. Response-time distribution for a processor-sharing system. SIAM J. Appl. Math. 1985, 45, Neuts, M.F. Structured Stochastic Matrices of M /G/1 Type and Their Applications; Marcel Dekker: New York, Neuts, M.F. Matrix-analytic methods in the theory of queues. In Advances in Queueing: Theory, Methods and Open Problems, Dshalalow, J.H., Ed.; CRC Press: Boca Raton, FL, 1995; Neuts, M.F.; Rao, B.M. Numerical investigation of a multiserver retrial model. Queueing Systems 1990, 7, Núñez-Queija, R. Processor-sharing Models for Integrated-service Networks. Ph.D. Thesis, Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands, Núñez-Queija, R. Sojourn times in a processor sharing queue with service interruptions. Queueing Systems 2000, 34, Núñez-Queija, R. Sojourn times in non-homogeneous QBD processes with processor sharing. Stochastic Models 2001, 17, O Donovan, T.M. Conditional response times in M /M /1 processor-sharing models. Opens Res. 1976, 24, Parekh, A.K.; Gallager, R.G. A generalized processor sharing approach to flow control in integrated services networks: the single-node case. IEEE/ACM Trans. Networking 1993, 1, Parekh, A.K.; Gallager, R.G. A generalized processor sharing approach to flow control in integrated services networks: the multiple node case. IEEE/ACM Trans. Networking 1994, 2, Puha, A.L.; Williams, R.J. Invariant states and rates of convergence for a critical fluid model of a processor sharing queue. Ann. Appl. Probab. 2004, 14, Ramanan, K.; Reiman, M.I. Fluid and heavy traffic diffusion limits for a generalized processor sharing model. Ann. Appl. Probab. 2003, 13, Ramaswami, V. The N /G/1 queue and its detailed analysis. Adv. Appl. Prob. 1980, 12, Ramaswami, V. The sojourn time in the GI /M /1 queue with processor sharing. J. Appl. Probab. 1984, 21,

25 530 Li et al. 49. Rege, K.M.; Sengupta, B. The M /G/1 processor-sharing queue with bulk arrivals. In Modelling and Performance Evaluation of ATM Technology; Elsevier: Amsterdam, 1993; Seneta, E. Complementation in stochastic matrices and the GTH algorithm. SIAM J. Matrix Anal. Appl. 1998, 19, Sengupta, B. An approximation for the sojourn-time distribution for the GI /G/1 processorsharing queue. Stochastic Models 1992, 8, Sengupta, B.; Jagerman, D.L. A conditional response time of the M /M /1 processor-sharing queue. AT&T Tech. J. 1985, 64, Stewart, W.J. Introduction to the numerical solution of Markov chains; Princeton University Press, Tan, X.; Yang, Y.; Knessl, C. The conditional sojourn time distribution in the GI /M /1 processorsharing queue in heavy traffic. Queueing Systems 1993, 14, van Uitert, M.; Borst, S.C. A reduced-load equivalence for generalized processor sharing networks with long-tailed input flows. Queueing Systems 2002, 41, Yang, Y.; Knessl, C. Conditional sojourn time moments in the finite capacity GI /M /1 queue with processor-sharing service. SIAM J. Appl. Math. 1993, 53, Yashkov, S.F. Processor-sharing queues: some progress in analysis. Queueing Systems 1987, 2, Yashkov, S.F. Mathematical problems in the theory of processor-sharing queues. J. Soviet Math. 1992, 58, Zhang, Z.L. Large deviations and the generalized processor sharing scheduling for a multiplequeue system. Queueing Systems 1998, 28,