Discriminatory Processor Sharing Queues and the DREB Method

Transcription

1 Lingnan University From the SelectedWorks of Prof. LIU Liming 2008 Discriminatory Processor Sharing Queues and the DREB Method Z. LIAN, University of Macau X. LIU, University of Macau Liming LIU, Hong Kong Polytechnic University Available at:

2 Stochastic Models, 24:19 40, 2008 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / DISCRIMINATORY PROCESSOR SHARING QUEUES AND THE DREB METHOD Z. Lian, 1 X. Liu, 1 and L. Liu 2 1 Faculty of Business Administration, University of Macau, Macau SAR, China 2 Department of Logistics, Hong Kong Polytechnic University, Hung Hom, Hong Kong SAR, China We study a discriminatory processor-sharing (DPS) queue with a Markovian arrival process (MAP) and K classes of customers. This system can be modeled into a K -dimensional quasi-birth-and death (QBD) process by the standard matrix-analytical approach, but such a process is computationally difficult to handle. We present a different formulation by which the K -dimensional QBD process is reduced to a single-dimensional level-dependent QBD process with expanding blocks. We derive analytical expressions and use them to compute the sojourn time and joint queue length distributions efficiently. This method allows us to study numerically a wide range of important open queueing problems that are common in computer and communication systems, such as various priority queues, queueing networks, and other dependent queues. Keywords Dimensions reduction; Discriminatory processor sharing; DREB method; Joint queue length; Level-dependent; Multi-dimensional QBD; Priority queues; Sojourn time. Mathematics Subject Classification Primary 90B22; Secondary 68M INTRODUCTION In this paper, we study a discriminatory processor-sharing (DPS) queue with a Markovian arrival process and K classes of customers. By the standard matrix-analytic method, this system can be modeled into a multidimensional quasi-birth-and-death (QBD) process where every dimension has infinitely many states. While such QBD processes often arise from practical systems, such as tandem queues, shortest queues, and other Received July 12, 2006; Accepted October 01, 2007 Address correspondence to Z. Lian, Faculty of Business Administration, University of Macau, Macau SAR, China via Hong Kong; lianzt@umac.mo

3 20 Lian et al. dependent queues, the literature that provides numerically implementable solutions to such problems is limited and it remains an important challenge both analytically and numerically to the research communities. We introduce in this paper a different approach by which our DPS problem is formulated into single-dimensional and level-dependent QBD process, which can be analyzed exactly. The matrix-analytic method has been very successful in handling problems that can be modeled into multi-dimensional Markov chains in which all but one dimension have finite numbers of states. In Neuts language, (Refs. [21,22] ), the dimension with infinitely many states is usually called the level, while the other finite dimensions are called phases. The Q matrix, i.e., the generator, constructed accordingly is an infinite matrix with finite blocks (i.e., each element of the infinite matrix is a finite matrix). In particular, many problems can be formulated into QBD, M /G/1-type, or GI /M /1-type processes with finite blocks, and for which efficient solution approaches have been developed, e.g., Neuts [21,22].On the other hand, when the blocks of the Q matrix are infinite, the situation becomes extremely complicated. Much of the existing theory and methods breakdown, since they are based on manipulations (such as multiplications) of finite blocks. Neuts [23] elaborated the difficulties involved in handling QBD processes with infinite blocks while identified it as a promising new direction. In one of the earliest works on QBD processes with infinite blocks, Ramaswami and Taylor [28] focused on the analysis of matrices G k and R k. They provided explicit expressions for G k and R k and gave the necessary and sufficient conditions for a vector x to be a right eigenvector of G k for all k and for x T to be a left eigenvector of R k. Takahashi et al. [36] gave a set of sufficient conditions for the stationary probability vector of a QBD process with infinite blocks to have a geometric tail along the level direction. Similar conditions are obtained by Miyazawa [18] and by Foley and McDonald [8] for the shortest-queue model. Haque et al. [10] identified a set of sufficient conditions equivalent to those of Ref. [36] They provided a practical method to verify whether the stationary probability vector of a specific model satisfies this set of sufficient conditions. Kroses et al. [14] considered a two-node tandem Jackson network with infinite buffers. The authors show that the decay rate of the stationary probability vector along the level direction may not be equal to the convergence norm of the R matrix. This shows that the decay rate of the probability vector in a QBD with infinite blocks may not be the same as the limit of the decay rates of the stationary probability vectors in the corresponding QBD with finite blocks obtained by truncation of the original QBD. This work supports the view expressed in Ref. [23] that the truncation approach may not work well for QBD processes with infinite blocks. Generally speaking, truncation may cause asymmetric information

4 Discriminatory Processor Sharing Queues 21 loss. To achieve better accuracy, the remaining blocks size and generator matrix size have to be sufficiently large to the extend that numerical computation may become infeasible when the number of dimensions is greater than two. Clearly, new approaches are needed to handle QBD processes with infinite blocks. We now consider processor-sharing queues. Processor-sharing queues are important models for computer and communication systems. Early works on processor-sharing queues were motivated by applications of multiuser mainframe computers. Recent interests in processor-sharing queues stem from their applications to communication networks and web servers, for example, modeling congested links with Transfer Control Protocol (TCP) traffic and job schedulers in web servers. Under a processorsharing service discipline, a server can be shared equally by all customers in the system. Specifically, when n customers are present in a system, each customer occupies 1/n of the total service capacity. This special processor-sharing discipline is usually referred to as egalitarian processorsharing (EPS). For many computer and communication systems, a more interesting and useful service discipline is discriminatory processor-sharing (DPS), by which the capacity of a system is shared unequally according to customers priorities. The DPS discipline for a single-processor system with K job classes is defined by a vector of weights 1,, K, based on which the fraction of the processor capacity assigned to each job class is determined. Using the DPS discipline, we give preference to one or more classes of jobs at the expense of others. Processor-sharing queues have been studied in the literature for more than three decades since the ground breaking work by Kleinrock [12], who introduced both the EPS and DPS disciplines and derived the conditional mean sojourn times in an M /M /1 processor-sharing queue. For the EPS discipline, conditional or unconditional mean sojourn times and or variances were derived through the Laplace Stieltjes transform or approximations by Coffman et al. [6] and O Donovan [25] for the M /M /1 system; Asare and Foster [2], Schassberger [32], and Ott [26] for the M /G/1 system; Ramaswami [27] and Knessl [13] for the GI /M /1 system; and Grishechkin [9] and Sengupta [33] for the GI /G/1 system. In addition, Braband [3,4] considered the waiting time distributions in an M /M /N processor-sharing queue, and by formulating QBD processes, Nunez- Queija [24] studied a Markovian processor-sharing queue; Masuyama and Takine [17] provided a recursive formula to compute the stationary sojourn time distribution in the MAP /M /1 processor-sharing queue; and Zwart and Boxma [38] provided an asymptotic analysis of the sojourn time in the M /G/1 processor sharing queue. Sericola et al. [34] studied the M /PH /1 Queue, and Li et al. [15] studied an MAP /M /1 generalized processorsharing queue with batch arrivals.

5 22 Lian et al. For the M /M /1 DPS queue, Fayolle and Mitrani [7] obtained the unconditional expected sojourn times, Rege and Sengupta [30] derived the moments of the queue-length distribution, while Kim and Kim [11] obtained the second and some higher moments of the sojourn time. Furthermore, Mitra and Weiss [19] and Morrison [20] provided some asymptotic analysis of a two-node DPS queueing network while Rege and Sengupta [29] obtained some properties of the sojourn time in the M /G/1 DPS queue. The above brief review shows that the existing results on DPS queues are limited. As mentioned earlier, this lack of progress can be attributed to a difficulty common to many stochastic models, i.e., a QBD process with infinite blocks. In this paper we tackle this difficulty with a different approach. We first define the DPS queue in Section 2 and show that it leads to a QBD with infinite blocks by the standard formulation. We then present a new formulation approach in Section 3. We call this approach the DREB method (see definition in Section 3), by which the stationary state process of any multi-dimensional QBD, and of the DPS queue as well, has essentially a single-dimensional level-dependent Quasi-Birth- Death (LDQBD) infinitesimal generator with expanding blocks instead of infinite blocks. LDQBD is an important model and has appeared in the literature for quite some time. For example, Lian and Deng [16] showed n that n = R 0 i=0 i and derived a closed form expression for R i when the blocks have some special features. Later, Ramaswami and Taylor [28] and Bright and Taylor [5] provided an effective algorithm to compute the R i matrix for the general LDQBD process. In Section 4, we derive an LDQBD for the stationary state process of the DPS queue. We provide an explicit expression of the stationary probability vector in terms of R N matrices which can then be computed using Bright and Taylor s algorithm [5]. Similarly, we construct an LDQBD with expanding blocks to model the sojourn time of the DPS queue in Section 5. We derive an explicit expression for the tail distribution of the sojourn time and then use the Rectangular Iterative Algorithm (RIA) from Shi et al. [35] to compute the distribution numerically. In Section 6, we provide some numerical examples to illustrate the effectiveness of our method. We conclude the paper in Section 7. One obvious advantage of our method is that there is no loss of information in the computation process since there is no need for block truncation. We believe that used together with some existing (but perhaps not yet well known) algorithms, the DREB method will significantly improve our ability to model and analyze many complex computer, communication, and industrial systems. 2. THE DPS QUEUE We first define a MAP /M /1 DPS system in which K classes of customers share the service of a single processor.

6 2.1. Customer Arrivals Discriminatory Processor Sharing Queues 23 For i = 1,, K, an arriving customer belongs to class i with probability p i. The overall customer arrivals of all classes follow a Markovian arrival process with an infinitesimal generator D = D 0 + D 1 in state space J = 1,, m, where D 0 = (dj,k 0 ) m m and D 1 = (dj,k 1 ) m m. All the off-diagonal elements of D 0 and all the elements of D 1 are nonnegative. The transitions associated with D 1 are called type-1 transitions. A customer arrives only at a type-1 transition epoch. Let z denote the stationary probability vector of the underlying irreducible Markov chain. It is uniquely determined by z(d 0 + D 1 ) = 0 and ze = 1, and the mean arrival rate of the MAP is = zd 1 e. We further assume that D 1 contains positive elements, and thus > The DPS Discipline Let n = (n 1,, n K ) be a K -dimensional row vector where n i represents the number of class-i customers in the system. Let = ( 1,, K ) T be a K -dimensional column vector and g i (n) be the capacity allocation function, i.e., the fraction of the processor s capacity given to class-i customers (as a whole) { (ni i )/n, if n 0, g i (n) = 0, if n = 0, i = 1,, K (1) The processing times of class-i customers are i.i.d. according to an exponential distribution with mean 1 i, i = 1,, K. Thus, when there are n i class-i customers in the system (n i 0), the processing completion rate of a class-i customer is i i /n The Markov Chain Let N i (t) be the number of class-i customers in the system at time t, i = 1,, K, and N(t) = (N 1 (t),, N K (t)). Let J (t) be the phase of the MAP at time t. Then N(t), J (t), t 0 is a multi-dimensional continuoustime Markov chain with a state space (n, j): n is a K -dimensional vector of nonnegative integers and 1 j m. Arranging states in the standard ascending or descending order, we can formulate N(t), J (t), t 0 into a multi-dimensional QBD. For example, arrange the states of N 1 (t), N 2 (t), J (t) as follows: (0, 0, 1),, (0, 0, m); (0, 1, 1),, (0, 1, m); (0, 2, 1),, (0, 2, m); ; (1, 0, 1),, (1, 0, m); (1, 1, 1),, (1, 1, m); (1, 2, 1),, (1, 2, m); ; (2, 0, 1),, (2, 0, m); (2, 1, 1),, (2, 1, m); (2, 2, 1),, (2, 2, m); ;

7 24 Lian et al. (3, 0, 1),, (3, 0, m); (3, 1, 1),, (3, 1, m); (3, 2, 1),, (3, 2, m); ; ; ; We then have a 2-dimensional level-dependent QBD process with the generator matrix B (0) A (0) C (1) B (1) A (1) Q = C (2) B (2) A (2), (2) in which B (0), A (0), C (i), B (i), A (i), i = 1, 2, are infinite blocks. Currently, we do not have an effective computational method to handle such QBD. In order to derive the stationary probability distribution, all the blocks are truncated so that a QBD with finite blocks is obtained in which the last rows of all B (i) and A (i), i = 0, 1,, must be modified to satisfy that new Q is still a generator matrix. The problem is that before the computation of the algorithm is well into its later stage, we do not know where to truncate the blocks so that a given error requirement can be satisfied. 3. THE DREB METHOD In this section, we introduce the DREB method, a formulation scheme by which any multi-dimensional QBD by the above standard formulation can be reformulated into a single-dimensional LDQBD with finite but expanding blocks. Here, DREB refers to dimension-reduction and expanding blocks and the reason for this name will be clear as we proceed. We consider continuous-time stochastic processes, but the results in this section can be easily extended to discrete-time stochastic processes. Denote set E = 0, 1, 2, and let n = (n 1,, n K ) E K. Definition 3.1. The level of n is defined as the summation of all its elements, i.e., N = K n i=1 i For any level N, there is a fixed number of permutations of the values of the K elements. Each such permutation is called a state (corresponding to the level N ). For N 0, denoted by V (N, K ) the number of possible states of a K -dimensional vector in level N. It is easy to see that V (N, K ) = (N + K 1)!, N!(K 1)! For example, V (N,2) = N + 1 and V (N,3) = (N + 2)(N + 1)/2.

8 Discriminatory Processor Sharing Queues 25 Definition 3.2. Let n (1) = ( ) n (1) 1,, n (1) K and n (2) = ( ) n (2) 1,, n (2) K, n (1), n (2) E K. We say that the order of n (1) is higher than the order of n (2) if there exists a j in [1, K ] such that for all i < j, n (1) i = n (2) i and n (1) j > n (2) j. We now define a way to list all the possible states of n. LASD Sequencing: (1) All the states with the same level are listed in the same row; (2) Rows are listed ascending in the level; and (3) States in a row are listed descending in the order. The listing of all possible states of E K according to the above three rules is called Level-Ascending and State-Descending (LASD) Sequencing. LASD example: Consider a three-dimensional state space E 3. By LASD Sequencing, all states in E 3 are listed as follows. (0, 0, 0); (1, 0, 0), (0, 1, 0), (0, 0, 1); (2, 0, 0), (1, 1, 0), (1, 0, 1), (0, 2, 0), (0, 1, 1), (0, 0, 2); (3, 0, 0), (2, 1, 0), (2, 0, 1), (1, 2, 0), (1, 1, 1), (1, 0, 2), (0, 3, 0), (0, 2, 1), (0, 1, 2), (0, 0, 3);, Let e i be a vector of all zero elements except the ith position, which is occupied by 1. We give a definition of an K -dimensional QBD process. Definition 3.3. Let N(t), J (t), t 0 is a continuous-time Markov chain with a state space (n, j) : n E K, j = 1,, m. We say N(t), J (t), t 0 is a K -dimensional QBD if the only nonzero elements are in the blocks representing transitions n n, n n + e i, and n + e i n. We are now in a position to present the DREB Theorem. Theorem 3.1 (The DREB Theorem). A multi-dimensional QBD process can always be reformulated into a single-dimensional LDQBD process with finite and expanding blocks by listing all the states according to LASD Sequencing. Proof. By Definition 3.3, the only possible transitions in the original infinitesimal generator are n n, n n + e i, and n + e i n. Let us arrange all the states of the multi-dimensional QBD by LASD Sequencing. Then, by Definition 3.1, N E, the only possible transitions are N N, N N + 1, and N + 1 N in the corresponding infinitesimal generator Q. N E, let C N +1 be the block of states that transit from

9 26 Lian et al. level N + 1 to level N ; B N the block of states that transit from level N to level N ; and A N the block of states that transit from level N to level N + 1 Obviously, we have B 0 A 0 C 1 B 1 A 1 Q = C 2 B 2 A 2 (3) The number of states in level N is finite and increasing in N. Thus, the blocks in Q are level-dependent, finite, and expanding with N. Remark 3.1. LASD Sequencing is a natural way to list all the states of a multi-dimensional vector/process and it usually comes with a physical meaning. In the context of a multi-dimensional Markov process for example, if X i represents the number of customers in stage i of a K -stage serial queueing network, then the level represents the number of customers in the entire network. In the DPS queue, the level represents the number of customers in the system from all K classes. 4. THE JOINT QUEUE LENGTH DISTRIBUTION Using LASD Sequencing, the process N(t), J (t), t 0 defined in Section 2 has an LDQBD infinitesimal generator with expanding blocks. For example, when K = 2: Level 0 : (0, 0, 1),, (0, 0, m); Level 1 : (1, 0, 1),, (1, 0, m); (0, 1, 1),, (0, 1, m); Level 2 : (2, 0, 1),, (2, 0, m); (1, 1, 1),, (1, 1, m); (0, 2, 1),, (0, 2, m); Level 3 : (3, 0, 1),, (3, 0, m); (2, 1, 1),, (2, 1, m); (1, 2, 1),, (1, 2, m); (0, 3, 1),, (0, 3, m);,, Notice that the last element in each vector is the phase of the corresponding MAP. Returning to the general case, the state transition diagrams of the LDQBD (i.e., equation (3)) are given below for ne = N 0, i = 1,, K, and j, k = 1,, m: In A N : (n, j) (n + e i, k) at p i d 1 j,k ; In C N +1 : (n + e i, j) (n, j) at g i (n + e i ) i ;

10 Discriminatory Processor Sharing Queues 27 In B N : (n, j) (n, k), j k, at d 0 j,k, (n, j) (n, j) at d 0 j,j and K g i (n) i All the remaining elements of Q not mentioned above are zeros. Specially, when K = 2, for all N 0, g 1 (N + 1, 0) 1 I m g 2 (N,1) 2 I m C N +1 = B N D0 = D 0 g 1 (N,1) 1 I m N + 1 rows of blocks g 2 (1, N ) 2 I m i=1 g 1 (1, N ) 1 I m g 1 (0, N + 1) 1 I m, (4) g 1 (N,0) 1 I m 2 i=1 g i(n 1, 1) i I m, 2 i=1 g i(1, N 1) i I m g2(0, N )2Im p 1 D 1 p 2 D 1 A N = (5) N + 1 rows of blocks (6) p 1 D 1 p 2 D 1 Remark 4.1. The size of the blocks grow up quite fast with N when m(n +K 1)! m(n +K )! K is large. For example, the block A N has rows and N!(K 1)! (N +1)!(K 1)! columns. However, there are at most mk non-zero elements in each row of each block. To save the memory space, we can usually adopt sparse matrices in MATLAB. Let i be the traffic intensity of class-i jobs. We have i = p i / i. We assume that the stability condition K i=1 i < 1 holds. To compute the stationary probability vector of Q, we first use the RG-factorization technique to derive the needed expressions (see, for example, Li et al. [15] ). Let Q = (I R U )U (I G L ), (7)

11 28 Lian et al. where O R 0 O R 1 R U = O R 2, (8) U = diag(u 0, U 1, U 2 ), (9) O G 1 O G L = G 2 O (10) Comparing (7) with (3), we can easily obtain, for all N 0 U N = B N + R N C N +1, (11) R N = A N U 1 N +1, (12) where R N and G N are, respectively, the minimal nonnegative solution to and A N + R N B N +1 + R N R N +1 C N +2 = 0 (13) C N + B N G N + A N G N +1 G N = 0 (14) The theorem below gives the stationary probability vector = ( 0, 1, 2, ) of the LDQBD Q in terms of the RG-factorization, where N = ( n,1,, n,m ), n E K satisfying ne = N is the probability vector for level N. Theorem 4.1. The stationary probability vector of Q is given by { 0 = z 0, N = 0 R (N ), n 1 (15) where R (N ) = N 1 i=0 R i, z 0 is the stationary probability vector of the generator U 0 and the scalar is uniquely determined by N =0 N e = 1, i.e., z 0 N =0 R (N ) e = 1 (16)

12 Discriminatory Processor Sharing Queues 29 Proof. We now solve the equation Q = 0or (I R U )U (I G L ) = 0 (17) Let x = (I R U ), (18) where x = (x 0, x 1, x 2, ). Then { 0 = x 0, N = x N + N 1 R N 1, N 1 (19) If follows from (17) and (18) that xu (I G L ) = 0, which is equivalent to x N U N x N +1 U N +1 G N +1 = 0, N 0 (20) By Theorem 4.5 in Anderson [1], the matrix U 0 = B 0 + R 0 C 1 is positive recurrent. It is easy to check that the vector (z 0,0,0,) is a nonzero solution to the system of equations (20). Hence, x 0 = z 0 and x N = 0 for N 1. Therefore, using (19), we have the stated result. Let n = lim t + P N (t) = n for n E K. It follows from Theorem 4.1 that n, the joint queue length probability distribution in the steady state, can be obtained by n = ( n,1,, n,m )e (21) With Theorem 4.1, the computation of the stationary probability vector is easy by following Lemma 1 in Bright and Taylor [5], which provides an efficient algorithm (B&T Algorithm) to calculate R N. The authors also pointed out a way to approximate the stationary probability vector by truncating the sum in (16). 5. THE SOJOURN TIME DISTRIBUTION For convenience of presentation, we only consider the sojourn time of the class-1 customers. For the sojourn times of other classes, say that of class-i, we only need to replace 1 by i in the corresponding expressions obtained in this section, i.e., we treat class-i as class-1. Let W be the sojourn time of a class-1 customer and W n,j (x) = P W > x N(t) = n, J (t) = j be the probability that the sojourn time of a class-1 customer is greater than x, given that this customer observes n customers in the system (excluding this customer) and the MAP transits

13 30 Lian et al. to phase j at the arrival epoch. Denote W n (x) = (W n,1 (x),, W n,m (x)) and N (x) = W n (x), n E K satisfying ne = N T. It is easy to see that N (0) = e. By standard probabilistic arguments, the vector sequence N (x) satisfies the following Kolmogorov s Backward Equations (see, for example, Ross [31] ). d dx 0(x) = (D 0 1 I ) 0 (x) + A 0 1 (x), (22) d dx N (x) = C N N 1 (x) + B N N (x) + A N N +1 (x) (23) where B N and A N are the same as in (3), and the nonzero elements in C N are shown in the following transition diagrams for all n satisfying ne = N, j = 1,, m, and i = 2,, K : (n, j) (n e 1, j) at n 1 1 n 1 g 1 (n) 1 and (n, j) (n e i, j) at g i (n) i. We now define (x) = ( 0 (x) T, 1 (x) T, 2 (x) T, ) T, (24) B 0 A 0 C 1 B 1 A 1 Q = C 2 B 2 A 2, (25) where B 0 = D 0 1 I. Equations (22) and (23) can then be rewritten as d dx (x) = Q (x) (26) Note that Q is a defective infinitesimal generator of a continuoustime Markov chain with state space (n, j), n E K \(0,,0), j = 1,, m. Therefore, the conditional sojourn time of a class-1 customer given an initial state (n, j) is equivalent to the first passage time from state (n, j) to an implicit absorbing state. Formally, the solution of (26) is (x) = exp( Qx)e (27) Below we provide an algorithm to calculate (x). Applying the uniformization technique (see, for example, Ref. [37] ), we have, from (27) (x) = ( + ) N x N N =0 N! e (+)x [ I + Q ] N e, (28) +

14 Discriminatory Processor Sharing Queues 31 where ˆ= max d1,1 0,, d m,m 0 and ˆ= sup n E K g K i=1 i(n) i = max 1,, K. Let T = (T n,k ) = [ I + +] Q, where Tn,k is a block matrix in level n. We have T 0,0 T 0,1 T 1,0 T 1,1 T 1,2 T = T 2,1 T 2,2 T 2,3 (29) Let (x) = P W > x for x 0 be the probability that the sojourn time of an arrival class-1 customer with an initial state probability vector y = y n is greater than x, where y n is a subvector of the initial probability when the numbers of customers are n. Applying the law of total probabilities to (28). We obtain (x) = y (x) = y ( + ) N x N e (+)x T N e (30) N! N =0 Remark 5.1. The phase of the MAP immediately after arrival points is a discrete time Markov chain with transition matrix (D 0 ) 1 D 1, and its stationary distribution J can be easily calculated by J [I + (D 0 ) 1 D 1 ]=0. Therefore, if a class-1 customer enters the system with n 0 = (n1 0,, n0 K ) other customers present in the system at the epoch of arrival, the initial probability vector y = y n n E K ; y n = J if n = n 0, y n = 0 otherwise. Remark 5.2. When y equals to, the stationary probability vector of the system, (30) is actually the tail probability of the remaining sojourn time of a class-1 customer in the steady state. We will calculate (x) using the Rectangle Iterative Algorithm (RIA) from Ref. [35] For N > 0, define ỹ N = (y 0, y 1,, y N ), (31) T 0,0 T 0,1 T 1,0 T 1,1 T 1,2 T N,N +1 =, (32) T N,N 1 T N,N T N,N +1 T (N ) N 0 = T N0,N 0 +1 T N0 +1,N 0 +2 T N0 +N 1,N 0 +N (33)

15 32 Lian et al. Lemma 5.1. For any >0, there exists a number N 0 such that 0 yt N e ỹ N0 T (N ) N 0 e < (34) Proof. We denote y = (ỹ N0, ỹ N0 +1, ) and T = ( T N,N +1 ) O O T N +1,N +2 By definition, we have T e e, and hence N > 0, T N +1,N +2 e e. Since y i=0 ie = 1, for any small >0, we can always find a number N 0 0so that N 0 y i=0 ie > 1, i.e., ỹ N0, e <. Therefore, N 0, [ N0 +N T N e =ỹ N0 T (N ) N 0 e +ỹ N0, ỹ N0 T (N ) N 0 e +ỹ N0, e < ỹ N0 T (N ) N 0 e + i=n 0 +1 T i,i+1 ]e Obviously, yt N e ỹ N0 T (N ) N 0 e. Therefore, yt N e N0 T (N ) e <. Algorithm 1. The RIA for (x). Step 1: Given an error >0 and a positive number X, let = /[2 + ( + )X ]; Step 2: Find N 0, so that N 0 y N =0 N e > 1 ; Step 3: Let M = min { max([2( + )ex ], [log 2 (1/)]), inf { N ỹ N0 T (N ) N 0 e }} ; Step 4: x [0, X ], calculate N 0 N0,M (x) = M ( + ) n x n n=0 n! e (+)x ỹ N0 T (n) N 0 e (35) Theorem 5.1. error. N0,M (x) approximates (x) for x [0, X ] with a uniform Proof. This is directly from Theorem 1 of Ref. [35] and Lemma 5.1. Remark 5.3. For an MAP /M /1-EPS queue, Masuyama and Takine [17] use a recursive formula to approximate the sojourn time distribution. The error in Ref. [17] depends on x. Using RIA, our approximation for the

16 Discriminatory Processor Sharing Queues 33 sojourn-time distribution has a uniform error bound for all x,0< x < X. We would also like to point out that using RIA, we can compute the transient-state joint-queue length distribution easily, and replacing the stationary distribution in (30) by the transient-state queue length distribution, we can also compute the time independent sojourn time distributions. 6. NUMERICAL EXAMPLES We use some simple numerical examples to compare the complexity of DREB method with that of the truncation method, and illustrate the use of the algorithms presented in Section 5. We also discuss the impact of different capacity allocations on the stationary queue length and expected sojourn time. The first example is an M /M /1-DPS system with K = 2. We set = 08, p 1 = p 2 = 05, 1 = 2 = 10, 1 = 02, and 2 = 08. The second example is an M /M /1-DPS system with K = 3. We set = 10, p 1 = p 2 = 04, p 3 = 02, 1 = 2 = 3 = 12, 1 = 07, and 2 = 3 = 015. Denoted by N the maximum level (equivalent to K in Ref. [5] ). That is, to satisfy the tolerance = 0001 (see Algorithm 2 in Ref. [5] ), we need to calculate R 0, R 1,, RN obtain the stationary probability vector. We found that N = 25 in the first example and N = 24 in the second example, that is, the maximum block size for AN is N + 1 N + 2 = in the first example and (N + 1)(N + 2)/2 (N + 2)(N + 3)/2 = in the second example. TABLE 1 The joint probabilities of the queue length in M /M /1 (K = 2) n 1 \n Pr (n 1 ) Pr (n 2 ) Expected queue lengths En 1 = 2729, En 2 = 1176

17 34 Lian et al. TABLE 2 The joint probabilities of the queue length in MAP /M /1 n 1 \n Pr (n 1 ) Pr (n 2 ) Expected queue lengths En 1 = 1657, En 2 = 0727 TABLE 3 The stationary probability distribution in M /M /1 (K = 3) Level n 2 + n 3 \n Expected queue lengths En 1 = , En 2 = , En 3 =

18 Discriminatory Processor Sharing Queues 35 Now we consider the truncation method. Since the last rows of all B i and A i, i = 0, 1, are modified after the truncation, some information are lost. The B&T Algorithm in Ref. [5] is also needed to handle the corresponding LDQBD with finite blocks. To satisfy the tolerance = 0001, the size of each truncated block must also be 25 25, and N is 25 in the first example (K = 2). In the second example (K = 3), the size of each block is = , and N = 24. We may say that the complexities of both DREB method and the truncation method are O(N K +1 ). However, the unfortunate fact for the truncation method is that we do not have an effective method to determine the suitable truncating size in advance that satisfies the given tolerance but is not too large. Next, we will illustrate the distributions of the queue length and the sojourn time. We consider one more example, the MAP /M /1-DPS system with K = 2. ( ) ( ) D 0 =, D 1 =, and hence again = 08. We set 1 = 2 = 12, 1 = 01, and 2 = 09. Tables 1 3, Figures 1 3 (after rounding up) show the queue-length distributions with two or three classes of customers. From Figures 1 and 2, we can see that the tails of the queue for class-1 customers are heavier than that for class-2. From Figure 3, we can see that the tails of the queue for class-2 and class-3 customers are heavier than that for class-1, as expected for the lower priority classes. Now we calculate the (remaining) sojourn time distribution of class-1 customers in example 1 and example 3 (K = 2), we set p 1 = 1/6, p 2 = 5/6 and, 1 = 2 = 1, and compare different capacity allocations with 1 = 02, FIGURE 1 Probabilities on queue length for different classes of customers (M /M /1, K = 2).

19 36 Lian et al. FIGURE 2 Probabilities on queue length for different classes of customers (MAP /M /1, K = 2). 0.5 or 0.8. We assume that the system probability vector in a class-1 customer arrival is the system probability distribution in the steady state. Figures 4 and 5, and Table 4 show the probabilities that the sojourn times of class-1 customers are greater than x. For example, we can see from Table 4, in the M /M /1 system, the probabilities that the sojourn times of class-1 customers are greater than 11.1 are 0.25, 0.15 and 0.10 when 1 = 02, 0.5, and 0.8, respectively. In the MAP /M /1 system, the probabilities that the sojourn times of class-1 customers are greater than 11.1 are 0.25, 0.12 and 0.07 when 1 = 02, 0.5, and 0.8, respectively. FIGURE 3 Probabilities on queue length for different classes of customers (M /M /1, K = 3).

20 Discriminatory Processor Sharing Queues 37 FIGURE 4 Probabilities on sojourn times of class-one customers for different (M /M /1). Remark 6.1. Since the number of states grow very fast in the state space when K > 3, it becomes increasingly more time consuming to compute either the joint stationary probability distribution or the sojourn time distribution. One alternative is to calculate the expectations or variances of some performance measures instead, for example, the expected queue length of each class, the expected sojourn time of the first class. We can also re-model the system to reduce the number of the customers classes. FIGURE 5 Probabilities on sojourn times for different (MAP /M /1).

21 38 Lian et al. TABLE 4 Tail probabilities on sojourn times for M /M /1 and MAP /M /1 systems M /M /1 MAP /M /1 x\ CONCLUDING REMARKS In computer and communication systems, as well as manufacturing and supply chain systems, many modeling and control problems are still open because the models that can be constructed are numerically intractable. In this paper, we present a simple and novel formulation approach. We show that the DPS queue with a multi-class arrival process can be completely evaluated under this formulation and with some existing numerical methods. We are also working on other well known open problems relevant to many important applications, such as queueing networks with non-poisson external demands and the general shortest queue problems. We believe our approach will stimulus further research in many areas, such as queueing theory, computer and communication systems, and the Internet, to tackle many existing and new problems. ACKNOWLEDGMENTS We thank Professor Peter Taylor and two anonymous referees for their constructive comments and suggestions on a pervious version of this paper that have been very helpful to us for improving the paper. The first and second authors are supported in part by the Research Committee of the University of Macau through RG052/04-05S/LZT/FBA. The third author is supported in part by Hong Kong Research Council through PolyU6145/04E. REFERENCES 1. Anderson, W.J. Continuous-Time Markov Chains, an Applications Oriented Approach. Springer-Verlag, Asare, B.K.; Foster, F.G. Conditional response times in the M /G/1 processor-sharing system. Journal of Applied Probability 1983, 20, 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,

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