Ann Oper Res (2012) 201:307–323 DOI 10.1007/s10479-012-1254-7
Computation of the steady state distribution for multi-server retrial queues with phase type service process Che Soong Kim · Vilena V. Mushko · Alexander N. Dudin
Published online: 8 November 2012 © Springer Science+Business Media New York 2012
Abstract We consider a multi-server retrial queueing system with the Batch Markovian Arrival Process and phase type service time distribution. Such a general queueing system suits for modeling and decision making in many real life objects including modern wireless communication networks. Behavior of such a system is described by the level dependent multi-dimensional Markov chain. Blocks of the generator of this chain, which is the block structured matrix of infinite size, can have large size if the number of servers is large and distribution of service time is not exponential. Due to this fact, the existing in literature algorithms allow to compute key performance measures of such a system only for a small number of servers. Here we describe the algorithm that allows to compute the stationary distribution of the system for larger number of servers and numerically illustrate its advantage. Importance of taking into account correlation in the arrival process is numerically demonstrated. Keywords Performance · Queues applications · Queues theory
1 Introduction Retrial queueing models play an important role in performance evaluation and capacity planning of many telecommunication networks including wireless communication networks and ad hoc networks and they are intensively studied in the literature, for references see, e.g.,
C.S. Kim Department of Industrial Engineering, Sangji University, Wonju, 220-702, Republic of Korea e-mail:
[email protected] V.V. Mushko · A.N. Dudin () Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Ave., Minsk, 220030, Belarus e-mail:
[email protected] V.V. Mushko e-mail:
[email protected]
308
Ann Oper Res (2012) 201:307–323
the books Artalejo and Comez-Corral (2008), Falin and Templeton (1997), survey paper Gomez-Corral (2006), and the bibliography in Artalejo (2010). The most general up to now multi-server retrial queue was studied in Breuer et al. (2002) where the arrival process was described by the Batch Markovian Arrival Process (BMAP), service times were assumed having PH (phase type) distribution and arbitrary dependence of the total retrial rate from the orbit on the number of customers in the orbit. The cases when the total retrial rate approaches to a finite constant or to infinity when the number of customers in an orbit increases were considered. In Breuer et al. (2002), stability and instability conditions for this model were derived and the numerically stable algorithm for computing the steady state distribution of the queueing system was elaborated. In Klimenok et al. (2007), the model of Breuer et al. (2002) was generalized to the case when the customer may leave the system forever without service after the each unsuccessful retrial, i.e., the customers are non-persistent. Expressions for the main performance measures were presented and an extensive numerical study of the model was implemented. In the paper Breuer et al. (2005), the results of Breuer et al. (2002) were applied to the problem of analyzing performance of hot spots in airports. This application showed shortcomings of the software based directly on the results from Breuer et al. (2002) if the service time has more general distribution than the exponential one. The main shortcomings are the long calculation time and high requirements to the RAM of a computer. This is evidently explained as follows. Investigation of the BMAP/PH/N retrial system in Breuer et al. (2002) was based on consideration of the state dependent continuous time multi-dimensional Markov chain whose components include the number of customers in the orbit, the number of busy servers, the current state of the continuous time Markov chain, which governs customers arrival, the current states of the continuous time Markov chains, which govern service process at each server of the system. Thus, the generator of this Markov chain has infinite dimension and, what may be even worse is, the dimension of the finite blocks of this generator, which describe the change of the number of customers in the orbit, is big. The main reason of the big size of a block is that if the number of busy servers is equal to N and the continuous time Markov chain, which governs service process in a server, has M states, then the total number of states of service processes in all N servers is equal to M N . Due to these shortcomings, the careful analysis of hot spots in airport presented in Breuer et al. (2005) is restricted to the case of one physical channel (a single TDMA frame in one radio frequency channel) divided into eight logical channels (slots) one of which is the management channel and seven channels are used for providing the service for customers. Analysis for more than one radio frequency channel based on the results from Breuer et al. (2002) is performed in Breuer et al. (2005) only under assumption that the holding time has an exponential distribution which is the very particular case of the PH distribution. However, as it is reported, e.g., in Pattavina and Parini (2005), assumption about the exponential distribution of holding times in mobile communication networks does not fit measurements made in real networks, while the another partial case of the PH distribution, the so called hyperexponential distribution, fits these measurements well. So, analysis of the BMAP/PH/N retrial queue in case of non-exponentially distributed holding time and the number of servers at least up to 21 is of a high practical importance. In this paper, we describe the algorithm which can be used for careful analysis of hot spots, and wireless networks in general, in the case of two and three radio frequency channels (up to 21 logical channels) with the PH distribution of holding time. A disadvantage of application of the phase type distributions to description of service time distribution in multi-server queues due to very high dimension of the underlying Markov chain is well known in the literature. However, as we already mentioned, the use
Ann Oper Res (2012) 201:307–323
309
of phase type distribution allows one to approximate many real-world service time requirements and currently PH is the popular descriptor of service time distribution. An elegant way to overcome the mentioned above disadvantage was offered in Ramaswami and Lucantoni (1985), Ramaswami (1985). Instead of keeping track on the states of the service process at each server separately it is enough to count how many servers have the respecting state of the underlying Markov chain. The profit from this manner of keeping information about the service process can be very essential when the number of states of the service underlying Markov chain is small. Say, if this number is equal to 2, then, conditional on that currently N servers are operating, one needs to keep track only on the number of servers in which service process is at the first phase, i.e., one needs to know only number in the set {0, . . . , N } instead of 2N possible combination of numbers 1 and 2 if one counts the states of the service process in all servers separately. Although the idea of Ramaswami and Lucantoni (1985), Ramaswami (1985) is interesting and well known for a long time, we can refer only to the paper Naoumov et al. (1996) were the idea was effectively applied to consideration of a queueing model with a finite buffer. It is well known that retrial queues are much more difficult subject of the research comparing to the queues with buffers. In our paper, we combine advantages of Ramaswami and Lucantoni (1985), Ramaswami (1985) relating to the effective description of the states of the service processes with advantages of a powerful and numerically stable procedure for computing the stationary distribution of the state inhomogeneous Markov chains (so called multi-dimensional asymptotically quasi-Toeplitz Markov chains—AQTMC) offered in Klimenok and Dudin (2006). The rest of the paper is organized as follows. In Sect. 2, the mathematical model is described. The process of the system states is defined in Sect. 3. In Sect. 4, ergodicity conditions are presented. Algorithm for computing the stationary distribution of the system states and the key performance measures is outlined in Sect. 5. In Sect. 6, some numerical results are presented. Section 7 concludes the paper.
2 Mathematical model We consider a multi-server queueing model having N independent and identical servers. Service times distribution is of PH type. It means the following. The service process is directed by the continuous time Markov process ηt , t ≥ 0. The state of this process at the service beginning epoch is defined according to the probabilistic row vector β = (β1 , . . . , βM ). Further transitions of the process ηt , t ≥ 0, are defined by the matrix S of dimension M × M. The diagonal entries of the matrix are negative and −Sη,η defines the parameter of the exponentially distributed sojourn time of the process in the state η, |Sη,η | < ∞, η = 1, M. The non-diagonal entries of the matrix S define the intensities of transition of the process ηt , t ≥ 0, within the state space {1, . . . , M}. The value − M η =1 Sη,η defines the intensity of the transition of the process ηt , t ≥ 0, from the state η into the absorbing state. The epoch of the transition of the process ηt , t ≥ 0, to the absorbing state defines the service completion epoch. Denote S0 = −Se. Here e is a column vector of appropriate size consisting of 1’s. It is assumed that all the entries of the column vector S0 are non-negative and at least one of them is positive. More information about the PH process, its properties and particular cases can be found in Neuts (1981). The primary customers arrive to the system according to a BMAP. We denote the directing process of the BMAP by νt , t ≥ 0. The state space of this irreducible continuous time Markov chain νt , t ≥ 0, is {0, 1, . . . , W }. The behavior of the BMAP is completely
310
Ann Oper Res (2012) 201:307–323
k characterized by the matrix generating function D(z) = ∞ k=0 Dk z , |z| < 1. The matrix Dk characterizes the intensities of transition of the process νt , t ≥ 0, which are accompanied by generating a batch of k customers, k ≥ 0. The matrix D(1) represents the generator of the process νt , t ≥ 0. The average arrival rate λ is defined as λ = θ D (1)e where θ is the invariant vector of the stationary distribution of the process νt , t ≥ 0. The vector θ is the unique solution to the system θ D(1) = 0, θ e = 1. Here 0 is the row vector of appropriate size consisting of 0’s. The intensity λb of batch arrivals is defined as λb = θ (−D0 )e. The variance −2 −1 v of the inter-arrival times is equal to 2λ−1 b θ (−D0 ) e − λb . The coefficient of variation 2 cvar of intervals between batch arrivals is given by (cvar ) = 2λb θ(−D0 )−1 e − 1, while the correlation coefficient ccor of intervals between successive batch arrivals is calculated by −1 ccor = λ−1 D(1) − D0 (−D0 )−1 e − λ−2 b θ (−D0 ) b /v. Such an arrival process was introduced as a versatile Markovian point process (VMPP) by M.F. Neuts in the 70s. The original development of the VMPP contained extensive notations; however these notations were simplified greatly in Lucantoni (1991) and ever since this process bears the name BMAP. The class of BMAPs includes many input flows considered previously, such as stationary Poisson (M), Erlangian (Ek ), Hyper-Markovian (HM), PhaseType (PH), Markov Modulated Poisson Process (MMPP). Generally speaking, the BMAP is correlated, so it is ideal to model correlated and bursty traffic in modern telecommunication networks. If the arriving batch of the primary customers meets several servers being idle, the primary customers occupy the corresponding number of the servers. If the number of the idle servers is insufficient (or all servers are busy) the rest of the batch (or all the batch) goes to the so called orbit. These customers are said to be repeated customers. Repeated customers try their luck later, until they will be served. Each customer staying in the orbit makes the repeated attempts in random intervals having length exponentially distributed with parameter α, α > 0, independently of the other customers. We assume that the repeated customers can be non-persistent. It means that after each unsuccessful retrial the customer leaves the system unserved with probability 1 − p (0 ≤ p ≤ 1) or comes back to the orbit with probability p. The orbit capacity is assumed to be unlimited.
3 Process of the system states Let • • • •
it , it ≥ 0, be the number of customers presenting in the orbit, nt , nt = 0, N , be the number of busy servers, νt , νt = 0, W , be the state of the directing process of the BMAP, (η) (η) (η) ht , be the number of servers at phase η, ht ∈ {0, . . . , nt }, η = 1, M, M η=1 ht = nt , nt = 0, N , at the epoch t , t ≥ 0.
(M) It is clear that the process ζt = {it , nt , νt , h(1) }, t ≥ 0, is the Markov chain. t , . . . , ht (1) , ν , h , . . . , h(M) }, t ≥ 0, is equal to J = Note that the dimensionality of the process {n t t t t N+M ¯ ¯ W M where W = W + 1. Note, that the dimensionality of the process, which describes the behavior of the finite components of the Markov chain in Breuer et al. (2002), is equal N+1 to W¯ MM−1−1 what may be essentially greater than J , see, e.g., Table 2 in Sect. 6. Denote the stationary state probabilities of this Markov chain by (M) = hM , p(i, n, ν, h1 , . . . , hM ) = lim P it = i, nt = n, νt = ν, h(1) t = h1 , . . . , ht t→∞
Ann Oper Res (2012) 201:307–323
311
i ≥ 0, ν = 0, W , hη ∈ {0, . . . , n}, η = 1, M,
M
hη = n, n = 0, N.
η=1
The sufficient condition for existence of these limits will be given below. Enumerate the states of the process ζt , t ≥ 0, in direct lexicographic order of components i, n, ν and then in reverse lexicographic order of components h1 , . . . , hM . Such kind of ordering allows to use the results of Ramaswami and Lucantoni (1985), Ramaswami (1985). Corresponding to this enumeration, let us compose the row vectors pi of the stationary state probabilities p(i, n, ν, h1 , . . . , hM ), i ≥ 0, corresponding to the number i, i ≥ 0, of customers in the orbit. Also form macro-vector p = (p0 , p1 , . . . pi , . . .), and denote by Q the infinitesimal generator of the Markov chain ζt , t ≥ 0. It is well-known that the vector p satisfies the system of equilibrium equations: pQ = 0,
pe = 1.
Lemma 1 The generator Q of the Markov chain ζt , t ≥ 0, has the following form: ⎛ ⎞ Q0,0 Q0,1 Q0,2 Q0,3 · · · ⎜ Q1,0 Q1,1 Q1,2 Q1,3 · · · ⎟ ⎜ ⎟ ⎜ Q2,1 Q2,2 Q2,3 · · · ⎟ Q=⎜ O ⎟ ⎜ O O Q3,2 Q3,3 · · · ⎟ ⎝ ⎠ .. .. .. .. .. . . . . . where the blocks Qi,j are calculated as follows: ⎧ O, n = 0, n − 2, n = 2, N ; ⎪ ⎪ ⎪ ⎪ ˜ ⊗ L (N, S), n = n − 1, n = 1, N ; I ⎨ W¯ N−n i ≥ 0, (Qi,i )n,n = D0 ⊕ An (N, S) + IW¯ ⊗ (n) ⎪ ⎪ ⎪ − iα(1 − pδ )I , n = n, n = 0, N; n,N W¯ K(n) ⎪ ⎩ Dn −n ⊗ Pn,n (β), n = n + 1, N, n = 0, N − 1, ⎧ ⎨IW¯ ⊗ Pn,n (β), n = n + 1, n = 0, N − 1; (Qi,i−1 )n,n = iα (1 − p)IW¯ K(N) , n = n = N ; i ≥ 1, ⎩ O, otherwise, ⎧ ⎨DN+k−n ⊗ Pn,N (β), n = N, n = 0, N − 1; i ≥ 0, k ≥ 1 (Qi,i+k )n,n = Dk ⊗ IK(N) , n = n = N ; ⎩ O, otherwise, where • • • • •
O is a zero matrix of the appropriate dimension, Ij is an identity matrix of dimension j , ⊗ is the symbol of Kronecker product of matrices, ⊕ is the symbol of Kronecker sum of matrices, the diagonal matrices that guarantee Qe = 0,
(n) , n = 0, N , are 0 O 1, n = N ; • S˜ = S0 S , δn,N = 0, n = N, • K(n) = n+M−1 0, N , , n = M−1
(1)
312
Ann Oper Res (2012) 201:307–323
Table 1 The intensities of the possible transitions of the Markov chain State 1
(i, n + k, ν , h +
k
l=1 e
(jl ) )
Transition rate
Condition
(Dk )ν,ν βj1 . . . βjk
n = 0, N − 1, k = 1, N − n
2
(i, n, ν , h)
(D0 )ν,ν
3
(i, n, ν, h)
(D0 )ν,ν +
4
(i, n, ν, h − e(η) + e(η ) )
5 6
M
η=1 hη Sη,η − iα(1 − pδn,N )
n = 0, N , ν = ν n = 0, N
Sη,η
n = 0, η = η
(i, n − 1, ν, h − e(η) )
(S0 )η
n = 0
(i − 1, n + 1, ν, h + e(η) )
iαβη
n = N , i = 0
7
(i − 1, n, ν, h)
iα(1 − p)
n = N , i = 0
8
(i + k, N, ν , h)
(DN +k−n )ν,ν βj1 . . . βjN−n
n = 0, N − 1, k ≥ 1
(Dk )ν,ν
n = N, k ≥ 1
• Pn,n (β) = Pn (β)Pn+1 (β) . . . Pn −1 (β), 0 ≤ n < n ≤ N , ˜ are described in Ramaswami and Lucantoni • matrices Pn (β), An (N, S), LN−n (N, S) (1985), Ramaswami (1985) along with the recursive algorithms for their calculation. Lemma 1 is proved by means of analysis of transitions of the Markov chain ζt , t ≥ 0, during the infinitesimal time interval. The possible transitions are as follows: 1. Transition of the BMAP directing process νt , t ≥ 0, from state ν to state ν with generating the k-size batch of customers that immediately start a service. 2. Transition of the BMAP directing process νt , t ≥ 0, from state ν to state ν without generating customers. 3. Exit from the state (i, n, ν, h) where h = (h1 , . . . , hM ). 4. A phase shift of one server from η to η without the service completion. 5. A service completion by a server being in phase η. 6. Retrial customer joins the phase η of service. 7. Retrial customer leaves the system. 8. The k-size batch of customers joins the orbit. The intensities of the corresponding transitions from the state (i, n, ν, h) to the resulting system states are given in Table 1 where e(j ) is the vector of dimension M filled with 0’s except the j -th element which is equal to 1. Taking into account transition rates presented in Table 1 and using the matrices Pn (β), ˜ introduced in Ramaswami and Lucantoni (1985), Ramaswami An (N, S), LN−n (N, S) (1985), we prove the statement of the Lemma 1.
4 Ergodicity conditions It is easy to see that the Markov chain ζt , t ≥ 0, describing the behavior of the system does not possess the state homogeneity property and so it does not belong to the class of so called M/G/1 type Markov chains extensively studied by M. Neuts, see Neuts (1989). But it can be verified that this Markov chain belongs to the class of multi-dimensional asymptotically quasi-Toeplitz Markov chains, see Breuer et al. (2002) and Klimenok and Dudin (2006). So, results from Breuer et al. (2002) and Klimenok and Dudin (2006) can be used to derive stability and instability conditions and to compute the ergodic distribution.
Ann Oper Res (2012) 201:307–323
313
Theorem 1 If p < 1 then the stationary distribution of the Markov chain ζt , t ≥ 0, exists for any set of parameters of the queueing system. If p = 1 then the stationary distribution of the Markov chain ζt , t ≥ 0, exists if inequality ρ = λ/μ¯ < 1
(2)
holds good, where the value μ¯ is given by formula ˜ μ¯ = yL0 (N, S)e
(3)
where, in turn, the row vector y is the unique solution to the system ˜ N−1,N (β) = 0, y AN (N, S) + (N) + L0 (N, S)P ye = 1.
(4)
The stationary distribution of the Markov chain ζt , t ≥ 0, does not exist if inequality ρ>1 holds good. Proof It follows from Klimenok and Dudin (2006) that to derive stability condition for the Markov chain ζt , t ≥ 0, we have to consider the discrete time Markov chain embedded at all epochs of transitions of Markov chain ζt (so called jump Markov chain) and then analyze its limiting behavior. It is well known that the one-step transition probability matrix P of jump Markov chain also has a structure like given by (1) and the blocks Pi,j of this Markov chain are expressed via the blocks Qi,j of the generator Q as follows: Pi,j = Ri−1 Qi,j ,
j ≥ i − 1, j = i, i > 0,
Pi,i = Ri−1 Qi,i + I,
i≥0
where the matrix Ri is the diagonal matrix with the positive diagonal entries defined by modulus of the diagonal entries of the matrix Qi,i . Then we compute the limits Yk = limi→∞ Pi,i+1−k , k ≥ 0. Stability condition isderived in k Klimenok and Dudin (2006) in terms of the matrix generating function Y (z) = ∞ k=0 Yk z , |z| < 1. If the matrix Y (1) is irreducible, the sufficient condition for ergodicity of the asymptotically quasi-Toeplitz Markov chain is given by the inequality xY (1)e < 1
(5)
where the vector x is the unique solution to the system xY (1) = x,
xe = 1.
(6)
Sufficient condition for non-ergodicity is given by the inequality xY (1)e > 1. If the matrix Y (1) is reducible, the sufficient condition for ergodicity of the asymptotically quasi-Toeplitz Markov chain is given in terms of irreducible blocks of the normal form of the reducible matrix Y (z) by the set of inequalities similar to (6). For more details see Klimenok and Dudin (2006).
314
Ann Oper Res (2012) 201:307–323
We will distinguish the cases when the customers in orbit are not absolutely persistent, i.e., probability p that the customer will return to the orbit after unsuccessful attempt is less than 1, and the case of absolutely persistent customers when p = 1. Consider first the case 0 ≤ p < 1. It can be verified that in this case Ri = C + iα Iˆ + iα(1 − p)I¯, C = − diag diag (D0 )ν,ν , ν = 0, W ⊕ (n) , n = 0, N , and the matrix generating function Y (z) has the following form: Y (z) = I˜β + I¯,
⎛
⎞ O IW¯ ⊗ P0,1 (β) O ··· O ⎜O ⎟ O O IW¯ ⊗ P1,2 (β) · · · ⎜ ⎟ ⎜ ⎟ . . . . .. .. .. .. I˜β = ⎜ .. ⎟, . ⎜ ⎟ ⎝O O O · · · IW¯ ⊗ PN−1,N (β) ⎠ O O O ··· O ⎞ ⎛ O ··· O O IW¯ K(0) ⎟ ⎜ O · · · O O I ¯ K(1) W ⎟ ⎜ ⎟ ⎜ . . . . . .. .. .. .. I¯ = I − Iˆ. Iˆ = ⎜ .. ⎟, ⎟ ⎜ ⎠ ⎝ O O O · · · IW¯ K(N−1) O O ··· O OW¯ K(N) We see that the function Y (z) does not depend on z, so inequality (5) holds good for any set of the system parameters. In the case p = 1 we have Ri = C + iα Iˆ, i ≥ 0, and it is possible to verify that Y (z) = I˜β + I¯z + I¯C −1 Q∗ + Q∗∗ (z) z where the matrix Q∗ and the matrix generating function Q∗∗ (z) are given by ⎧ O, n = 0, n − 2, n = 2, N ; ⎪ ⎪ ⎨ ˜ ∗ n = n − 1, n = 1, N; I ¯ ⊗ LN−n (N, S), Q n,n = W (n) ⎪ ⎪D0 ⊕ An (N, S) + IW¯ ⊗ , n = n, n = 0, N; ⎩ Dn −n ⊗ Pn,n (β), n = n + 1, N , n = 0, N − 1, ⎧∞ k−N+n ⊗ Pn,N (β), n = N, n = 0, N − 1; ⎪ ⎨k=N−n+1 Dk z ∗∗ ∞ k Q (z) n,n = n = n = N ; k=1 Dk z ⊗ IK(N) , ⎪ ⎩ O, otherwise. Such a structure allows us to use Theorem 6 from Klimenok and Dudin (2006) to derive the sufficient condition for ergodicity of the asymptotically quasi-Toeplitz Markov chain ζt , t ≥ 0. By this theorem, the sufficient condition is expressed in terms of block Y˜ (z) =
˜ zI + (C ∗ )−1 D(z)
˜ (C ∗ )−1 (IW¯ ⊗ L0 (N, S))z
IW¯ ⊗ PN−1,N (β)
O
Ann Oper Res (2012) 201:307–323
315
of the matrix Y (z) as
det zI − Y˜ (z) z=1 > 0
(7)
where C ∗ = − diag (D0 )ν,ν , ν = 0, W ⊕ (N) ,
˜ D(z) = zD(z) ⊕ z AN (N, S) + (N) .
Using the block structure of the matrix Y˜ (z) inequality (7) can be written in the following form:
−1 ¯ det C ∗ zW K(N−1) × det −z D(z) ⊕ AN (N, S) + (N) ˜ IW¯ ⊗ PN−1,N (β) > 0. − IW¯ ⊗ L0 (N, S) z=1
Taking into account that ˜ IW¯ ⊗ PN−1,N (β) D(1) ⊕ AN (N, S) + (N) + IW¯ ⊗ L0 (N, S) is a generator and det(C ∗ )−1 > 0 we reduce the last inequality to the form: det T (z) z=1 > 0
(8)
where ˜ IW¯ ⊗ PN−1,N (β) . T (z) = −z D(z) ⊕ AN (N, S) + (N) − IW¯ ⊗ L0 (N, S)
(9)
Following the proof of Corollary 1 in Klimenok and Dudin (2006) we can show that inequality (8) is equivalent to the following inequality xT (1)e < 0
(10)
where x is the unique solution of the system xT (1) = 0,
xe = 1.
(11)
Taking into account (9), inequality (6) and system (11) can be represented in the form x D(1) + D (1) ⊕ AN (N, S) + (N) e < 0, ˜ N−1,N (β) = 0, x D(1) ⊕ AN (N, S) + (N) + L0 (N, S)P
(12) xe = 1.
(13)
By direct substitution, it can be verified that solution x of the system (13) is given by x=θ ⊗y where the vector y is the unique solution to system (4) and that inequality (13) is equivalent to inequality (2). Theorem is proved.
316
Ann Oper Res (2012) 201:307–323
5 Algorithm for calculating the stationary distribution and the key performance measures The Markov chain ζt , t ≥ 0, belongs to the class of multi-dimensional asymptotically quasiToeplitz Markov chains. So, the effective and stable algorithm for computing the stationary probabilities, which is presented in Klimenok and Dudin (2006), can be applied. The algorithm is based on censoring technique, see, e.g., Kemeni et al. (1966). The algorithm consists of the following principal steps. 1. Calculate the matrix G as the minimal nonnegative solution of the matrix equation G = Y (G). This equation is the Neuts’ equation for the M/G/1 type Markov chain having Y (z) as the generating function of its transition probability matrices and iterative methods for its solution were extensively discussed in literature. 2. For appropriately chosen sufficiently large integer i0 , calculate the matrices Gi0 −1 , Gi0 −2 , . . . , G0 using the equation of the backward recursion Qi+1,i +
∞
Qi+1,n Gn−1 Gn−2 . . . Gi = O,
n=i+1
i = i0 − 1, i0 − 2, . . . , 0, with the boundary condition Gi = G, i ≥ i0 . 3. Calculate the matrices ¯ i,l = Qi,l + Q
∞
Qi,n Gn−1 Gn−2 . . . Gl ,
l ≥ i, i ≥ 0.
n=l+1
4. Calculate the matrices Φl using the recurrent formulae l−1 Φi Q¯ i,l (−Q¯ l,l )−1 , Φl = Q¯ 0,l +
l ≥ 1.
i=1
5. Calculate the vector p0 as the unique solution of the system ∞ ¯ p0 Q0,0 = 0, p0 e + Φl e = 1. l=1
6. Calculate the vectors pl as follows: pl = p0 Φl , l ≥ 1. As soon as the vectors pi , i ≥ 0, have been calculated, we are able to calculate different performance measures of the system. ∞ • average number Ls of busy servers is defined by Ls = N n=1 n[P (1)]n e, P (1) = i=0 pi . Here [P (1)]n is a part of the vector P (1) corresponding the states of Markov chain when there are n busy servers, n = 0, N ; • average number Lo of customers in the orbit is calculated by Lo = ∞ i=0 ipi e; • average number L of customers in the system is determined by L = Lo + Ls ; • probability P0 of an empty orbit is computed by P0 = p0 e; • probability Phit that an arbitrary customer reaches a server immediately upon arrival is defined by n N 1 (k − n)Dk ⊗ IK(N−n) e; P (1) N−n Phit = λ n=1 k=0
Ann Oper Res (2012) 201:307–323
317
Table 2 Comparison of computation time for calculating the steady state distribution N
Computation time
J
J in BDK
Computation time in BDK
1
6
0h0m0s
6
0h0m0s
2
12
0h0m1s
14
0h0m2s
3
20
0h0m5s
30
0h0m14s
4
30
0h0m13s
62
0h2m9s
5
42
0h0m37s
126
0h19m13s
6
56
0h1m22s
254
2h46m52s
7
72
0h2m52s
510
32h16m26s
• probability Ploss that an arbitrary customer will be lost is calculated by Ploss = 1 −
N 1 ˜ e. P (1) n IW¯ ⊗ LN−n (N, S) λ n=1
6 Numerical example We present results of two numerical experiments. The aim of the first experiment is to compare the running time on computer of the algorithm proposed in the current paper and algorithm presented in Breuer et al. (2002). Let the matrices Dk , k ≥ 0, which define the BMAP, are given by −7 0.000006 6.8909925 0.1090015 D0 = , D= , 0.000005 −0.3 0.2289013 0.0710937 ∗ Dk = Dq k−1 (1 − q)/ 1 − q k , k = 1, k ∗ , q = 0.8, k ∗ = 3. This BMAP has fundamental rate λ = 8.9633889. The intensity λb of batch arrivals is equal to 4.8386436. The squared coefficient of variation cvar = 10.34250142. The correlation coefficient ccor = 0.1. The individual intensity of retrial α is assumed to be equal to 90. The nominal service time distribution is characterized by irreducible representation 1.9918642 . The mean service rate is μ = 0.9959321. (β, S) where β = (1, 0), S = −1.9918642 0 −1.9918642 In the following examples, the matrix S is scaled to provide, for a given value of N , the trafλ = 0.6. fic intensity ρ = Nμ N+1 As it was mentioned above, the size J of the blocks of the generator is equal to W¯ 1−M 1−M N+M in Breuer et al. (2002) while this size in the current paper is equal to J = W¯ M . The size of the blocks essentially affects the running time of algorithms. For the numerical experiments, we exploited PC AMD Athlon (tm) 64 3700+, 2.21 GHz, RAM 2 GB, under Microsoft Windows XP. Table 2 contains comparison of computation time for calculating the steady state distribution for the number of servers from the set {1, . . . , 7} for the algorithm elaborated in this paper with algorithm given in the paper Breuer et al. (2002), referred to as BDK, along with information about the corresponding block sizes J . It is seen from Table 2 that for N = 7 computation time by means of algorithm from this paper is less than 3 minutes while this time for the algorithm from Breuer et al. (2002) is
318
Ann Oper Res (2012) 201:307–323
Table 3 Computation time for different values of N for the present algorithm
N
J
Computation time
7
72
0h2m52s
8
90
0h5m35s
9
110
0h10m17s
10
132
0h17m48s
11
156
0h29m13s
12
182
0h48m15s
13
210
1h17m51s
14
240
1h45m47s
15
272
2h43m33s
16
306
3h52m45s
17
342
5h21m16s
18
380
8h45m59s
19
420
12h26m18s
20
462
16h35m1s
21
506
21h58m24s
more than 32 hours. Huge difference in computation time is explained by the big difference in the size J of matrix blocks involving into the algorithm. For N > 7, the program based on the algorithm from Breuer et al. (2002) failed to reach the result. Computation times along to the sizes J for N from interval [7, 21] for the algorithm from this paper are given in Table 3. It is interesting to note that the block size J is equal to 510 for N = 7 and algorithm from Breuer et al. (2002). We have almost the same block size J (J = 506) in our algorithm for N = 21. But the computation time by our algorithm is equal to 21h58m24s versus computation time 32h16m26s for algorithm from Breuer et al. (2002). This fact is explained by more fast convergence of matrices Gi to the limiting matrix G in the algorithm from this paper. The purpose of the second experiment is to show dependence of some key performance measures of the system on probability p that a customer returns to the orbit after unsucλ . We will consider the average number cessful retrial and on the traffic intensity ρ = Nμ Lo of customers in the orbit, probability Phit that an arbitrary customer reaches a server immediately upon arrival, and probability Ploss that an arbitrary customer will be lost. Simultaneously, we illustrate the necessity of taking into account correlation in arrival process. Let the number of servers be N = 9. The individual retrial intensity α is assumed to be equal to 111. We consider three different arrival processes having the same fundamental rate λ = 11.126204. The arrival process coded as M is the stationary Poisson process. The arrival process coded as MX is the batch stationary Poisson process. It is defined by parameters D0 = −5, ∗ D = 5, Dk = Dq k−1 (1 − q)/(1 − q k ), k = 1, k ∗ , q = 0.8, k ∗ = 4. The intensity λb of batch arrivals is equal to 5. The processes M and MX have the coefficient of correlation of interarrival times ccor = 0. The BMAP input coded as BMAP is defined by the matrices D0 , Dk which are given by ⎛
⎞ −13.33652 0.58866 0.61739 −2.4469 0.423 ⎠ , D0 = ⎝ 0.69276 0.68235 0.41443 −1.63565
Ann Oper Res (2012) 201:307–323
319
Fig. 1 Dependence of the average number Lo of customers in the orbit on the probability p for different arrival processes
Fig. 2 Dependence of the probability Phit that an arbitrary customer reaches a server immediately upon arrival on the probability p for different arrival processes
⎛
⎞ 11.548588 0.363178 0.218704 0.865978 0.080862 ⎠ , D = ⎝ 0.3843 0.285213 0.04255 0.211111 ∗ Dk = Dq k−1 (1 − q)/ 1 − q k , k = 1, k ∗ , q = 0.8, k ∗ = 4. The intensity λb of batch arrivals is also equal to 5. The coefficient of correlation ccor = 0.1. Service time distribution is characterized by an irreducible representation (β, S) where 24.724898 . The mean rate of service is μ = 12.362449. β = (1, 0), S = −24.724898 0 −24.724898 Figures 1, 2, 3 illustrate the dependence of Lo , Phit and Ploss on the probability p for ρ = 0.8 for three different arrival processes. Figures 4, 5, 6 illustrate the dependence of Lo , Phit and Ploss on the probability p for the BMAP arrival process under three different values of the system load ρ. Figures 7, 8 illustrate the dependence of Lo and Phit on the system load ρ for p = 1 (absolutely persistent customers) for three different arrival processes. Figure 9 illustrates the dependence of probability Phit on the number N of servers for p = 1 (absolutely persistent customers) for three different arrival processes. This probability
320 Fig. 3 Dependence of probability Ploss that an arbitrary customer will be lost on the probability p for different arrival processes
Fig. 4 Dependence of the average number Lo of customers in the orbit on the probability p for different loads of the system
Fig. 5 Dependence of the probability Phit that an arbitrary customer reaches a server immediately upon arrival on the probability p different loads of the system
Ann Oper Res (2012) 201:307–323
Ann Oper Res (2012) 201:307–323 Fig. 6 Dependence of probability Ploss that an arbitrary customer will be lost on the probability p for different loads of the system
Fig. 7 Dependence of the average number Lo of customers in the orbit on the load ρ of the system for different arrival processes
Fig. 8 Dependence of the probability Phit that an arbitrary customer reaches a server immediately upon arrival on the load ρ of the system for different arrival processes
321
322
Ann Oper Res (2012) 201:307–323
Fig. 9 Dependence of the probability Phit that an arbitrary customer reaches a server immediately upon arrival on the number N of servers for different arrival processes
approaches value 1 when the number of servers increases. But the rate essentially depends on the correlation in the arrival process. From the figures, we can conclude the following: performance measures of the system become worse when the system load increases, the customers are more persistent and the arrival flow is correlated. Figures confirm that ignorance of a correlation and a batch arrival leads to too optimistic prediction of the system performance.
7 Conclusion Matrix notation for several independent Markov processes in parallel by V. Ramaswami in combination with algorithm for calculation of stationary distribution for Asymptotically Quasi-Toeplitz Markov chains from Klimenok and Dudin (2006) allows effectively calculate the performance measures of the BMAP/PH/N retrial queue. Results can be used for performance evaluation and capacity planning of any systems, which can be modeled by the multi-server retrial queues, wireless communication networks in particular. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2010-0003269). This paper was also supported by research funds of Sangji University in 2011.
References Artalejo, J. R. (2010). Accessible bibliography on retrial queues: progress in 2000–2009. Mathematical and Computer Modelling, 51, 1071–1081. Artalejo, J. R., & Comez-Corral, A. (2008). Retrial queueing systems: a computational approach. Berlin: Springer. Breuer, L., Dudin, A. N., & Klimenok, V. I. (2002). A retrial BMAP/PH/N system. Queueing Systems, 40, 433–457. Breuer, L., Klimenok, V. I., Birukov, A. A., Dudin, A. N., & Krieger, U. (2005). Modeling the access to a wireless network at hot spots. European Transactions on Telecommunications, 16, 309–316. Falin, G. I., & Templeton, J. G. C. (1997). Retrial queues. London: Chapman & Hall. Gomez-Corral, A. (2006). A bibliographical guide to the analysis of retrial queues through matrix analytic techniques. Annals of Operations Research, 141, 163–191. Kemeni, J. G., Snell, J. L., & Knapp, A. W. (1966). Denumerable Markov chains. New York: Van Nostrand.
Ann Oper Res (2012) 201:307–323
323
Klimenok, V. I., & Dudin, A. N. (2006). Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory. Queueing Systems, 54, 245–259. Klimenok, V. I., Orlovsky, D. S., & Dudin, A. N. (2007). A BMAP/PH/N system with impatient repeated calls. Asia-Pacific Journal of Operational Research, 24, 293–312. Lucantoni, D. M. (1991). New results on the single server queue with a batch Markovian arrival process. Communications in Statistics. Stochastic Models, 7, 1–46. Naoumov, V., Krieger, U. R., & Wagner, D. (1996). Analysis of a multiserver delay-loss system with a general Markovian arrival process. Lecture Notes in Pure and Applied Mathematics, 183, 43–66. Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models. Baltimore: Johns Hopkins University Press. Neuts, M. F. (1989). Structured stochastic matrices of M/G/1 type and their applications. New York: Dekker. Pattavina, A., & Parini, A. (2005). Modelling voice call inter-arrival and holding time distributions in mobile networks. In: The 19th international teletraffic congress. Performance challenges for efficient next generation networks, Aug.–Sept., 2005 (pp. 729–738). Ramaswami, V. (1985). Independent Markov process in parallel. Communications in Statistics. Stochastic Models, 1, 419–432. Ramaswami, V., & Lucantoni, D. (1985). Algorithm for the multi-server queue with phase-type service. Communications in Statistics. Stochastic Models, 1, 393–417.
