Analysis of a tandem queueing model with GI service time at the first queue
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1 nalysis of a tandem queueing model with GI service time at the first queue BSTRCT Zsolt Saffer Department of Telecommunications Budapest University of Technology and Economics, Budapest, Hungary safferzs@hitbmehu In this paper we consider the analysis of a tandem queueing model M/G/ -> /M/ In contrt to the vt majority of the previous literature on tandem queuing models we consider the ce with infinite buffers and with GI service time at the first queue The system can be described by an M/G/-type Markov process at the departure epochs of the first queue We derive the steady-state probability vector of the number of customers in the second queue and the vector factorial moments of the steady-state number of customers in the first queue, both at the above embedded epochs The steady-state mean sojourn time of the customers in the system is also obtained It turns out that the behavior of the second queue at the above embedded epochs is closely related to two G/M/- type Markov processes Utilizing the specific structure of one of these Markov processes enables to interpret the results This gives an insight into the behavior of this tandem queuing model and can be a be for developing approximations for it Tandem queuing model is commonly used in telecommunication networks and it is applied also to manufacturing systems Categories and Subject Descriptors G3 [Mathematics of Computing]: Probability and Statistics Queueing theory General Terms Theory, Performance Keywords queueing theory, tandem queueing model, M/G/-type Markov process, G/M/-type Markov process Wuyi Yue Department of Intelligence and Informatics Konan University Kobe , JPN yue@konan-uacjp INTRODUCTION Tandem queueing model consists of sequentially interconnected queues and it is a special ce of open queueing networks The tandem queueing model h several application are This model is a common model of telecommunication networks (see eg in []) nother application area of it is the manufacturing systems and their optimal control ([2]) Tandem queueing models with infinite and finite buffers h a huge literature The vt majority of the literature deals with the ce of Markovian service times, for which numerous approximation methods have been developed, like the one bed on the decomposition of the finite buffer system into single-server stations ([3], [4]) or the product-form approximation of model with infinite buffers ([5]) Tandem queueing model with deterministic service times is analyzed by Boxma and Resing in [6] very general model with GI service time at the first server and with finite intermediate buffer is investigated and proposed model of a call center by Dudin et al in [7] In contrt to the above references, in this paper we consider the ce with infinite buffers and with GI service time at the first queue This system can be described by an M/G/-type Markov chain embedded at the customer departure epochs of the first queue We apply standard probability-generating function (PGF) techniques and matrix analytic methods for the analysis of the system The contribution of this paper is the analysis of the tandem queueing model M/G/ -> /M/ We derive the governing equation of the system in terms of vector generating functions (vector GF) and matrix generating functions This relation characterizes the evolution of the system at the embedded epochs Bed on it we derive the steady-state probability vector of the number of customers at the second station and the vector factorial moments of the steady-state number of customers at the first station, both at the embedded epochs It turns out that at the embedded epochs the behavior of the second queue is closely related to two G/M/-type Markov processes Utilizing the specific structure of one of these Markov processes enables to interpret these results This gives an insight into the behavior of this tandem queuing model, which can serve a be for developing further approximations The the steady-state mean sojourn time of the customers in the system is also given The rest of the paper is organized follows In section 2 we introduce and explain the model In section 3 we deal with the structural characteristics of the model well we introduce several notations The steady-state number of
2 customers at the embedded epochs is investigated in section 4 In section 5 we derive the steady-state mean sojourn time of the customers in the system Finally the discussion of the steps of the computational procedure closes the paper in section 6 2 MODEL DESCRIPTION We consider a tandem queueing model consisting of two stations fter getting service at first station each customer moves to the second one Each of the stations h its own queue and one server unit Both queues have infinite buffers, ie all customers arriving to the system will be served at both stations The customers arriving to the first station according to Poisson process with rate λ The customer service times at the first station are independent and identically distributed B, B(s), e b, b (2) denote the service time rv, its Laplace-Stieljes transform (LST) and its first two moments, respectively Each customer departure from the server unit of the first station imply an immediate customer arrival to the second station The distribution of the customer service time at second station is exponential with parameter µ The service during the service period is work conserving and non-preemptive at both stations The customers during their staying at first station, ie during their arrival, waiting, service or departure, are called -customers Similarly, the customers during their staying at second station, ie during their arrival, waiting, service or departure, are called 2-customers ccording to this the naming of the customers moving from the first station to the second one changes from -customers to 2-customers We define D-epoch the epoch just after the customer departure from the first station and just before the arrival of the same customer to the second station We also define the -exhaustion epoch the D-epoch at the start of the idle period at first station The server utilization at the first station is ρ = λb We also use the notations ρ 2 = λ and µ ρ max 2 = We call this tandem queueing model also bµ M/G/ -> /M/ tandem queueing model We sume that the model is stable On this tandem queueing model we impose the following sumptions: The arrival rate and the mean customer service times at both stations are positive and finite, ie < λ <, < b < and < /µ < 2 The arrival process and each of the customer service times are mutually independent 3 The customers are served in First-In-First-Out (FIFO) order The stability condition of this tandem queueing model is well-known and it is given eg in [8]) The second station is stable if and only if the average arrival rate of the 2-customers, which is the same the average output rate of the -customers, does not exceed the constant service rate at second station However the average output rate of the -customers equals the arrival rate to the first station due to the stability of the first station Thus the stability condition of the second station is given λ < µ, which is equivalent with ρ 2 < When Y denotes a matrix, then [Y] j,l stands for its j, l-th element Similarly [y] j denotes the j-th element of vector y When Y(z), b z is a matrix generating function, Y denotes its value at z =, ie, Y = Y() b When by(z), z is a vector GF, y (k) denotes its k-th (k ) factorial moment, ie, y (k) = dk by(z) dz k z= and y denotes its value at z =, ie, y = by() We also use notation y (k) = y (k) e 3 STRUCTURL CHRCTERISTICS OF THE SYSTEM 3 Description of the system at D-epochs We describe the system and its evolution at D-epochs Let N (l) and N 2(l) be the number of customers at the first and at the second station at the l-th D-epoch, for l >, respectively The sequence {(N (l), N 2(l)), l > } is a bivariate homogeneous embedded Markov chain on the state space ({,, }, {,, }) We say that the chain is in state (i, k) when N (l) = i and N 2(l) = k We also say that the chain is in level i when N (l) = i Let p i,k (j, n) denote the probability of transition from state (i, k) to state (j, n) in this Markov chain, ie p i,k (j, n) = P {N (l + ) = j, N 2(l + ) = n N (l) = i, N 2(l) = k}, l, i, k, j, n In the following we consider the structure of the transition probability matrix of the above defined Markov chain We define γ u the probability of u 2-customer departures during the idle period of the first station Due to the Poisson arrival process this idle period is exponentially distributed Using it the probability γ u can be expressed γ u = Z t= (µt) u e µt λe λt dt u u! Furthermore we define the probability γ u ρ < and ρ 2 < () This can be explained follows The whole system is stable if and only if both stations are stable The first station is an M/G/ queue, therefore it is stable if and only if ρ < (see γu = γ k, u k=u We define the matrix Π
3 γ γ γ2 γ γ Π = γ 3 γ 2 γ γ B γ 3 γ 2 γ C We define β r,u the joint probability of r -customer arrivals and u 2-customer departures during a -customer service time The joint probability β r,u can be expressed β r,u = Z t= (λt) r r! e λt (µt) u e µt db(t) r, u u! Moreover we also define the joint probability β r,u β r,u = β r,k, r, u k=u We define the matrix Π d r βr, βr, β r, Π d r = β r,2 β r, β r, B β r,2 β r, β r, C The (k, n)-th element of matrix Π d r describes the joint probabilities of r -customer arrival and the k n transition in number of 2-customers during the first -customer service time after the idle period of the first station Next we define the matrix Π () r βr, β r, βr,2 β r, β r, Π + r = β r,3 β r,2 β r, β r, B β r,3 β r,2 β r, C The (k, n)-th element of matrix Π + r describes the joint probabilities of the k n transition in number of 2-customers during a -customer service time and r -customer arrival during that -customer service Let a r denote the probability of r -customer arrivals during a -customer service time for r, which can be expressed a r = β r,u = βr,, r (3) u= We define the special Toeplitz matrix Ω and the matrix E Ω =, B E = B Observe that, using also (3), these special matrices connect the matrices Π d r and Π + r Π () r = Π Π d r (2) The (k, n)-th element of matrix Π () r describes the joint probabilities of the k n transition in number of 2- customers from the -exhaustion epoch to the D-epoch at the end of the next -customer service and r -customer arrival during that -customer service It can be seen from their definitions that matrix Π h G/M/-type structure, while matrix Π d r is lower triangular Then it follows from (2) that Π () r h also G/M/-type structure We also define the matrix Π + r Π d r = Ω Π + r + a re, r (4) Furthermore (2) and (4) imply that Π () r = Π Ω Π + r + a rπ E, r (5) Let Π be the transition probability matrix of the above defined Markov chain in terms of block matrices, in which the (k, n)-th element of the (i, j)-th block matrix equals p i,k (j, n) for i, j, k, n It follows from the interpretation of matrices Π () r that the (, j)-th block matrix of Π, for j, equals Π () j
4 The (i, j)-th block matrix of Π, for i and j, describes the transitions, in which j i + -customer arrive during a -customer service time Hence the (i, j)-th block matrix of Π, for i and j, equals Π + j i+ Therefore the matrix Π is given Π () Π () Π () 2 Π () 3 Π + Π + Π + 2 Π + 3 Π = Π + Π + Π + 2 Π + Π + C Remark The transition matrix of the Markov chain embedded at D-epochs, Π, h M/G/-type structure on block level and its every block matrix h G/M/-type structure Setting z = in b Π + (z) gives matrix Π + It follows from the definitions of Π + r, b β u(z) and b β u(z) that the matrix Π + can be expressed β β β2 β β Π + = β 3 β 2 β β B β 3 β 2 β C Using several definitions gives that matrix Π + is stochtic, ie its every elements can be interpreted probability and Π + e = e, We define the partial matrix PGF b Π d (z) bπ d (z) = Π d rz r, z The definitions imply that the matrix b Π d (z) can be expressed bπ d (z) = bβ (z) bβ (z) β(z) b bβ 2 (z) β(z) b β(z) b bβ 3 (z) β2(z) b β(z) b β(z) b, C where e is an column vector having all elements equal to one 32 The evolution of the system We define the row vector q i, for i, by its k-th element [q i] k = lim l P {N (l) = i, N 2(l) = k}, for i, k The vector q i represents the steady-state joint probabilities of the number of customers at both stations for the ce when the number of -customers in the system is i We also define the corresponding steady-state vector GF bq(z) where bβ u(z) = β r,uz r, u, z, and bβ u(z) = βr,uz r, u, z bq(z) = q iz i, z i= Furthermore we define the row vector m the conditional steady-state probability vector of the number of customers at the second station at -exhaustion epoch, ie given that the number of -customers is The partial PGF matrix b Π () (z) is defined bπ () (z) = Π b Π d (z), z Furthermore we define the partial matrix PGF b Π + (z) This definition implies that the joint probability [q ] k equals [m] k P {the first station is idle at a D-epoch} for k This probability equals the reciprocal of the mean number of customers served during a service period at station This is ρ, since the first station h an M/G/ queue and its stochtic behavior is independent of the one of the second station Using it we get the following relationship between q and m bπ + (z) = Π + r z r, z q = ( ρ )m (6)
5 Theorem The governing equation of the stable M/G/ -> /M/ tandem queueing model satisfying sumptions - 3 is given in terms of bq(z) and m bq(z)(zi b Π + (z)) = ( ρ )m(z b Π () (z) b Π + (z)) (7) where I denotes the identity matrix Proof Writing the equilibrium equation of the Markov chain in partitioned form yields q i = q Π () i+ i + q k Π + i+ k, i (8) k= Multiplying both sides of (8) by z i, summing over i and rearrangement leads to q iz i = q Π () i z i + i= i= = q b Π () (z) + z i= q k Π + i+ k (9) z i i+ k= z k q k k= i+ k= z i+ k Π + i+ k Using the definitions of bq(z) and b Π + (z) in (9) gives 4 Determination of vector m We define matrix G, whose (k, l)-th elements is given the probability that starting from state (n +, k) in the Markov chain defined in subsection 3 the first state visited in level n is (n, l), n,, 2,, k, l Proposition In the stable M/G/ -> /M/ tandem queueing model satisfying sumptions - 3 the vector m is uniquely determined by the following system of linear equations:! mπ Ω G + E a rg r = m and me =, () where G can be computed from the following matrix equation: G = Π + r G r (2) Proof The M/G/-type structure of matrix Π ensures that matrix G can be computed by applying standard matrix-analytic method (see eg in []) This leads to Π + + Π + G + Π + r+g r+ = G (3) r= bq(z) = q b Π () (z) + z (bq(z) q) b Π + (z) () Rearranging (3) results the second statement of the theorem The sum of elements of vector m is, since it is a conditional probability vector, ie The statement of the theorem comes by applying (6) in () and rearranging it me = (4) Remark 2 The structure of relation (7) is similar to the one of the expression of the vector GF of the number of customers at customer departure epochs in the BMP/G/ queue (see eg in [9] In the M/G/ -> /M/ tandem queueing model the number of 2-customers play similar role the phe of BMP in the BMP/G/ queue 4 THE STEDY-STTE NUMBER OF CUSTOMERS T D-EPOCH In this section we derive several formul starting from the governing equation of the model The form of (7) implies that all of these formul can be given in terms of m Therefore first we give a method for determining the vector m Then the derivation of the steady-state probability vector of the number of 2-customers at D-epochs, q, follows In the lt part of the section we provide the expression of the vector factorial moments of the steady-state number of - customers at D-epochs, q (n) for n In order to establish a relation for m we use again a standard matrix-analytic argument The transition from a - exhaustion epoch to the next -exhaustion epoch brings vector m to itself Let us consider the transitions in the number of customers at the second station during the idle period and the succeeding first customer service time at the first station If the number of -customer arrivals during that first customer service time is r then the above transition can be described by matrix Π () r t the end of this transition there are r number of customers at the first station, ie the Markov chain is in level r Getting back from level r to level, ie to idle state of the first station, requires r consecutive level decrees Each of these level decrees can be described by matrix G Counting for all possible values of r results in a relation for vector m m Π () r G r = m (5)
6 pplying (5) in (5) yields mπ Ω Π + r G r + E a rg r! = m (6) Taking into consideration (4) and using (2) in (6) gives the first statement of the theorem 42 Steady-state probability vector of the number of 2-customers Lemma Let us consider the stochtic matrix Y and a Markov chain, whose transition probabilities are given by Y If the Markov chain irreducible and aperiodic then the matrix (I Y + ey) is nonsingular, where y is the steadystate probability vector of that Markov chain Proof If the Markov chain is irreducible and aperiodic then the probability of each state of the Markov chain after n-step transition always converge to a limit n, ie the steady-state probability vector exists and it is unique Furthermore the steady-state probability vector is a solution of the system of linear equation yy = y First we consider the ce, when the Markov chain is positive recurrent We apply an indirect argument In this ce y is a distribution and it is uniquely determined by the system of linear equations y(y I) = and ye =, where is the row vector having all elements equal zero Let us sume that matrix (I Y + ey) is singular, thus there exists a row vector x, which fulfills the homogenous system of linear equation x(i Y + ey) = Hence x(i Y + ey)e = also holds which together with y implies that xe =, ie the row sum of x is zero Using this it follows from the above homogenous system of linear equation that x(y I) = is also true This together with the uniqueness of y imply that x must be a constant multiplication of y In this ce the row sum of x must differ from zero since x and the row sum of y is not zero However this a contradiction it w shown that the row sum of x is zero, therefore matrix (I Y + ey) must be nonsingular Now consider the other ce when the Markov chain is null recurrent or transient Solving the system of linear equation yy = y by applying recursive substitution leads to y = y lim n (Y) n If the Markov chain is null recurrent or transient then the n-step transition probabilities to each state of the chain always converge to zero n, ie lim n (Y) n =, where is the zero matrix It follows that the only solution of the homogenous system of linear equation y(y I) = is y = and thus the matrix (Y I) must be nonsingular Due to y = the matrices (Y I) and (I Y + ey) are the same, which implies the statement for this ce Let π + be the steady-state probability vector of the Markov chain whose transition matrix is stochtic matrix Π + This steady-state probability vector exists and it is unique, since the above Markov chain h one irreducible cls of aperiodic states The former can be seen from the structure of matrix Π + and the later follows from β > which itself comes from model sumption Theorem 2 In the stable M/G/ -> /M/ tandem queueing model satisfying sumptions - 3 the expression of the steady-state probability vector q can be characterized The steady-state probability vector q is given a sum of two terms, in which the first term is the steady-state probability vector of the number of customers at the customer arrival epochs of the virtual G/M/ queue with interarrival time B and mean service time µ (characterized by matrix Π + ) and the second term represents the dependency on the difference of two G/M/-type structure matrices, (Π () Π + ) 2 The steady-state probability vector of the number of customers at the customer arrival epochs of the above virtual G/M/ queue, π +, can be given in dependency of the stability of that virtual G/M/ queue If ρ max 2 < then the above virtual G/M/ queue is stable and the probability vector π + represents a geometrical distribution, for which [π + ] k = ( σ +)σ k +, k, (7) where σ + is the only root of the equation in the region < σ + < σ + = e B(µ µσ +) (8) If ρ max 2 then the above virtual G/M/ queue is instable and the probability vector π + is given π + = (9) 3 Furthermore the expression of q is given by q = π + + ( ρ )m(π () Π + )(I Π + + eπ + ) (2) Proof We start from the governing equation of the system Setting z = in (7) yields q(i Π + ) = ( ρ )m(π () Π + ) (2) dding qeπ + to both sides of (2) yields q(i Π + + eπ + ) = qeπ + + ( ρ )m(π () Π + ) (22)
7 Utilizing the irreducibility and aperiodicity of the Markov chain whose transition matrix is stochtic matrix Π +, Lemma can be applied Hence the matrix (I Π + + eπ + ) is nonsingular Multiplying both sides of (22) by (I Π + + eπ + ) gives where an empty sum is and q (n) = d n e B(λ λz) ( ρ )( z) eb(λ λz) z dz n, (25) z= q = qeπ + (I Π + + eπ + ) + ( ρ )m(π () Π + )(I Π + + eπ + ) (23) Using π + (I Π + +eπ + ) = π + in (23) and applying qe = results in the third statement of the theorem The first statement of the theorem comes from the structure of (2) and from the definition of vector π + The second statement is well-known from the theory of standard G/M/ queue (see eg in [8]) If ρ max 2, then the Markov chain whose transition matrix is stochtic matrix Π + becomes instable, ie it becomes either null recurrent or transient, which can be checked by applying Foster s criterion on matrix Π + Remark 3 s the mean length of the idle period of station tends to then matrix Π () becomes equal to matrix Π + s a consequence of it the second term in (2) representing the dependency on the difference of these matrices vanishes 43 The vector factorial moments of the steady-state number of -customers Let q () denote the value of bq(z) at z =, ie q () = bq() Corollary In the stable M/G/ -> /M/ tandem queueing model satisfying sumptions - 3 the vector factorial moments of the steady-state number of -customers is given a sum of two terms, from which the first term is the multiplication of the n-th factorial moment of the number of customers in an M/G/ queue corresponding to the first station of the model and the steady-state probability vector of the number of customers at the customer arrival epochs of the virtual G/M/ queue with interarrival time B and mean service time µ (characterized by matrix Π+ ), π + Furthermore the expression of q (n), for n, is given by ie q (n) is the n-th derivative of the Pollaczek-Khinchine transform formula at z = Proof The corollary can be proved by taking the n-th derivative of (7) and applying the same technique used in the proof of theorem 2 5 STEDY-STTE MEN SOJOURN TIME The sojourn time of a -customer is defined the time elapsed from its arrival to station until its D-epoch Let Wl, denote the sojourn time of a -customer that arrives the l-th into station, l The sojourn time of a 2-customer is defined the time elapsed from the D-epoch just before its arrival to station 2 until its departure from station 2 Let Wl,2 denote the sojourn time of a 2-customer that arrives the l-th into station 2, l Furthermore we define the sojourn time of a customer in the system the time elapsed from its arrival to station until its departure from station 2 Let Wl denote the sojourn time of a customer that arrives the l-th into the system, l It follows that Wl can be given W l = W l, + W l,2, l (26) We define the steady-state mean sojourn times of the - customers (E[W ]), of the 2-customers (E[W2 ]) and the customers in the system (E[W ]) E[W ] = lim l E[Wl,], E[W2 ] = lim l E[Wl,2] and E[W ] = lim l E[Wl ], respectively Furthermore we define the conditional steadystate mean sojourn time of a 2-customer, given that the number of 2-customers at the D-epoch just before the arrival of that 2-customer is n, E[W 2 n] = lim l E[W l,2 N 2(l) = n], n Theorem 3 In the stable M/G/ -> /M/ tandem queueing model satisfying sumptions - 3 the steadystate mean sojourn time of the customers in the system is given q (n) = q (n) π + (24) + nq (n ) (Π +() I)(I Π + + eπ + )! n 2 n + q (k) Π +(n k) (I Π + + eπ + ) k k= + ( ρ )m(π ()(n) Π +(n) )(I Π + + eπ + ) + n( ρ )mπ ()(n ) (I Π + + eπ + ), E[W ] = b + λb(2) 2( ρ + ) µ + n 2 = n 2[q] n2! (27) Proof Taking lim l of the expectation of (26) and conditioning E[W2 ] on the number of 2-customers at the D- epoch just before the arrival of the tagged 2-customer leads to
8 E[W ] = E[W ] + [q] n2 E[W2 n 2] (28) n 2 = The steady-state mean sojourn time of the -customers is the steady-state mean sojourn time in the standard M/G/ queue, which is given (see eg [8]) E[W ] = b + λb(2) 2( ρ ) (29) Given that the number of 2-customers at a D-epoch is n 2, the conditional mean sojourn time in the second station is the sum of n 2 times the mean service time of a 2-customer and the service time of the actually arriving 2- customer This is because the remaining service time of the 2-customer h also exponential distribution due to the exponential service time at station 2 Therefore E[W2 n 2] can be given by E[W2 n 2] = n2 +, n (3) µ The theorem comes by applying (29) and (3) in (28) and rearranging it Remark 4 pplying Little s law in (27) results in the expression of the steady-state mean total number of customers, ie the sum of the mean number of -customers and 2-customers, in the system, E[Q], E[Q] = ρ + λ2 b (2) 2( ρ ) + ρ2 + n 2 = n 2[q] n2! (3) Note that the steady-state total number of customers differs from the steady-state total number of customers at D-epochs, since the number of 2-customers at an arbitrary epoch and at D-epochs are different This difference is also reflected in the lt term of (3) 6 THE COMPUTTIONL PROCEDURE To keep the computation of the infinite dimensional vectors and matrices tractable we apply an upper limit on the number of 2-customers in the system, ie k, and an upper limit on the number of -customer arrivals during a -customer service time, ie r R This results in finite number of elements in these vectors and matrices well finite number of matrices Π + r and Π () r, which results in finite number of required operations among others in computing matrix G from (2) and vector m from () The proper value of and R depends on the required precision and can be determined in an iterative manner until the difference of consecutive values of eg the probabilities [q] k, for every k, becomes less than the specified error The detailed investigation of this and other existing truncation methods and the evaluation of their impact to the computational procedure is out of scope of this work and it could be a topic of future research Taking into account the above truncation consideration in the computational steps of the steady-state mean sojourn time of this model, they can be summarized follows: Determination of σ +, the only root of the equation (8) in the region < σ + < 2 Calculation of matrix G by means of successive substitution in the nonlinear matrix equation equation (2) starting with G = 3 Computation of vector m from the system of linear equations () 4 Calculation of the steady-state probability vector q by applying (7) and (2) 5 Calculation of the steady-state mean sojourn time of the customers in the system, E[W ], by applying (27) 7 REFERENCES [] D Bertsek and R Gallager Data Networks (2nd Edition) Prentice Hall, 99 [2] D G Pandelis Optimal control of flexible servers in two tandem queues with operating costs Probability in the Engineering and Informational Sciences, 22:7 3, 28 [3] Heindl Decomposition of general tandem queueing networks with mmpp input Performance Evaluation, 44 [4] M van Vuuren, I J B F dan, and E Resing-Ssenb Performance analysis of multi-server tandem queues with finite buffers and blocking OR Spektrum, 27:35 339, 25 [5] G Cale, P G Harrison, and M Vigliotti Product-Form pproximation of Tandem Queues via Matrix Geometric Methods In 6th International Workshop on the Numerical Solution of Markov Chains (NSMC 2), September 2 [6] O J Boxma and J C Resing Tandem queues with deterministic service times nnals of Operations Research, 49 [7] N Dudin, C S Kim, V I Klimenok, and O S Taramin dual tandem queueing system with a finite intermediate buffer and cross traffic In 5th International Conference on Queueing Theory and Network pplications (QTN 2) Beijing, jul 2 [8] L Kleinrock Queuing Systems Vol I: Theory John Wiley, 975 [9] D L Lucantoni New results on the single server queue with a batch markovian arrival process Stochtic Models, 7: 46, 99 [] G Latouche and V Ramwami Introduction to Matrix Geometric Methods in Stochtic Modeling S-SIM Series on Statistics and pplied Probability SIM, Philadelphia, P, 999
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