Tandem queueing system with infinite and finite ... - Semantic Scholar

European Journal of Operational Research 235 (2014) 170–179

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution Chesoong Kim a,⇑, Alexander Dudin b, Olga Dudina b, Sergey Dudin b a b

Sangji University, Wonju, Kangwon 220-702, Republic of Korea Belarusian State University, 4, Nezavisimosti Ave., Minsk 220030, Belarus

a r t i c l e

i n f o

Article history: Received 17 July 2013 Accepted 6 December 2013 Available online 15 December 2013 Keywords: Queueing Tandem queueing system Marked Markovian arrival process Phase-type distribution Laplace–Stieltjes transform

a b s t r a c t A tandem queueing system with infinite and finite intermediate buffers, heterogeneous customers and generalized phase-type service time distribution at the second stage is investigated. The first stage of the tandem has a finite number of servers without buffer. The second stage consists of an infinite and a finite buffers and a finite number of servers. The arrival flow of customers is described by a Marked Markovian arrival process. Type 1 customers arrive to the first stage while type 2 customers arrive to the second stage directly. The service time at the first stage has an exponential distribution. The service times of type 1 and type 2 customers at the second stage have a phase-type distribution with different parameters. During a waiting period in the intermediate buffer, type 1 customers can be impatient and leave the system. The ergodicity condition and the steady-state distribution of the system states are analyzed. Some key performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution of type 2 customers is derived. Numerical examples are presented. Crown Copyright Ó 2013 Published by Elsevier B.V. All rights reserved.

1. Introduction Tandem queueing systems are very important part of the queueing theory that takes into account the possibility that a customer may need service from several sequentially arranged servers. The overwhelming majority of the results in the theory of tandem queues is obtained for systems with stationary Poisson arrival process and exponential service time distribution. Tandem queueing systems with correlated arrival flows were investigated not so extensively. For background information and overview of the present state of the art in the study of tandem queueing systems with correlated arrival flows and phase-type service time distribution see, e.g., Gomez-Corral (2002a, 2002b), Klimenok, Breuer, Tsarenkov, and Dudin (2005), Gomez-Corral and Martos (2006), Klimenok, Kim, Tsarenkov, Breuer, and Dudin (2007), Kim, Park, Dudin, Klimenok, and Tsarenkov (2010), and Kim, Dudin, Dudin, and Dudina (2013a). The papers Gomez-Corral (2002a, 2002b) and Gomez-Corral and Martos (2006) are devoted to the MAP=PH=1 ! =G=1 system with blocking. The tandem queue BMAP=G=1=N ! =PH=1=M with losses is considered in Klimenok et al. (2005). The tandem-queue of the BMAP=G=1 ! =PH=1=M type with losses and feedback is investigated in Klimenok et al. (2007). Analysis of the BMAP=G=1 ! =PH=1=M tandem queue

⇑ Corresponding author. Tel.: +82 33 730 0464. E-mail addresses: [email protected] (C. Kim), [email protected] (A. Dudin), [email protected] (O. Dudina), [email protected] (S. Dudin).

with retrials and losses is provided in Kim et al. (2010). In Kim et al. (2013a), a tandem queueing system with more than one server in each stage, Markovian arrival flow, and phase-type service time distribution at the second stage is considered. In this paper, we analyze a tandem queueing system with two types of customers. Type 1 customers arrive to the first stage that represents a multi-server queue without buffer. After service completion at the first stage, type 1 customers leave the system or move to the second stage. Type 2 customers arrive to the second stage directly. The second stage is a multi-server queue with a finite buffer for type 1 customers and an infinite buffer for type 2 customers. Type 1 customers have non-preemptive priority over type 2 customers. They can be impatient and leave the intermediate buffer without service. In order to study the effect of variation and correlation within the arrival process as well as the correlation of arrivals of different types of customers, we consider a Marked Markovian arrival process ðMMAPÞ to model customer arrivals. Additionally, we assume that the service time of different types customers has a phase-type ðPHÞ distribution with different parameters. This allows us to take into account the variance of the service time. Note that using a phase-type distribution significantly increases the dimension of the system state space and complicates the investigation of the system. To overcome these difficulties to some extent, we use the method proposed by Ramaswami (1985) and Ramaswami and Lucantoni (1985), for reducing the dimension of the state spaces of systems with a phase-type service time distribution,

0377-2217/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2013.12.012

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and consider a generalized phase-type distribution of the service times at the second stage instead of considering two separate phase-type distributions for each type of customers, that greatly facilitates the investigation of the system. The queueing system under consideration is quite general and can be applied for modeling many real-world systems. Let us introduce several examples of such systems: (i) Contact-center with an IVR (Interactive Voice Response). Type 1 customers can be interpreted as calls and type 2 customers are the text requests (e’mails). The first stage server represents an IVR and the second stage server is a contact center operator. E’mail requests arrive to the second stage directly and are always patient. Callers are firstly serviced by an IVR. If the caller cannot solve his (her) problem by using an IVR, he (she) can request to connect with an operator and move to the second stage. Type 1 customers can be impatient. The service times of e’mail requests and calls are different. (ii) Manufacturing system. The system processes two types of details. Type 1 details need some preprocessing while type 2 details do not require such a preprocessing. After the preprocessing, type 1 details should be processed during a limited time (e.g., if the preprocessing includes heating of details, type 1 details should be processed before they become cool). (iii) Database. The system processes queries from external users, who need preliminary identification and are non-patient, along with the requests from the internal users to retrieve or update information. (iv) Information transmission system. More important information should be encrypted prior transmission and transmitted more urgent than some less important information which is sent without encryption. (v) Medical system. Emergency surgery center has several rooms for anesthesiology and operation theatre. Depending on the type of injury, some arriving patients need more urgent treatment after a preliminary general anesthetic, while others may wait longer time and do not need general anesthetic at all. The paper is organized as follows. In Section 2, the mathematical model is described. In Section 3, the process of the system states is considered. The ergodicity condition and the stationary state distribution are analyzed in Section 4. The expressions for the main performance measures of the system are given in Section 4. The Laplace–Stieltjes transform of the sojourn time distribution of an arbitrary type 2 customer is presented in Section 5. Section 6 contains numerical examples. Section 7 concludes the paper.

2. Mathematical model We consider a tandem queueing system with two types of customers and two intermediate buffers. The structure of the system under study is presented in Fig. 1. Customers arrive to the system according to the MMAP. The customers in the MMAP are heterogeneous and have different types. The arrival of customers is directed by the stochastic process mt ; t P 0, which is an irreducible continuous-time Markov chain with the state space f0; 1; . . . ; Wg. The sojourn time of this chain in the state m is an exponentially distributed with the positive finite parameter kðmÞ . When the sojourn time in the state m expires, with ð0Þ probability pm;m0 the process mt jumps to the state m0 without generðlÞ ation of a customer, m; m0 ¼ 0; W; m – m0 , and with probability pm;m0 0 the process mt jumps to the state m with a generation of type l

customer, l ¼ 1; 2; m; m0 ¼ 0; W. The notation m ¼ 0; W means that the parameter m takes values in the set f0; 1; . . . ; Wg. The behavior of the MMAP is completely characterized by the matrices D0 , D1 ; l ¼ 1; 2, defined by the entries ðD1 Þm;m0 ¼ kðmÞ pm;m0 , ðlÞ

ðlÞ

ðlÞ

m; m0 ¼ 0; W, l ¼ 1; 2, and ðD0 Þm;m ¼ kðmÞ , m ¼ 0; W, ðD0 Þm;m0 ¼ kðmÞ pð0Þ m;m0 , ð2Þ m; m0 ¼ 0; W, m – m0 . The matrix Dð1Þ ¼ D0 þ Dð1Þ þ D represents 1 1 the generator of the process mt ; t P 0. The average total arrival intensity k is defined by   ð1Þ ð2Þ k ¼ h D1 þ D1 e where h is the invariant vector of the stationary distribution of the Markov chain mt ; t P 0. The vector h is the unique solution to the system hDð1Þ ¼ 0, he ¼ 1. Hereinafter e denotes a column vector consisting of 1’s, and 0 is a zero row vector. The average arrival intensity kl of type l customers is ðlÞ

defined by kl ¼ h D1 e, l ¼ 1; 2. The squared integral (without differentiating the types of customers) coefficient of variation cv ar of the intervals between customer arrivals is defined by cv ar ¼ 2khðD0 Þ1 e  1. The ðlÞ squared coefficient of variation cv ar of inter-arrival times of type l customers is defined by

 1 ðl0 Þ cðlÞ e  1; v ar ¼ 2kl h D0  D1

0

0

l – l; l ; l ¼ 1; 2:

The integral coefficient of correlation ccor of two successive intervals between arrivals is defined by

ccor ¼ ðkhðD0 Þ1 ðDð1Þ  D0 ÞðD0 Þ1 e  1Þ=cv ar : ðlÞ

The coefficient of correlation ccor of two successive intervals between type l customers’ arrivals is computed by

   1  1 ðl0 Þ ðlÞ ðl0 Þ =cðlÞ cðlÞ ¼ k h D þ D D D þ D e  1 0 0 l cor v ar ; 1 1 1 0

0

l – l; l ; l ¼ 1; 2: We assume that type 1 customers arrive to the first stage, while type 2 customers arrive to the second stage directly, not entering the first stage. The first stage is described by an R-server queueing system without buffer. The service time for each server at this stage has an exponential distribution with the parameter l. After receiving service at the first stage, type 1 customer proceeds to the second stage of the tandem with probability q, 0 6 q 6 1, or leaves the system forever (is lost) with the complementary probability. The second stage represents an N-server queue with a finite buffer of capacity K for type 1 customers (buffer 1) and an infinite buffer for type 2 customers (buffer 2). If there is a free server at the second stage during an arbitrary customer arrival epoch, the customer occupies this server immediately. If all servers are busy and buffer 1 is not full during an arbitrary type 1 customer arrival epoch, the customer is admitted to this buffer. Otherwise, type 1 customer leaves the system forever (is lost). If all servers at the second stage are busy during a type 2 customer arrival epoch, this customer moves to buffer 2.

Fig. 1. Structure of the system.

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We suggest that type 1 customers have priority, i.e., if there is at least one type 1 customer in buffer 1 during the service completion epoch, the next service will be provided to type 1 customer. Otherwise, type 2 customer is taken for service if he (she) is presenting in buffer 2. For customers of the same type the service discipline is First-In-First-Out. At the second stage, type 1 customers can be impatient, i.e., the customer leaves buffer 1 after an exponentially distributed time described by the parameter a, 0 < a < 1, after arrival, due to a lack of service. The service time of type l customer for the second stage server has a phase-type distribution PHl with an irreducible representation ðbl ; Sl Þ; l ¼ 1; 2. This service time can be interpreted as the time ðlÞ until the underlying Markov process mt ; t P 0, with the finite state space f1; . . . ; M l ; M l þ 1g reaches the single absorbing state M l þ 1 conditioned on the fact that the initial state of this process is selected among the states f1; . . . ; M l g according to the probabilistic row vector bl ; l ¼ 1; 2. The transition rates of the process ðlÞ mt within the set f1; . . . ; M l g are defined by the sub-generator Sl and the transition rates into the absorbing state (what leads to service completion) are given by the entries of the column vector ðlÞ S 0 ¼ Sl e; l ¼ 1; 2. Note that the representation ðbl ; Sl Þ is irreducðlÞ ible if the matrix Sl þ S 0 bl is irreducible, l ¼ 1; 2. The service time distribution function has the form R1 Al ðxÞ ¼ 1  bl eSl x e, the Laplace–Stieltjes transform ðLSTÞ 0 esx ðlÞ

dAl ðxÞ of this distribution is bl ðsI  Sl Þ1 S 0 , Re s > 0; l ¼ 1; 2. The mean service time of type l customer at the second stage is calcuðlÞ

lated by b1 ¼ bl ðSl Þ1 e; l ¼ 1; 2. The squared coefficient of varia 2 ðlÞ ðlÞ ðlÞ tion of type l customer is given by cv ar ¼ b2 = b1  1 where ðlÞ

b2 ¼ 2bl ðSl Þ2 e. 3. The process of system states Let us note that the queueing system under consideration is complicated for analysis. So, the task of the optimal choice of the process of the system states is extremely important. The behavior of the queueing system under study can be described in terms of the following regular irreducible continuoustime Markov chain:

n o ð1Þ ð1Þ ðn Þ ð1Þ ðminfit ;Ngnt Þ nt ¼ it ; r t ; kt ; nt ; mt ; xt ; . . . ; xt t ; ft ; . . . ; ft ;

t P 0;

where it is the number of customers at the second stage, it P 0, r t is the number of customers at the first stage, r t ¼ 0; R, kt is the number of type 1 customers in buffer 1, kt ¼ 0; maxf0; minfit  N; Kgg, nt is

mt is mt ¼ 0; W, xtðmÞ is

the number of type 1 customers in service, nt ¼ 0; minfit ; Ngg, the state of the underlying process of the MMAP,

the state of mth server that serves type 1 customer, m ¼ 1; nt , ðmÞ t

x

¼ 1; M1 ,

ðlÞ ft

is the state of lth server that serves type 2 cusðlÞ nt , f t

tomer, l ¼ 1; minfit ; Ng  ¼ 1; M2 , during the epoch t; t P 0. ð1Þ It is evident that the Markov chain nt has a huge dimension of the state space even for small dimensions of underlying MMAP and PH processes. Computation of the performance measures of the queueing system under consideration even on advanced computer seems to be extremely difficult task. To reduce the dimension of the ð1Þ nt

stochastic process we use the method proposed by Ramaswami (1985) and Ramaswami and Lucantoni (1985), for reducing the dimension of the state space of systems with a phase-type service time distribution. The main idea of this method is the following. Inð1Þ

ðnt Þ

stead of the consideration of the components xt ; . . . ; xt

and

ð1Þ ðminfit ;Ngnt Þ , it is proposed to consider the components ft ; . . . ; ft ð1Þ ðM 1 Þ ð1Þ ðM Þ and ft ; . . . ; ft 2 , which mean the following: t ;...; t

x

x

ðmÞ

 xt is the number of servers at the phase m of service of type 1 P 1 ðmÞ ðmÞ customers, m ¼ 1; M 1 ; xt ¼ 0; nt ; M ¼ nt ; m¼1 xt ðlÞ  ft is the number of servers at the phase l of service of type 2 customers, PM2 ðlÞ ðlÞ l ¼ 1; M 2 ; ft ¼ 0; minfit  nt ; Ng; l¼1 ft ¼ minfit  nt ; Ng. Then to reduce the dimension of the process of system states inð1Þ stead of the Markov chain nt we should consider the Markov chain

n o ð2Þ ð1Þ ðM Þ ð1Þ ðM Þ nt ¼ it ; r t ; kt ; nt ; mt ; xt ; . . . ; xt 1 ; ft ; . . . ; ft 2 ;

t P 0:

ð2Þ

Despite on the fact that the Markov chain nt has more suitable dimension of the state space for investigation in comparison to ð1Þ Markov chain nt , its complicated structure is not conducive to the analysis. In order to greatly simplify the investigation of the queue under study, instead of considering the service times of type 1 and type 2 customers we propose to consider a generalized processing time having distribution which we call as a generalized phase-type distribution with an irreducible representation ðbð1Þ ; bð2Þ ; SÞ. The time having such a distribution can be interpreted as the time until the underlying Markov process gt ; t P 0, with the finite state space f1; . . . ; M; M þ 1g, where M ¼ M 1 þ M 2 , reaches the single absorbing state M þ 1. The initial state of this process is selected among the states f1; . . . ; Mg depending on the type of a customer who is chosen for service. If an arbitrary type 1 customer is chosen for service, the initial state of this process is selected according to the probabilistic row vector bð1Þ ¼ ðb1 ; 0M2 Þ and, if type 2 customer is chosen for the service, the initial state is selected according to the probabilistic row vector bð2Þ ¼ ð0M1 ; b2 Þ. The transition rates of the process gt within the set f1; . . . ; Mg are defined   S1 O by the sub-generator S ¼ and the transition rates into O S2 the absorbing state (which lead to the service completion) are given by the entries of the column vector S 0 ¼ Se. If we use a generalized phase-type distribution, we do not consider the service process of each type of customers separately. Using of a generalized phase-type distribution instead of consideration of two separate phase-type distributions does not change the dimension of the state space of the Markov chain, but allows us to eliminate from the consideration of the component nt (the number of type 1 customers in service) that greatly simplifies the investigation of the system under study and allows us to construct the generator, obtain the ergodicity condition, and find sojourn time distribution more or less easily. ð2Þ So, instead of the Markov chain nt we assume that the behavior of the system under study is described in terms of the regular irreducible continuous-time Markov chain ð1Þ

ðMÞ

nt ¼ fit ; r t ; kt ; mt ; gt ; . . . ; gt g;

t P 0;

ðmÞ t

where g is the number of servers at the phase m of generalized P ðmÞ ðmÞ service, m ¼ 1; M, gt ¼ 0; minfit ; Ng, M ¼ minfit ; Ng, during m¼1 gt the epoch t, t P 0. 4. Ergodicity condition and stationary probabilities For further use throughout this paper, we introduce the following notation:  I is the identity matrix;  and  indicate the symbols of Kronecker sum and product of matrices, respectively;  W ¼ W þ 1, R ¼ R þ 1, K ¼ K þ 1, K i ¼ maxf0; minfi  N; Kgg þ1, iP 0;  iþM1  Ti ¼ , i ¼ 0; N; ¼ ðiþM1Þ! i!ðM1Þ! M1

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 C l ¼ diagf0; 1; . . . ; l  1g;  I is the square matrix of size K with all zero entries except the entry ðIÞ0;0 ¼ 1; b  I l , l ¼ R; K, is the square matrix of size l with all zero entries except the entry ðbI l Þ ¼ 1; l1;l1



Iþ l ,

l ¼ R; K, is the square matrix of size l with all zero entries   except the entries Iþ l i;iþ1 , i ¼ 0; l  2, which are equal to 1;

Q i;i1 ¼ IRW  LNi ðN; ~SÞ; 1 6 i 6 N;   Q i;i1 ¼ IR  EiNþ1  IW  L1 þ EiNþ1  IW  L2 þ aC iNþ1 EiNþ1  IWT N ; N < i 6 N þ K;   Q i;i1 ¼ Q 0 ¼ IR  IK  IW  L1 þ I  IW  L2 þ aC K IK  IWT N ; C R IR C R IR

ð1Þ

Q i;iþ1 ¼ ql

 IW  Pi ðb Þ þ IR 

E iNþ1  D1  IT N ;  EþiNþ1  IWT N þ IR  e

 El , l ¼ 2; K þ 1, is the matrix of size l  ðl  1Þ with all zero entries except the entry ðEl Þ0;0 ¼ 1;

where

l ¼ 2; K þ 1, is the matrix of size l  ðl  1Þ with all zero   entries except the entries E l i;i1 , i ¼ 1; l  1, which are equal to 1;  e E l , l ¼ 1; K, is the matrix of size l  ðl þ 1Þ with all zero entries except the entries ð e E l Þi;i , i ¼ 0; l  1, which are equal to 1.



Let us enumerate the states of the Markov chain nt , t P 0, in the reverse lexicographic order of the components gð1Þ ; . . . ; gðMÞ and in the direct lexicographic order of the components m; k and refer to ði; rÞ as macro-state consisting of WK i T minfi;Ng states. Let Q be the generator of the Markov chain nt ; t P 0, consisting of the blocks Q i;j , which, in turn, consist of the matrices ðQ i;j Þr;r0 of the transition rates of this chain, from the macro-state ði; rÞ to the macro-state ðj; r0 Þ; r, r 0 ¼ 0; R. The diagonal entries of the matrices Q i;i are negative, and the modulus of the diagonal entry of the blocks ðQ i;i Þr;r defines the total intensity of leaving the corresponding state of the Markov chain nt , t P 0. Analysing all transitions of the Markov chain nt , t P 0, during an interval of an infinitesimal length and rewriting the intensities of these transitions in the block matrix form we obtain the following result. Lemma 1. The infinitesimal generator Q ¼ ðQ i;j Þi;jP0 of the Markov chain nt , t P 0, has a block-tridiagonal structure:

0

Q 0;0 Q 0;1 O O B B Q 1;0 Q 1;1 Q 1;2 O B B B O Q 2;1 Q 2;2 Q 2;3 B B . .. .. .. B . B . . . . Q ¼B B O O O B O B B O O O O B B B O O O O B @ .. .. .. . . ... .

...

O

O

O

...

O

O

O

...

O

O

O

..

.. .

.. .

.. .

.

. . . Q NþK;NþK1 Q 1 Q 2 ...

O

Q0 Q1

... .. .

O .. .

O .. .

Q0 .. .

O

O

...

1

C O O ...C C C O O ...C C C .. .. C . ...C . C: C O O ...C C Q2 O . . . C C C Q1 Q2 . . . C C A .. .. .. . . .

The non-zero blocks Q i;j ; i; j P 0, have the following form:

Q i;i ¼ lC R  IK i WT minfi;Ng þ IRK i  D0  IT minfi;Ng þ IRK i W   ð1Þ  ðAminfi;Ng ðN; SÞ þ Dðminfi;NgÞ Þ þ bI R þ IþR  IK i  D1  IT minfi;Ng þ ð1  qÞlC R IR  IK i WT minfi;Ng  aIR  C K i  IWT minfi;Ng ; 0 6 i < N þ K; Q i;i ¼ Q 1 ¼ lC R  IKWT N þ IRK  D0  IT N þ IRKW  ðAN ðN; SÞ þ DðNÞ Þ   ð1Þ þ bI R þ IþR  IK  D1  IT N þ ð1  qÞlC R IR  IKWT N  aIR  C K  IWT N þ qlC R IR  bI K  IWT N ;

i P N þ K;

 Pi ðb Þ;

C R IR



IþK

 IWT N þ IRK 

ð2Þ D1

 IT N ;

i > N þ K;

0 6 i < N;

ð2Þ

Q i;iþ1 ¼ Q 2 ¼ ql

E l ,

ð2Þ

Q i;iþ1 ¼ ql

matrix of size l with all zero entries  I l , l ¼ R; K, is the square   except the entries I , l i;i1 i ¼ 1; l  1, which are equal to 1;

K þ 1, is the matrix of size l  ðl þ 1Þ with non-zero  Eþ l , l ¼ 1;   entries Eþ l i;iþ1 , i ¼ 0; l  1, which are equal to 1;

ð2Þ D1

N 6 i < N þ K;

i P N þ K;

SÞPN1 ðbð1Þ Þ; L1 ¼ L0 ðN; e

Dð0Þ ¼ 0;

DðiÞ

L2 ¼ L0 ðN; ~SÞPN1 ðbð2Þ Þ; ¼ diagfAi ðN; SÞe þ LNi ðN; e SÞeg;

i ¼ 1; N:

ðlÞ

The detailed description of the matrices Pi ðb Þ, l ¼ 1; 2, i ¼ 0; N  1; Ai ðN; SÞ and Li ðN; e SÞ, i ¼ 0; N, and the algorithms for their calculation can be found in Kim, Dudin, Taramin, and Baek (2013b). Proof. The entries of the blocks Q i;iþ1 , i P 0, define the intensities of the transitions of the Markov chain nt , t P 0, that increase the number of customers at the second stage by one. This can happen only if type 1 or type 2 customer arrives to the second stage. Let us consider three possible cases: (a) The number of customers at the second stage i < N (there is a free server). There are the following events that increase the number of customers at this stage:  Arrival of type 1 customer to the second stage after receiving service at the first stage. The entries of the matrix qlC R I  IW  Pi ðbð1Þ Þ define the intensities of R the event that the number of customers at the first stage decreases by one (a customer completes service at this stage), type 1 customer proceeds to the second stage and starts service at this stage.  Arrival of type 2 customer. The intensities of this event ð2Þ are defined by the entries of the matrix IR  D1  Pi ðbð2Þ Þ. Note that the matrix Pi ðbðlÞ Þ, l ¼ 1; 2, defines the transition proban o ð1Þ ðMÞ , t P 0, at the epoch of starting bilities of the process gt ; . . . ; gt new service of type l customer given that i servers are busy. (b) The number of customers at the second stage N 6 i < N þ K (all servers are busy and an arriving customer can join the buffer). The following events are possible:  Type 1 customer enters the second stage and joins buffer 1. The entries of the matrix qlC R I  Eþ iNþ1  IWT N define R the intensities of the event that the number of customers at the first stage decreases by one, the number of type 1 customers in buffer 1 increases by one.  Type 2 customer enters the second stage and joins buffer 2. The intensities of this event are defined by the entries ð2Þ of the matrix IR  e E iNþ1  D1  IT N . (c) The number of customers at the second stage i P N þ K. There are the following possible events:  Arrival of type 1 customer to the second stage from the first stage. Then the number of customers at the second stage increases only if an arriving customer joins buffer 1 (if buffer 1 is not full). The intensities of this event are given by the entries of the matrix qlC R I  Iþ  IWT N . R K  Arrival of type 2 customer to the second stage. The intensities of this event are given by the entries of the matrix ð2Þ IRK  D1  IT N . Since the component k has the state space f0; . . . ; maxf0; minfi  N; Kggg, then for i P N þ K the state space of the component k does not depend on i, and the blocks Q i;iþ1 also do not depend on i and equal the matrix Q 2 .

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The entries of the blocks Q i;i1 , i P 1, define the intensities of the transitions of the Markov chain nt , t P 0, that decrease the number of customers at the second stage by one. This can happen if ðaÞ an arbitrary customer completes service or ðbÞ type 1 customer leaves the system due to impatience. If there are no customers in buffer 1 and buffer 2, i.e., i 6 N, only event ðaÞ is possible. Its intensities are given by the entries of the matrix IRW  LNi ðN; e SÞ. Note that the matrix LNi ðN; e SÞ defines the n o ð1Þ ðMÞ , t P 0, intensities of the transitions of the process gt ; . . . ; gt during the service completion epoch given that i servers are busy during this epoch. If the number of customers in the buffers does not exceed K, i.e., i ¼ N þ 1; N þ K, then both events are possible. The intensities of event ðaÞ are given by the entries of the matrix IR  E iNþ1  I W  L1 if there are type 1 customers in buffer 1, and by the entries of the matrix IR  EiNþ1  IW  L2 , otherwise. The intensities of event ðbÞ are given by the entries of the matrix IR  aC iNþ1 E iNþ1  I WT N . By analogy, if the number of customers in the buffers exceeds K, i.e., i > N þ K, then both events are also possible and their  IW intensities are given by the entries of the matrices IR  I K L1 , IR  I  IW  L2 and IR  aC K I  IWT N , respectively. The blocks K Q i;i1 , i > N þ K, are equal to the matrix Q 0 , since the state space of the component k does not depend on i for i > N þ K. The non-diagonal entries of the blocks Q i;i , i P 0, define the intensities of the transitions of the chain nt , t P 0, that do not lead to the change of the number of customers at the second stage. If the number of customers at the second stage i < N þ K, the following events are possible: ðaÞ the transition of the process mt without generation of customers; ðbÞ the transition of the process mt with generation of type 1 customer; ðcÞ loss of type 1 customer after service at the first stage and ðdÞ the transition of the process n o ðMÞ , t P 0, which does not lead to the service complegð1Þ t ; . . . ; gt tion at the second stage. The intensities of events ðaÞ; ðbÞ; ðcÞ and ðdÞ are given by the non-diagonal entries of the matrices ð1Þ

 IK i  D1  IT minfi;Ng , ð1  qÞlC R I  IK WT IRK  D0  IT minfi;Ng , Iþ R R i

i

minfi;Ng

and IRK W  Aminfi;Ng ðN; SÞ, respectively. i

The diagonal entries of the matrix Q i;i are negative and the modulus of each entry defines the total intensity of leaving the corresponding state of the chain nt ; t P 0. The diagonal entries of þ I  D0  IT þ bI  IK the matrix lC  I R

K i WT minfi;Ng

RK i

minfi;Ng

R

i

ð1Þ

D1  IT minfi;Ng þ IRK W  ðAminfi;Ng ðN; SÞ þ Dðminfi;NgÞ Þ þ ð1  qÞlC R I R i

IK WT i

minfi;Ng

 aIR  C K i  IWT

minfi;Ng

, i < N þ K, define, up to sign, the

total intensity of leaving the corresponding state of the chain. Note that the matrix Ai ðN; SÞ defines the intensities of the n o ð1Þ ðMÞ transitions of the process gt ; . . . ; gt , t P 0, which do not lead to the service completion, while the moduli of the diagonal entries of the matrix DðiÞ define the total intensity of leaving the corresponding states of this process given that i servers are busy. For i P N þ K, the blocks Q i;i do not depend on i and are equal to the matrix Q 1 . Since the probability that more than one arrival can occur and more than one customer can leave the system during the interval of an infinitesimal length is negligible, the blocks Q i;j are zero matrices for ji  jj > 1, thus the generator Q has a block-tridiagonal structure. h

It follows from Neuts (1981) that the necessary and sufficient ergodicity condition of the quasi-birth-and-death process is the fulfillment of the inequality

yQ 0 e > yQ 2 e;

ð1Þ

where the row vector y is the unique solution to the following system of linear algebraic equations

yðQ 0 þ Q 1 þ Q 2 Þ ¼ 0; ye ¼ 1: Let us suppose that the vector y has the form ðy0 ; y1 ; . . . ; yR Þ. Note that the matrix Q 0 þ Q 1 þ Q 2 has a block-tridiagonal structure and we propose the following numerically stable algorithm for calculation of the sub-vectors yr , r ¼ 0; R. Step 1. Calculate the matrices Qr;r , Qr;rþ1 and Qr;r1 which are the diagonal, the updiagonal and the subdiagonal blocks of the matrix Q 0 þ Q 1 þ Q 2 respectively:

  ð1Þ ð2Þ Qr;r ¼ IK  D0 þ dR;R D1 þ D1  IT N þ IKW  ðAN ðN; SÞ þ DðNÞ Þ   þ IK  IW  L1 þ I  IW  L2  lrIKWT N þ aC K IK  IK  IWT N ; r ¼ 0; R; ð1Þ

Qr;rþ1 ¼ IK  D1  IT N ; r ¼ 0; R  1;   Qr;r1 ¼ qlr bI K þ IþK  IWT N þ ð1  qÞlrIKWT N ;

where di;j is the Kronecker delta. Step 2. Calculate the matrices Ar ; r ¼ 0; R  1, using the backward recursion

AR1 ¼ QR1;R ðQR;R Þ1 ; Ar ¼ Qr;rþ1 ðQrþ1;rþ1 þ Arþ1 Qrþ2;rþ1 Þ1 ; r ¼ R  2; R  3; . . . ; 0: Step 3. Calculate the matrices F r using the recurrent formulas

F 0 ¼ I;

F r ¼ F r1 Ar1 ;

r ¼ 1; R:

Step 4. Calculate the sub-vector y0 as the unique solution to the following system

y0 ðQ0;0 þ A0 Q1;0 Þ ¼ 0;

y0

R X F r e ¼ 1: r¼0

Step 5. Calculate the sub-vectors yr ¼ y0 F r ; r ¼ 1; R. If ergodicity condition (1) of the Markov chain nt is fulfilled, then the stationary probabilities of the system states pði; r; k; m;

gð1Þ ; . . . ; gðMÞ Þ, i P 0, r ¼ 0; R, k ¼ 0; maxf0; minfi  N; Kgg, m ¼ 0; W, gðmÞ ¼ 0; minfi; Ng, m ¼ 1; M, exist. Let us form the row vectors pði; r; k; mÞ of these probabilities enumerated in the reverse lexicographic order of the components gð1Þ ; . . . ; gðMÞ . Then let us form the row vectors

pði;r;kÞ ¼ ðpði; r; k; 0Þ; pði;r;k;1Þ;. .. ; pði;r;k; WÞÞ; k ¼ 0; maxf0; 2 fi  N; Kgg;

pði;rÞ ¼ ðpði;r; 0Þ; pði; r;1Þ;. .. ; pði;r;maxf0;minfi  N; KggÞÞ; r ¼ 0; R;pi ¼ ðpði;0Þ; pði;1Þ;. .. ; pði;RÞÞ; i P 0: It is well known that the probability vectors pi ; i P 0, satisfy the following system of linear algebraic equations:

ðp0 ; p1 ; . . . ; pi ; . . .ÞQ ¼ 0; Corollary 1. The Markov chain nt ; t P 0, belongs to the class of continuous-time quasi-birth-and-death processes, see, e.g., Neuts (1981).

r ¼ 1; R;

ðp0 ; p1 ; . . . ; pi ; . . .Þe ¼ 1

where Q is the infinitesimal generator of the Markov chain nt , t P 0. To solve this system the numerically stable algorithm that takes

C. Kim et al. / European Journal of Operational Research 235 (2014) 170–179

into account that the matrix Q has a block-tridiagonal structure, which is presented in Dudin, Kim, and Dudina (2013), could be proposed. 5. Performance measures As soon as the vectors pi , i P 0, have been calculated, we are able to find various performance measures of the system. ð1Þ The probability P loss that an arbitrary type 1 customer will be lost at the first stage is computed by ð1Þ

Ploss ¼

1   1X pði; RÞ IK i  Dð1Þ 1  IT minfi;Ng e: k1 i¼0

The average number N ð1Þ of busy servers at the first stage is computed by

Nð1Þ ¼

1 X R X

rpði; rÞe:

i¼0 r¼1

ð1Þ

The average intensity kout of flow of customers, who receive service at the first stage, is computed by ð1Þ

kout ¼ Nð1Þ l: The average number N ð2Þ of busy servers at the second stage is computed by

Nð2Þ ¼

1 X

minfi; Ngpi e:

i¼1

The average number N buffer of type 1 customers in buffer 1 is 1 computed by 1 X R minfiN;Kg X X

Nbuffer ¼ 1

i¼Nþ1 r¼0

kpði; r; kÞe:

k¼1

The average number N buffer of type 2 customers in buffer 2 is 2 computed by 1 X R minfiN;Kg X X

Nbuffer ¼ 2

i¼Nþ1 r¼0

ði  N  kÞpði; r; kÞe:

k¼0

The average number Lsystem of customers in the system is computed by system

L

1 X R X ¼ ði þ rÞpði; rÞe: i¼0 r¼0

The average intensity kout of flow of customers, who receive service at the second stage, is computed by

kout ¼

1 X

pi ðIRK i W  Lmaxf0;Nig ðN; eSÞÞe:

i¼1 ð2Þ

The probability P loss that an arbitrary type 1 customer will be lost at the second stage is computed by ð2Þ

Ploss ¼ 1 

kout  k2 ð1Þ

qkout

:

The probability P 2ent that an arbitrary type 1 customer will be loss lost at the entrance to the second stage is computed by 1 X R  1 X ð1Þ P2ent ¼ kout r lpði; r; KÞe: loss i¼NþK r¼1

The probability P2imp that, after arrival to the second stage, an loss arbitrary type 1 customer will go to the buffer and leave it due to impatience is computed by

175

ð2Þ

2imp Ploss ¼ Ploss  P2ent loss :

The average intensity kout type 1 of flow of type 1 customers, who receive service at the second stage, is computed by

kout

type 1

  ð2Þ ð1Þ ¼ 1  Ploss qkout :

6. Distribution of sojourn and waiting times of type 2 customer Because type 1 customers have a priority, they leave the system due to impatience and restriction on the total number of type 1 customers at the second stage, the upper bound of their sojourn time is more or less clear. So, we concentrate on the consideration of the sojourn time of type 2 customers. We will derive the distribution of an arbitrary type 2 customer’s waiting and sojourn time in terms of the LST. Let WðxÞ be the distribution function of the sojourn time of an arbitrary type 2 cusR1 tomer in the system under study and wðsÞ ¼ 0 esx dWðxÞ, Re s > 0, be its LST. To derive the expression for the LST wðsÞ we use the method of collective marks (method of additional event, method of catastrophes) (see, e.g., Kesten & Runnenburg (1956) and van Danzig (1955)). Let us tag an arbitrary type 2 customer and keep track of its staying in the system. We interpret the variable s as the intensity of some virtual stationary Poisson flow of catastrophes. Thus, wðsÞ has the meaning of the probability that no catastrophe arrives during the sojourn time of the tagged type 2 customer. Let wðs; n; r; k; m; gð1Þ ; . . . ; gðMÞ Þ be the probability that catastrophe will not arrive during the tagged type 2 customer’s sojourn time in the system conditioned on the fact that, during the given moment, the position of the tagged customer in buffer 2 is n; n P 1, the number of customers at the first stage is r; r ¼ 0; R, the number of type 1 customers in buffer 1 is equal to k; k ¼ 0; K, the state of the process mt is m, the states of the proð1Þ ðMÞ cesses gt ; . . . ; gt are gð1Þ ; . . . ; gðMÞ respectively, t P 0. Let us enumerate the probabilities wðs; n; r; k; m; gð1Þ ; . . . ; gðMÞ Þ in the lexicographic order of components as indicated above and form the column vectors wðs; n; r; kÞ from these probabilities. Theorem 1. The LST wðsÞ of distribution of an arbitrary type 2 customer’s sojourn time in the system is computed by

wðsÞ ¼ k1 2

" N1 X

1 ð2Þ pi ðIR  Dð2Þ 1  I T i Þeb2 ðsI  S2 Þ S 0

i¼0

# 1 X R minfiN;Kg   X X : pði; r; kÞ Dð2Þ  I þ wðs; i  N  k þ 1; r; kÞ TN 1 i¼N r¼0

k¼0

Proof. The following situations are possible during the arrival epoch of the tagged type 2 customer:  There is an idle server during the arrival epoch of the tagged customer and this customer immediately starts getting service. PN1 ð2Þ The probability of this event is k1 i¼0 pi ðIR  D1  I T i Þe. In 2 this case, the probability that no catastrophe arrives during the sojourn time is equal to the probability that no catastrophe arrives during the service time of type 2 customer and is given ð2Þ as b2 ðsI  S2 Þ1 S0 .  All servers are busy during the arrival epoch and the tagged customer joins buffer 2. The probability of this event is   P1 PR PminfiN;Kg k1 pði; r; kÞ Dð2Þ 2 i¼N r¼0 1  I T N e. In this case, the k¼0 probability that no catastrophe arrives during the sojourn time of the tagged type 2 customer under the fixed values of the components n; r; k is equal to wðs; i  N  k þ 1; r; kÞe.

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Using the law of total probability one can verify the validity of the theorem. h So, for calculation of the LST wðsÞ we must compute the vectors wðs; n; r; kÞ. Let us introduce the column vectors

wðs; n; rÞ ¼ ðwðs; n; r; 0Þ; . . . ; wðs; n; r; KÞÞT ; wðs; nÞ ¼ ðwðs; n; 0Þ; . . . ; wðs; n; RÞÞT : Theorem 2. The column vectors wðs; nÞ; n P 1, are computed by

wðs; 1Þ ¼ ðsI  XÞ1 aðsÞ; wðs; nÞ ¼ ðsI  XÞ1 Wwðs; n  1Þ;

n > 1;

where

h  i ð2Þ D0 þ D1  ðAN ðN; SÞ þ DðNÞ Þ     ð1Þ þ bI R þ IþR  IK  D1  IT N þ aIR  C K IK  IK  IWT N þ IR h   i  IK  IW  L1 þ lC R IR  q bI K þ IþK þ ð1  qÞIK  IWT N ;

X ¼ lC R  IKWT N þ IRK 

ðsI þ XÞwðs;nÞ þ ð1  dn;1 ÞWwðs;n  1Þ þ dn;1 aðsÞ ¼ 0T ; n P 1:

W ¼ IR  I  IW  L2 ;

ð3Þ

It can be verified that the diagonal entries of the matrix ð2Þ

~  ððIW  L2 ÞeÞb2 ðsI  S2 Þ1 S 0 ; aðsÞ ¼ eR  e

X  sI dominate in all rows of this matrix. So the inverse matrix

~ is the column vector of size K with all zero entries except the enand e ~Þ0 ¼ 1. try ðe Proof. Based on a probabilistic sense of the LST and the law of total probability, the vectors wðs; n; r; kÞ can be found from the following system of linear algebraic equations: 1

wðs; n; r; kÞ ¼ ½ðs þ ka þ r lÞI  D0  ðAN ðN; SÞ þ DðNÞ Þ  ð1Þ  ð1  dr;R ÞD1  IT N wðs; n; r þ 1; kÞ   ð1Þ ð2Þ þ dr;R D1 þ D1  IT N wðs; n; r; kÞ

exists. Thus, based on (3) one can verify the validity of the theorem. h Corollary 2. The average sojourn time V soj of an arbitrary type 2 customer is calculated by

" ð2Þ

V soj ¼k1 b1 2

N1  1 X R minfiN;Kg  X X X pi IR  Dð2Þ pði; r; kÞ 1  IT i e  i¼N r¼0

i¼0

k¼0

#   ð2Þ 0  D1  IT N w ðs; i  N  k þ 1; r; kÞjs¼0 ;

where the column vectors w0 ðs; n; r; kÞjs¼0 are calculated as the blocks of the vector w0 ðs; nÞjs¼0 which can be computed as follows

þr lðð1  qÞ þ dk;K qÞwðs; n; r  1; kÞ þð1  dk;K Þr lqwðs; n; r  1; k þ 1Þ

w0 ðs; 1Þjs¼0 ¼ X1 ½e  a0 ðsÞjs¼0 ;

ð2Þ

þdk;0 dn;1 ðIW  L2 Þeb2 ðsI  S2 Þ1 S 0

w0 ðs; nÞjs¼0 ¼ X1 ½e  Ww0 ðs; n  1Þjs¼0 ;

þdk;0 ð1  dn;1 ÞIW  L2 wðs; n  1; r; 0Þ

 þð1  dk;0 ÞðIW  L1 þ kaIWT N Þwðs; n; r; k  1Þ ; n P 1; r ¼ 0; R; k ¼ 0; K:

cases when type 1 customer leaves the system after the service completion at the first stage. In this case the number of customers at the first stage decreases by one. The fourth term corresponds to the case when type 1 customer is admitted to the second stage. In this case the number of customers at the first stage decreases by one, and the number of type 1 customers in buffer 1 (the component k) increases by one. The fifth term correspond to the case when during the service completion epoch at the second stage there are no type 1 customers in buffer 1, the tagged customer has the first position in buffer 2 and is chosen ð2Þ for service. The number b2 ðsI  S2 Þ1 S 0 defines the probability that catastrophe will not arrive during the service time of the tagged customer. The sixth term corresponds to the case when during the service completion epoch at the second stage there are no type 1 customers in buffer 1 and there are type 2 customers that arrive to the system before the tagged customer. In this case the position of the tagged customer in buffer 2 (component n) decreases by one. Finally, the seventh term corresponds to the case when type 1 customer leaves the buffer (starts service or leaves the buffer due to impatience). System (2) can be rewritten into the matrix form as

0

Here a ðsÞjs¼0 ¼

ð2Þ

Let us explain formula (2) in brief. The diagonal entries of the matrix in the square brackets in (2) are equal to the total intensity of the events which can happen after an arbitrary epoch: catastrophe arrival, the transition of the underlying process of the MMAP, the transition of any underlying process of the generalized PH service processes, abandon of type 1 customers from the buffer, and the service completion of type 1 customer at the first stage. The non-diagonal entries of the matrix in the square brackets in (2) are equal to the intensities of the transition of the MMAP underlying process without generation of a customer, the transition of one of generalized PH underlying processes which does not lead to the service completion (these events do not impact on the components n, r and k). The first term in the round brackets in (2) corresponds to the case when type 1 customer is admitted at the first stage of the system. In this case the number of customers at the first stage (the component r) increases by one. The second term in the round brackets in (2) corresponds to the cases when type 1 customer is rejected at the first stage upon arrival and type 2 customer is admitted at the second stage. In this case the values of the components n, r and k do not change. The third term corresponds to the

ð2Þ b1 eR

n > 1:

~  ððIW  L2 ÞeÞ. e

Proof. Formula for the calculation of the average sojourn time of an arbitrary type 2 customer is based on the definition V soj ¼ w0 ðsÞjs¼0 taking into account that wðs; nÞjs¼0 ¼ e; n  1. h Corollary 3. The average waiting time V wait of an arbitrary type 2 customer is calculated by ð2Þ

V wait ¼ V soj  b1 : 7. Numerical examples In Experiment 1, we investigate the impact of the coefficient of correlation in the arrival flow on the system performance measures. For this purpose, let us introduce three MMAPs defined by the ð1Þ ð2Þ matrices D0 ; D1 and D1 . All these MMAPs have the same average total arrival intensity k, the average intensity of type 1 customers k1 ¼ 34 k, the average intensity of type 2 customers k2 ¼ 14 k, but different coefficients of correlation. MMAPx defines the MMAP arrival process with the coefficient of correlation ccor ¼ x.

C. Kim et al. / European Journal of Operational Research 235 (2014) 170–179 0 The first process, as MMAP 3 coded ð2Þ 1  , is defined by the matrices ð1Þ D0 ¼ ðkÞ, D1 ¼ 4 k and D1 ¼ 4 k . It has the coefficients of corðlÞ relation ccor ¼ ccor ¼ 0 and the coefficients of variation ðlÞ cv ar ¼ cv ar ¼ 1, l ¼ 1; 2. In this case, the arrival processes of type 1 and type 2 customers are defined as the stationary Poisson processes. The second process, coded as MMAP0:2 , is defined by the matrices

 ; 0 0:04384   0:33566 0:00224 ; ¼k 0:00610 0:00485

D0 ¼ k ð2Þ D1



1:35162

0

ð1Þ

D1 ¼ k



1:00699 0:00673 0:01832 0:01457

ð1Þ

 ;

ð2Þ

has the coefficients of correlation ccor ¼ 0:17, ccor ¼ 0:07 and the ð1Þ ð2Þ coefficients of variation cv ar ¼ 12:34, cv ar ¼ 10:55, cv ar ¼ 5:21. The third process, coded as MMAP0:4 , is defined by the matrices



 ; 0:00101 0:11019   0:84057 0:00884 : ¼k 0:00303 0:02426

D0 ¼ k ð2Þ D1

3:39767

0

ð1Þ

D1 ¼ k

ð1Þ



2:52172 0:02654 0:00909

ð2Þ

0:07280

 ;

It has the coefficients of correlation ccor ¼ 0:38, ccor ¼ 0:28 and the ð1Þ ð2Þ coefficients of variation cv ar ¼ 12:39, cv ar ¼ 11:92, cv ar ¼ 9:20. We assume that the number of servers at the first stage R ¼ 8, the service intensity at the first stage l ¼ 0:8, the probability q ¼ 0:2, the number of servers at the second stage N ¼ 8, the intensity a ¼ 0:5, the buffer capacity for type 1 customers K ¼ 8. PH1 service process of type 1 customers at the second stage is characterized by the vector b1 ¼ ð1Þ and the matrix S1 ¼ ð1Þ. ð1Þ The mean service time b1 of type 1 customers at the second stage is equal to 1. PH2 service process of type 2 customers at the second stage is characterized by the vector b2 ¼ ð1Þ and the matrix ð2Þ S2 ¼ ð0:5Þ. The mean service time b1 of type 2 customers at the second stage is equal to 2.

177

Let us vary the average total arrival intensity k in the interval ½1; 15 in steps of 0.1. Figs. 2, 3 illustrate the dependence of the average number L of customers in the system, the average sojourn time V soj of type 2 customers, the average intensities kout of flow of customers and kout type 1 of flow of type 1 customers, who receive service at the ð1Þ second stage, the loss probability P loss of an arbitrary type 1 cusð1Þ tomer at the first stage, the average intensity kout of flow of customð2Þ ers, who receive service at the first stage, the loss probability P loss of an arbitrary type 1 customer at the second stage and the probability P2imp that an arbitrary type 1 customer will join buffer 1 and loss leave it due to impatience on the average arrival intensity k for different MMAPs. It is clearly evident from Figs. 2 and 3 that an increase in the average arrival intensity leads to an increase in the average intensities of output flows, the loss probabilities of type 1 customers and the average sojourn time of type 2 customers. So, with increasing average arrival intensity the quality of service becomes worse. Based on figures presented one can conclude that the coefficient of correlation in the arrival process has a profound impact on the system performance measures and taking into account the correlation in the arrival flow is extremely important for adequate modeling of the system performance. For example, when the coefficient of correlation increases, the average output intensities kout type 1 , ð1Þ kout and kout decrease essentially, while the loss probability of type 1 customers at the first stage increases essentially. Additionally, the fulfillment of the ergodicity condition depends on the coefficient of correlation in the arrival flow. In the example under consideration, the ergodicity condition is not fulfilled for k P 14:1 in case of MMAP0 , for k P 14:4 in case of MMAP0:2 and for k P 14:9 in case of MMAP0:4 . This finding can be explained as follows. As it was mentioned above, for arrival flows with small coefficients of correlation the loss probability of type 1 customers at the second stage is less than for arrival flows with high coefficients of correlation. Thus, for arrival flows with small coefficients of correlation

Fig. 2. Dependence of the average number of customers in the system, the average waiting time of type 2 customers, the average intensities kout of flows of customers and kout type 1 of type 1 customers, who receive service at the second stage of the system, on the average arrival intensity k for different MMAPs.

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C. Kim et al. / European Journal of Operational Research 235 (2014) 170–179

Fig. 3. Loss probability of an arbitrary type 1 customer at the first stage, the average intensity of flow of customers, which receive service at the first stage, the loss probability of an arbitrary type 1 customer at the second stage and the probability that an arbitrary type 1 customer will go to the buffer and leave it due to impatience as functions of the average arrival intensity for different MMAPs.

Fig. 4. Dependence of the average waiting time of type 2 customers, the average number of type 1 customers in the buffer 1 and the loss probabilities P 2ent and P 2loss on the loss intensity of impatience a for PH service processes with different coefficients of variation.

more type 1 customers arrive to the second stage thereby increasing the load of the second stage and lead to an earlier violation of the ergodicity condition. It also explains the dependence of the average number L of customers in the system and the average sojourn time V soj of an arbitrary type 2 customer on the average

arrival intensity k for k > 13. Note for example, that for the stationary Poisson arrival flows with k ¼ 13 the average sojourn time of an arbitrary type 2 customer V soj ¼ 5:23, while under k ¼ 14 the average sojourn time V soj ¼ 327:71, i.e., increasing the average arrival flow of 7.7% (change k from 13 to 14) leads to an increase in

C. Kim et al. / European Journal of Operational Research 235 (2014) 170–179

the average sojourn time V soj by more than 62 times. Prediction of such behavior without corresponding calculations is unrealistically difficult task, so this proves the necessity of the presented research. The numerical results also show that the following equalities hold true: ð2Þ

V soj  b1 ¼ V wait ¼ Nbuffer 1 ð1Þ kout

¼

2imp Ploss

a

Nbuffer 2 ; k2

:

The first equality is the famous Little formula. The second equality can be explained as follows. Let us rewrite it in the form ð1Þ aNbuffer ¼ kout P 2imp . The left and the right sides of the equality define 1 loss the intensity of type 1 customers leaving buffer 1 due to impatience, so the equality is correct. In Experiment 2, let us show the impact of the coefficient of variation in the service processes of type 1 and type 2 customers at the second stage on the main performance measures of the system. To this end, we consider three cases of PH service processes of type 1 and type 2 customers. The first case corresponds to the exponential service time distributions of type 1 and type 2 customers. We assume that the service time of type 1 customers is defined by the vector b1 ¼ ð1Þ and the matrix S1 ¼ ð1Þ. The service time of type 2 customers is defined by the vector b2 ¼ ð1Þ and the matrix S2 ¼ ð0:5Þ. These processes have the same coefficient of variation cv ar ¼ 1. In the second case, we assume that the service time distribution of type 1 customers is characterized by the vector b1 ¼ ð1; 0Þ and   2 2 , and the service time distribution of the matrix S1 ¼ 0 2 type 2 customers is characterized by the vector b2 ¼ ð1; 0Þ and   1 1 the matrix S2 ¼ . These processes have the same coeffi0 1 cient of variation cv ar ¼ 0:5. In the third case, we assume that the service time distribution of type 1 customers is characterized by the vector b1 ¼ ð0:1; 0:9Þ   0:23319 0:01163 and the matrix S1 ¼ , and the service 0:13989 2:54192 time distribution of type 2 customers is characterized by the vector   0:11659 0:00581 . b2 ¼ ð0:1; 0:9Þ and the matrix S2 ¼ 0:06994 1:27096 These processes have the same coefficient of variation cv ar ¼ 5. Note that in all cases under consideration the average service ð1Þ time of type 1 customers b1 ¼ 1 and the average service time of ð2Þ type 2 customers b1 ¼ 0:5, but the service processes have different values of the coefficient of variation cv ar . Let us fix the number of servers at the first stage R ¼ 5, the service intensity at the first stage l ¼ 0:8, the probability q ¼ 0:5, the number of servers at the second stage N ¼ 4, the buffer capacity for type 1 customers K ¼ 5. We assume that arrivals are defined by the MMAP0:4 arrival process presented in the first experiment with the arrival intensity k ¼ 1, the coefficient of correlation ccor ¼ 0:4 and the coefficient of variation cv ar ¼ 12:39. Let us vary the intensity of impatience a in the interval ½0:01; 4. Fig. 4 illustrates the dependence of the average waiting time V soj of type 2 customers, the average number N buffer of type 1 customers 1 in buffer 1, the probability P 2ent that an arbitrary type 1 customer loss will be lost at the entrance to the second stage, and the loss probð2Þ ability P loss of an arbitrary type 1 customer at the second stage on the intensity of impatience a for PH service processes with different coefficients of variation. As it is seen from Fig. 4, with growth of the intensity of impatience a the average sojourn time of type 2 customers decreases. This finding can be explained as follows. When the intensity of impatience increases, the loss probability of type 1 customers at

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ð2Þ

the second stage Ploss also increases. Because type 1 customers have a priority over type 2 customers, the loss of type 1 customers decreases the average waiting time (and also the average sojourn time) of type 2 customers. For the same reason the average number of type 1 customers in buffer 1 essentially decreases with an increase in the intensity of impatience which in turn leads to decrease in the loss probability of type 1 customers due to buffer 1 overflow. Another important conclusion from Fig. 4 is that the coefficient of variation in the service process affects the system performance measures. So, for adequate prediction of the system operation it is necessary to take into account the variation in the service process. 8. Conclusion A tandem queueing system with Marked Markovian arrival flow and generalized phase-type service time distribution is studied. The process of the system states is analyzed, and the ergodicity condition is derived. Some key performance measures are obtained. The Laplace–Stieltjes transform of the sojourn time distribution of type 2 customers is derived. The numerical results show the importance of taking into account the correlation in the arrival flow and the variance of the service time. Acknowledgements This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2011-0015214). The authors are very thankful to three anonymous referees for valuable remarks and suggestions. Their account led to improvement of presentation of the results. References Dudin, S., Kim, C., & Dudina, O. (2013). MMAPjMjN queueing system with impatient heterogeneous customers as a model of a contact center. Computers and Operations Research, 40, 1790–1803. Gomez-Corral, A. (2002a). On a tandem G-network with blocking. Advances in Applied Probability, 34, 626–661. Gomez-Corral, A. (2002b). A tandem queuewith blocking and Markovian arrival process. Queueing Systems, 41, 343–370. Gomez-Corral, A., & Martos, M. E. (2006). Performance of two-stage tandem queues with blocking: The impact of several flows of signals. Performance Evaluation, 63, 910–938. Kesten, H., & Runnenburg, J. Th. (1956). Priority in waiting line problems. Amsterdam: Mathematisch Centrum. Klimenok, V. I., Breuer, L., Tsarenkov, G. V., & Dudin, A. N. (2005). The BMAP=G=1=N ! =PH=1=M tandem queue with losses. Performance Evaluation, 61, 17–40. Klimenok, V. I., Kim, C. S., Tsarenkov, G. V., Breuer, L., & Dudin, A. N. (2007). The BMAP=G=1 ! =PH=1=M tandem queue with feedback and losses. Performance Evaluation, 64, 802–818. Kim, C. S., Park, S. H., Dudin, A. N., Klimenok, V. I., & Tsarenkov, G. V. (2010). Investigation of the BMAP=G=1 ! =PH=1=M tandem queue with retrials and losses. Applied Mathematical Modelling, 34, 2926–2940. Kim, C. S., Dudin, A. N., Dudin, S. A., & Dudina, O. S. (2013a). Tandem queueing system with impatient customers as a model of call center with Interactive Voice Response. Performance Evaluation, 70, 440–453. Kim, C. S., Dudin, S. A., Taramin, O. S., & Baek, J. (2013b). Queueing system MAPjPHjNjN þ R with impatient heterogeneous customers as a model of call center. Applied Mathematical Modelling, 37, 958–976. Neuts, M. (1981). Matrix-geometric solutions in stochastic models – An algorithmic approach. Baltimore: Johns Hopkins University Press. Ramaswami, V. (1985). Independent Markov processes in parallel. Communications in Statistics – Stochastic Models, 1, 419–432. Ramaswami, V., & Lucantoni, D. M. (1985). Algorithms for the multi-server queue with phase-type service. Communications in Statistics – Stochastic Models, 1, 393–417. van Danzig, D. (1955). Chaines de Markof dans les ensembles abstraits et applications aux processus avec regions absorbantes et au probleme des boucles. Annales de l’Institut Henri Pioncaré, 14, 145–199.