Elsevier

Applied Mathematics and Computation

Volume 314, 1 December 2017, Pages 154-172
Applied Mathematics and Computation

Analysis of unreliable BMAP/PH/N type queue with Markovian flow of breakdowns

https://doi.org/10.1016/j.amc.2017.06.035Get rights and content

Abstract

Unreliability of components is the inherent feature of many real world systems and its account is vital for correct prediction of performance measures of the system. Multi-server queueing model considered in this paper allows to evaluate characteristics of the systems under much more general assumptions about the probabilistic distributions describing behavior of the system than models known in existing literature. We analyse the multi-server queue with infinite buffer and the Batch Markovian Arrival Process (BMAP) of customers. The servers are identical and independent of each other. Service time of a customer has phase-type (PH) distribution. Servers are subject to breakdowns and repairs. Breakdowns occurrence moments are defined by the Markovian Arrival Process (MAP). The breakdown causes a failure of any server, which is not under repair. When a server fails the repair period starts immediately. The duration of this period has PH distribution. A customer whose service is interrupted occupies an idle server, if any, and continues his/her service. If he/she does not see an idle server, the customer goes to the buffer with some probability and permanently leaves the system with the complementary probability. We derive the constructive ergodicity condition and calculate the stationary distribution and the main performance characteristics of the system. Illustrative numerical examples are presented.

Introduction

Queueing. theory is well recognized mathematical tool for solving the problems of design, performance evaluation, capacity planning and optimization of various real world systems and processes in which some restricted resources are shared by numerous requests arriving at the random times. Especially wide applications this theory found in the area of electrical engineering and telecommunications. Existing literature in this theory is huge. This paper is devoted to analysis of a multi-server queueing system with unreliable servers. So, we will mention related works concerning only multi-server queues with servers breakdowns.

One of the first papers where such a queue is under study is [1]. In this paper queueing model of M/M/N type is considered. The system has N identical servers and an infinite buffer. Arrival flow of customers is defined by the stationary Poisson process, service time at each server has exponential distribution. Time till breakdown of any server and its recovering time also have exponential distribution. Behavior of the system is described by two-dimensional Markov chain one component of which describes the number of customers in the system while the second one defines the current number of non-broken servers. Ergodicity condition for this Markov chain is derived. The problem of computation of the stationary distribution of the chain is solved using the partial generating functions and reasonings of analyticity of these functions in the unit disc of the complex plane.

After that, due to the wide practical importance of analysis of multi-server queueing systems with unreliable servers, their analysis was extended in many directions. One of these directions suggests that recovering times of broken servers are not independent. There exists a finite pool of repairmen and recovering of a broken server can start only at presence of at least one repairman in this pool. If all repairmen are busy, requests for repair of the servers form a special queue. As important early work in this direction, the paper [2] should be mentioned. In the paper, again queueing model of M/M/N type is considered. All assumptions about arrivals, service, breakdowns and recovering times distribution are the same as in [1]. However, repairing of the servers is implemented by a finite number of repairmen. Behavior of the system is described by two-dimensional Markov chain with components having the same meaning as in [1]. Ergodicity condition for this Markov chain is derived. The stationary probabilities of the states of the chain are computed in the vector form. They define so called matrix-geometric distribution. This approach for analysis of two-dimensional Markov chains was further developed by authors of [2] to more general and complicated classes of Markov chains. Besides the analysis of the distribution of the Markov chain, the authors of [2] analyzed the stationary distribution of waiting time in the system. Results of numerical experiments are presented. The authors show that an approach to the analysis of such a system via computer simulation suffers in modeling the system when the number of available servers may become small even during quite short periods of time. For references to more recent papers devoted to analysis of multi-server queueing systems with unreliable servers, the papers [5], [6], [7], [8] and [9] can be recommended.

Shortcoming of models analyzed in [1] and [2], as well as in overwhelming majority of papers concerning the multi-server queueing systems with unreliable servers, from the point of view of potential applications, is imposing strict assumption about distributions of random variables defining the operation of the systems. As a rule, these distributions are assumed to be exponential. This assumption drastically simplifies mathematical analysis of the model but now rarely holds true in real world systems, especially telecommunication systems. Due to significant progress in analysis of two-dimensional Markov chains achieved first of all by Neuts, see, e.g., books [3] and [4], and his followers, the mathematical background for the analysis of queues with more complicated arrival flows and service and repair processes was created. This background allows to deal with multi-server queues with BMAP or MAP arrival process as models of arrival and breakdowns processes and the PH distribution of the service and repair times. Detailed information about a BMAP and the PH type distribution will be presented in the next section. Usage of BMAP instead of stationary Poisson process (which is a very special case of BMAP) allows to take into account not only the mean arrival rate, but also the variance of inter-arrival times and possible correlation of successive inter-arrival times. Usage of the PH distribution instead of exponential one (which is a very special case of the PH) allows to take into account not only the average service rate, but also the variance of service time. Because arrival processes in many real world systems exhibit correlation while service and repair times may by highly variable, investigation of queues of BMAP/PH/N type is important from the point view of potential applications. Likely, the first work devoted to the analysis of a reliable BMAP/M/N type queue with infinite buffer is [10]. The BMAP/PH/N/N type queue, which is essential extension of Erlang’s loss model, was analyzed in [11]. The BMAP/PH/N type queue with retrials of customers was analyzed in [12].

The literature on multi-server queueing systems with unreliable servers and arrival flow different from the stationary Poisson process and (or) service time distribution different from exponential one is not extensive. In the paper [13], published 30 years after [2], results of [2] for the M/M/N type unreliable queue are extended to the M/PH/N type queue. Such an extension was not trivial from theoretical point of view and, what is more essential, from computational point of view due to high dimension of multi-dimensional Markov chain that should be investigated to analyze the queueing model.

The main advantage of our paper comparing to other literature on multi-server queueing systems with unreliable servers consists of the analysis of the system under quite general assumptions about arrival processes of customers and breakdowns and distributions of service and repair times. The system, which is close to the system under consideration, has been investigated in [14]. But the model in [14] does not have a buffer and much less effective way to construct the multi-dimensional Markov chain is utilized there. We study the system of BMAP/PH/N type using another way to construct the multi-dimensional Markov chain. This way provides much lower dimension of the chain what has a drastic effect on feasibility of computer implementation of the algorithm for computation of steady state probabilities and performance measures of the system.

The rest of the paper is organized as follows. In Section 2, the mathematical model is described. The process of the system states as a multi-dimensional Markov chain is constructed in Section 3. A generator of this chain as a block structured matrix is presented here. An ergodicity condition for the Markov chain under study is derived in Section 4. This condition is intuitively tractable. In general case, it is necessary to solve a finite system of linear algebraic equations to check this condition. In case of exponential distribution of service and recovering times and stationary Poisson process of breakdown arrival, this condition is given in explicit form. An algorithm for the calculation of the steady state distribution of the Markov chain is described in brief in this section. Formulas for computation of performance measures of the system are given in Section 5. Section 6 contains numerical examples which give some insight into behavior of the system and justify investigation of unreliable queue with arrival flows more general than the stationary Poisson. Section 7 concludes the paper.

Section snippets

The mathematical model

We consider an N-server retrial queue with Batch Markovian Arrival Process (BMAP). The BMAP is defined by its underlying process νt, t ≥ 0, which is an irreducible continuous-time Markov chain with the finite state space {0,,W}, and the matrix generating function D(z)=k=0Dkzk,|z|1. The batches of customers enter the system only at the epochs of the chain νt, t ≥ 0 transitions. The matrices Dk, k ≥ 1 (nondiagonal entries of the matrix D0) define the intensities of the process νt, t ≥ 0

Process of the system states

It is intuitively clear that the process of the system operation can be described in terms of multi-dimensional continuous time Markov chain. Our first goal is to properly choose the components of this chain. Because we consider the multi-server system with phase-type distribution of service and repair times, it is necessary to carefully describe the service and repair processes in the system, i.e., the processes of change in service and repair phases at the working and failed servers.

Ergodicity condition. Steady state distribution

When the state space of the random process that describes behavior of the system is infinite, the mandatory step in analysis of such a process is validation of existence of the stationary mode of the system operation under the given set of the system parameters. In the case when such a random process is the Markov chain, the existence of the stationary mode of the system operation is equivalent to ergodicity of the Markov chain.

Theorem 1

The necessary and sufficient condition for ergodicity of the

Performance measures

Having the stationary distribution pi, i ≥ 0, been calculated we can find a number of stationary performance measures of the system. Below we give some of them. Nontrivial performance measures will be presented with brief explanations.

  • Mean number of customers in the queue Lqueue=i=1ipie.

  • Mean number of busy servers Nbusy=p0diag{I^n,n=0,N¯}e+i=1piI^Ne,where I^n=diag{rIaCr+M1M1Cnr+R1R1,r=0,n¯},n=0,N¯.

  • Mean number of customers in the system L=Lqueue+Nbusy.

  • Mean number of servers under

Numerical examples

In this section we present the results of three computational experiments on calculation of the system performance measures.

Experiment 1

The goal of the first experiment is to demonstrate how the input rate and correlation in BMAP impact on the system performance measures.

Let us introduce three MAPs having the same fundamental rate equal to 1 but different coefficients of correlation.

MAP1 has the coefficient of correlation ccorr=0.2 and is defined by the matrices D0=(1.3526000.04391),D=(1.34360.0090.02446

Conclusion

In this paper, we have analyzed an unreliable multi-server queue with quite general assumptions about the arrival processes of customers and breakdowns, service and repair time distributions what predefines an importance of the implemented analysis from the point of view of its potential applications. The stability condition for this system is presented in a simple algorithmic form. If an arrival flow of breakdowns is described by the stationary Poisson process and service and repair times have

Acknowledgments

This publication was financially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2014K2A1B8048465), by Belarusian Republican Foundation for Fundamental Research (Grant No. F15KOR-001) and by the Ministry of Education and Science of the Russian Federation (the Agreement number 02.a03.21.0008).

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