International Journal of Stochastic Analysis

Volume 2016, Article ID 9152701, 11 pages

http://dx.doi.org/10.1155/2016/9152701

## Analysis of a Priority Queue with Phase-Type Service and Failures

^{1}Department of Applied Mathematics and Cybernetics, Tomsk State University, 36 Lenina Avenue, Tomsk 634050, Russia^{2}Department of Applied Mathematics and Computer Science, Belarusian State University, 4 Nezavisimosti Avenue, 220030 Minsk, Belarus

Received 30 March 2016; Accepted 17 June 2016

Academic Editor: Lukasz Stettner

Copyright © 2016 Alexander Dudin and Sergei Dudin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a single server queue with two types of customers. We propose a discipline of flexible priority in access that combines the features of randomization and the threshold type control. We introduce a new class of distributions, phase-type with failures () distribution, that generalizes the well-known phase-type () distribution to the case when failures can occur during service of a customer. The arrival flow is described by the marked Markovian arrival process. The service time distribution is of type with the parameters depending on the type of a customer. Customers of both types can be impatient. Behavior of the system is described by the multidimensional Markov chain. Problem of existence and computation of the stationary distribution of this Markov chain is discussed in brief as well as the problem of computation of the key performance measures of the system. Numerical examples are presented that give some insight into behavior of the system performance measures under different values of the parameters defining the strategy of customers access to service.

#### 1. Introduction

In this paper, we consider a single server queueing model of information transmission system with two types of customers. Type-1 customers can be queued into the buffer having an infinite capacity. The buffer for type-2 customers has a finite capacity. Such kind of systems quite often describes behavior of various telecommunication systems where type-1 customers are interpreted as delay tolerant and type-2 customers are interpreted as loss tolerant; see, for example, [1]. Existence of two types of customers causes the necessity of managing the discipline of customers access to the server. Popular disciplines are priority (preemptive or nonpreemptive) disciplines in which a priority is given to one of types and customers of another type have a chance to enter service only when the priority customers are absent in the system.

However, such disciplines are not appropriate when it is necessary to provide more fair access of customers to the server. In particular, in some real world systems, for example, the vehicular ad hoc networks (VANET) that use the IEEE 802.11p protocol, it is necessary to share the available access time between a Control Channel and Service Channels. The Control Channel is used for the periodical dissemination of control information (beaconing) and traffic safety related information event messages. The Service Channels are used to disseminate noncritical information for infotainment applications; see, for example, [2]. So it is necessary to alternate access to the server by two types of customers, which cannot be achieved via the classical priority discipline.

Aiming to provide more fairness in access of customers of different type to the server, in this paper, we do not assume existence of an explicit priority given to one of the types. Instead, we suggest randomized choice of a customer for service when both queues are not empty at a service completion epoch. Priority can be given to one of types implicitly by means of fixing higher probability of a choice for service. If we set equal probabilities of the choice of customers of different types for service, in probabilistic sense, we create good conditions for frequent alternation of the type of a customer in service. However, equal probabilities may be not reasonable when the intensities of two flows are quite different. So it is important to optimally choose the probability defining the randomized access. The problem becomes even more complicated if impatience of customers should be taken into account and customers of one type are more impatient than the customers of another type.

In the model under study, we account impatience of customers. Because type-2 customers can be dropped (lost) also due to their buffer overflow, additionally we assume that randomization does not work and type-2 customer is always chosen for service if the number of type-2 customers in the buffer exceeds some predefined threshold. To the best of our knowledge such type of mechanisms of customers access to service was not considered in the literature.

One of the popular distributions of service time in queueing literature (and information transmission time in telecommunications literature) is so-called phase-type () distribution. The class of distributions is dense (in the sense of a weak convergence) in the set of all probability distributions of nonnegative variables. Thus, the distribution is very general and can be used for approximation of an arbitrary distribution; see [3]. However, in real life systems during some phases of service (information transmission), the server may fail. The failure occurrence implies the loss of a customer or the necessity of complete repetition of its service or resuming service from the phase at which the failure occurred. The use of the classical distribution does not allow taking into account the failures during the service process effectively. In this paper, we introduce and apply extension of the distribution, which we call (phase-type with failures) distribution, that allows taking the failures into account.

Last but not least, we consider the model with quite complicated marked Markovian arrival process (); see [4]. Such arrival process of heterogeneous customers is much more complicated than the superposition of the stationary Poisson processes. But it allows taking into account the bursty correlated nature of information transmission processes in modern telecommunication networks.

The rest of the paper is organized as follows. The distribution is introduced and briefly analyzed in Section 2. Mathematical model of the queueing system under study is described in Section 3. Process of the system states is defined in Section 4. Its generator as a block matrix is written down. The problem of ergodicity of this process is discussed in brief and the algorithm for computation of stationary distribution of the system states is chosen. In Section 5, expressions for computation of various performance measures of the system based on the known stationary distribution of the system states are derived. In Section 6, results of numerical experiments are presented. Section 7 concludes the paper.

#### 2. Phase-Type with Failures (PHF) Distribution

In this paper, we consider the queueing model of information transmission system. We assume that errors can occur during the transmission (service of a customer), which can cause the loss of information (customer) or necessity of retransmission (repetition, in full or partial extent, of customer’s service). Thus, to formally describe the service process of an arbitrary customer with account of possible errors, we introduce essential extension of well-known distribution; see, for example, [5]. We call this new distribution (phase-type with failures) distribution. In defining distribution, Neuts has interpreted the time having such a distribution as some sequence of random times called phases, durations of which have an exponential distribution. By analogy, we define distribution as some sequence of random times called stages each of which, in turn, consists of a random number of phases. distribution can characterize different positive random variables, for example, interarrival, interretrial, and service times, in some queueing system. Because in this paper we will use this distribution to model the service time in the queueing model described in the next section, for easier interpretation, we will speak here about the type distribution of a service time.

The type distribution of a service time is defined by the continuous-time Markov chain , with a finite state space . The states are assumed to be transient. The initial state of the process at the service beginning moment (which coincides with the moment of the first stage beginning) is chosen among the transient states with probabilities defined by the entries of the stochastic row vector The sojourn time of the chain in the state , is exponentially distributed with the parameter After this time expires, with probability , , , , , the process transits to the state . The intensities of the transition between transient states of the process are defined by the subgenerator that is defined by its entries as follows:

The states and are the absorbing states transition which corresponds to the end of the current stage of the service. The transition to the absorbing state means the end of the stage and successful completion of customer’s service. The intensities of the transition to the absorbing state are defined by the entries of the column vector :

The transition to the absorbing state means the end of the stage of the service due to a failure occurrence. The intensities of the transition to the absorbing state are defined by the entries of the column vector : Note that After the transition to the absorbing state , the following three scenarios are possible: (i) with probability , service of a customer is completed and this customer leaves the system permanently without successful service (is lost); (ii) with probability , the next stage of service starts and the initial state of the process , is again chosen among the transient states with probabilities defined by the vector ; (iii) with probability , the next stage of service starts and the initial state of the process is chosen as the state from which the transition to the absorbing state occurred. In both scenarios (ii) and (iii), the intensities of transition of the process within the set of the transient states and to the absorbing states are the same as at the first stage of service.

So service of a customer finishes when either the process transits to the absorbing state (this customer is considered as successfully served) or the process transits, after some stage, to the absorbing state and scenario (i) is realized (the customer is lost). In interpretation of the service time as transmission time of some information unit (e.g., a file), scenario (ii) means the necessity of complete retransmission of the unit while scenario (iii) means its retransmission from some point where the error occurs.

Thus, the distribution is defined by the set consisting of the row vector , the matrix , the column vector , and the probabilities and , while the classical distribution is defined only by the row vector and the matrix . The pair is called in [5] an irreducible representation of distribution. distribution can be treated as the special case of when By analogy with [5] we call the set an irreducible representation of distribution.

Let us describe some properties of distribution.

Let be Laplace-Stieltjes Transform () of distribution with an irreducible representation

Lemma 1. *The LST is defined by formula where the column vector function is defined by formula where where means the diagonal matrix with the diagonal entries defined by the entries of the vector , where *

Proof of Lemma 1 is implemented based on the known probabilistic interpretation of . Let us assume that, independently of the system operation, some virtual stationary Poisson flow of some virtual events, sometimes called catastrophes, arrives. Let be the intensity of this flow.

We define as the column vector entries, of which have the meaning of the probability that catastrophe will not arrive during the rest of the time having distribution with an irreducible representation conditioned on the fact that at the given moment the underlying Markov chain of this distribution stays at the state

Using this probabilistic interpretation and formula of total probability, it is easy to derive formula from which and formula of total probability the statement of Lemma 1 immediately follows.

*Remark 2. *It can be noted that of distribution with an irreducible representation coincides with of a classical distribution with an irreducible representation It is well known for the classical distribution that its is easily calculated given the irreducible representation. However, the inverse problem, to restore the irreducible representation given the values of , is very complicated and does not have a unique solution, while, namely, the components of the irreducible representation are necessary to write down the generator of the Markov chain describing behavior of the states of some queueing system with service process. Analogously, the noted coincidence of of distribution with an irreducible representation with of a classical distribution with an irreducible representation does not imply that we can avoid introduction of distribution and substitute this distribution by the classical distribution.

Corollary 3. *The moments of distribution with an irreducible representation are calculated by formula In particular, the expectation is given by .*

Let be of the distribution of the service time which is finished successfully (probability that the service time is finished successfully and no catastrophe arrives during this time) and let be of the distribution of the service time that is finished by a customer loss (probability that the service time is finished by a customer loss and no catastrophe arrives during this time).

Lemma 4. *The s and are defined by formulas *

Proof of Lemma 4 uses similar arguments as the proof of Lemma 1.

Corollary 5. *The probability that an arbitrary service will be finished successfully and the probability that an arbitrary service will be finished by a customer’s loss are given by *

Corollary 6. *The average service times () of an arbitrary customer conditioned on the fact that it is known that service is finished successfully (is failed) are computed as follows: *

#### 3. Mathematical Model

We consider a single server queueing system with an infinite buffer and a finite buffer of capacity , the structure of which is presented in Figure 1.