A repairable queueing model with two-phase service, start-up times and retrial customers
Introduction
The main characteristics of the queueing model analysed in this paper are (i) the retrial customers (jobs), (ii) the server breakdowns and repairs, (iii) the two-phase service and (iv) the start-up (system preparation) times.
Queueing systems with repeated attempts (retrials) are characterized by the feature that an arriving customer who finds the server unavailable, leaves the system, joins a pool of unsatisfied customers, the so-called retrial box, and repeats his demand for service after a random amount of time. Retrial queues have been widely used to model many problems in telephone switching systems, telecommunications networks and computer units. For a complete survey on this topic we refer Artalejo [3], Kulkarni and Liang [17], and the books of Falin and Templeton [13], and Artalejo and Gomez-Corral [5].
In most of the queueing literature, the server is assumed to be reliable and always available to customers. However, in practice, we often meet cases where the server may breakdown and has to be repaired. In queueing literature, there have been several works taking into account both retrial phenomenon and server breakdowns with repairs. As related works we mention the papers by Aissani [1], Aissani and Artalejo [2], Kulkarni and Choi [16], Wang et al. [25], Kumar et al. [20].
The assumption of a two-phase service provided by a single server has been proved useful to analyse many practical situations arising in packet transmissions, multimedia communications, central processors, etc. Such kind of systems have been discussed for the first time by Krishna and Lee [15] and Doshi [12], and more recently have been generalized to include models with vacations, N-policy, etc. (see [6], [9], [14]).
Wang [24], considered a two-phase queueing model with the assumptions of breakdowns and repairs, in which he assumed that the second optional service follows an exponential distribution. Kumar et al. [18], Artalejo and Choudhury [4], and Choudhury [7] are the first who imposed the concept of retrial customers in the two-phase models. Kumar et al. [18] generalize their previous work of a single service station [19] by considering now a two-phase service system where an arriving customer who finds the server unavailable joins the retrial box from where only the first customer can retry for service after an arbitrarily distributed time period while in the work of Choudhury [7] the investigated model includes Bernoulli server vacations and linear retrial policy. The common feature of the above papers is that there are no server breakdowns, no ordinary queue and all waiting customers join the retrial box. Choudhury and Deka [8], generalize the works by Wang [24] and Artalejo and Choudhury [4] by considering an M/G/1 retrial queue with second optional service channel which is subject to server breakdowns and repairs. Wang and Li [26] consider a similar model, where only the first retrial customer can retry for service after an arbitrarily distributed time period.
Recently Dimitriou and Langaris [11], considered a two-phase model where all arriving customers are queued up in a single ordinary queue. After the completion of the first phase service the customer either proceeds to the second phase or joins the retrial box from where he retries, after a random amount of time, to find the server available, and to complete his second phase of service. This system is suitable to model situations (production lines, communication networks, etc.) where a quality control and a possible separation of the serviced units, in high and low quality items, is needed.
It is easy to understand that, in any real situation of the kind described above, a special preparation (set-up) of the servicing procedure may be needed any time it starts serving a low quality unit. Moreover the servicing machine is naturally subject to failures and of course it needs repair to restart serving. The system in [11] cannot be used to model such a real situation, as it does not contain the important characteristics of start-up times and of breakdowns and repairs.
In this work we generalize the model of Dimitriou and Langaris [11], allowing server breakdowns and repairs in both phases of service, while in addition, the server needs a start-up time in order to start serving a retrial customer in the second phase of service. Our system can be used to model any situation with two stages of service where in the first stage a control and a separation of the serviced units, according to some quality standards or some measure of importance, must been taking place. If a unit satisfies these quality standards then it proceeds immediately to the second phase of service while if the quality of the unit is poor then it is removed from the system and repeats its attempt to receive a special second service later when the server is free from high quality units. Moreover the machine (server) is subject to breakdowns and repairs while a special preparation of the machine is needed to start serving the low quality unit. Such a situation often arises in packet transmissions, manufacturing systems, central processors, multimedia communications, etc. It is clear that the concepts of breakdowns, repairs, and the start-up period for the retrial customer, make our model here more realistic and applicable.
The article is organized as follows. A full description of the model is given in Section 2. Some very useful for the analysis, results on the customer completion time and server busy period are given in Section 3. In Section 4 the conditions for statistical equilibrium are investigated. The generating functions of the steady state probabilities are obtained in Section 5 and used to give, in Section 6, some important measures of the system performance. Finally in Section 7, numerical results are obtained and used to compare system performance under various changes of the parameters.
Section snippets
The model
Consider a queueing system consisting of two phases of service and a single server, who follows the customer in service when he passes from the first phase to the second. Customers arrive to the system according to a process with parameter , and are placed in a single queue waiting to be served. When a customer finishes his service in the first phase, he either goes to the second with probability or he joins, with probability , a retrial box from where he retries, independently to
Preliminary results
We agree from here on to denote in general by the Laplace–Stieltjes transform (LST) of any function .
Let us define now by the time interval from the epoch a customer starts his service in the second phase until the epoch the service is successfully completed. We have to point out here that the service period of a customer may be delayed due to server breakdowns. Let also be the number of new customers that arrive during . Note here that the interrupted customer
Stability conditions
Let be the number of , customers in the ordinary queue (not in service) and in the retrial box, respectively, at time and denote by
Consider also the time instants where is the epoch at which the server becomes idle for the th time, and let i.e. denote
Steady state probabilities
Let us assume that and so a state of statistical equilibrium exists for our model. Let also , . Define finally for where the elapsed duration of any random variable , and the corresponding generating functions
Performance measures
In the sequel we will use formulas for the generating functions obtained above, to derive expressions for the system performance. Thus by putting into relations (17), (18), (19), (20), (21), (22) we obtain easily
Numerical results
In this section we use the formulae derived previously to obtain numerical results in order to investigate the way, the probability of an idle server , the mean duration of the generalized busy period the mean duration of the generalized completion time and mainly the mean number of customers in the retrial box are affected when we vary the values of the parameters. We have to point out here the crucial role that plays on the opportunity of a customer to find
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2014, Applied Mathematical ModellingCitation Excerpt :As early studies on this topic, we refer Thirurengadan [11], Mitrany and Avi-Itzhak [12], and references therein. Recently, Choudhury and Ke [13], Choudhury and Tadj [14], Dimitriou [15], Dimitriou and Langaris [16], Falin [17], Ke [18–21], Ke et al. [22], Lee et al. [23], Yang et al. [24], and others considered the unreliable queueing systems with various features wherein one of the underlying assumptions is that a failed server is sent for repair at the repair shop and present customers in the system should wait for the server to be repaired without being served. However, in practical situations, the system should be equipped with a substitute (standby) server in preparation for possible main server failures.
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Availability and reliability analysis of a retrial system with warm standbys and second optional repair service
2023, Communications in Statistics - Theory and MethodsThe M/M/1 Repairable Queueing System with Variable Input Rates and Failure Rates
2022, IAENG International Journal of Applied MathematicsA simulation study on the necessity of working breakdown in a state dependent bulk arrival queue with disaster and optional re-service
2022, International Journal of Ad Hoc and Ubiquitous Computing