Optimization of multi-channel queuing systems with a single retail attempt: Economic approach

. by the authors; licensee Growing Science, Canada 20 20 ©

This paper deals with retrial queues of the |MQ/M/c/ꝏ|-type in which the rate of input flow depends on the number of sources of repeated calls and each call has only one retrial attempt. That is, if a call fails to enter the server facility at the retrial attempt, then it leaves the system without service. The existence conditions of stationary regime and the vector-matrix of the stationary probabilities of the service process are represented. This representation uses an approximation of the initial model by means of the truncated one and the directs passage to the limit. For these systems a threshold strategy for the rate of input flow is used. The multi-criterion optimization problem for finding optimal strategy of control is considered. The quality functionals of the optimization problem are represented through the stationary probabilities.

Introduction
Nowadays, the modern development of call centers, telecommunication and computer networks requires new models of queuing systems, which model real processes more fully. It is necessary to consider queuing systems that take into account the phenomenon of repeated calls. Such retrial queues can be used to solve the practical problems without loss of calculation accuracy. For retrial queues, it is assumed that calls that arrive in the system while all the servers are busy become sources of repeated calls and return their demands to get service after some time. Calls that had no access to servers at the moment of arrival are assumed to be in the orbit and independently apply for service in a random time (Falin et al., 1997;Kovalenko, Koba, 2010). All calls in the orbit are equivalent, disordered.

Literature review
Classical retrial queues have been analyzed in the monographs and articles where generating functions and truncated models have been used (Falin et al., 1997;Artalejo et al., 2008;Yakubiv et al., 2020;Kuznetsov et al., 2019;Pavlov., 2020). It made it possible to obtain a lot of results for such systems. Typically for retrial queues, it is assumed that each call can get its repeated demands until it receives service. This is only an approximation to real situations as the number of repeated attempts to the system is often limited. Systems with limited number of retrials attempts to begin service have been investigated numerically by using the algorithmic method and simulation experiments (Shin, Moon, 2008;Irtyshcheva et al., 2020;Harkusha et al., 2016;Hrynkevych et al., 2020). But research of these retrial queues also includes the search for conditions of the stationary mode existence and construction of algorithms for calculating the stationary distribution. It is quite relevant in nowadays, in particular, from the point of view of optimization these systems. The problem of choice of the optimal rate of the input flow has its practical meaning. The threshold strategy is used for an optimal control in retrial queues. It can be used in the case of the rate of input flow dependent on the number of calls in the orbit (the number of sources of repeated calls). Such modifications of retrial queues are more difficult for research.
In the present paper, we consider the case of the rate of input flow dependent on the number of calls in the orbit. Also we assumed that each call can make a single retrial attempt to begin service process. Under these conditions, the method of generating functions is not used in the analysis of service process. To investigate the characteristics of service process in stationary mode, the approximate approach has been developed. According to this approach, a truncated finite system is considered first, and then the obtained results are used for analysis of the original system. The aim of this paper is to solve the problem of control of the rate of input flow in case of threshold strategy using effective formulas for stationary probabilities.

Materials and Methods
The main model that we consider in this paper is a two-dimensional Markov chain This Markov chain models the process of service in the following system. Calls arrive at the input for service. If there is at least one free server at the moment of arrival, the call instantly occupies it and then leaves the system after the service. The service time is a random variable exponentially distributed with the parameter μ. If all the servers are busy, then the arriving call becomes a source of repeated call and tries again to obtain service in a random time that is exponentially distributed with the parameter v. The call that finds out all the servers busy at the time of retrial, leaves the system and does not obtain the service. It means that each call has only one retry to begin the service. The intensity of arrival flow is λj, j=0,1,… and depends on the number j of sources of repeated calls. Similar processes were analyzed by numerical methods in (Shin, Moon, 2008). According to the notation accepted in the queuing theory, we will represent the model under study by |Mg/M/c/ꝏ|. The character ꝏ at the last place means the absence of constraints for the number of sources of repeated calls. We also assume that λj, μ,v>0, j=0,1,…, which is always taken into account in practice. If this requirement is satisfied, we will say that the model parameters are non-degenerate. The conditions of the existence of stationary mode for , , , Using the theorem about the equality of flows of probabilities through the boundary of domain ( ) j S Q in stationary mode (Walrand, 1988), we get (7) In the general case, for an arbitrary number of sources of repeated calls, it is impossible to find explicit formulas for the stationary probabilities that satisfy the Eqs.
(3)- (7). An exception is one-channel systems, which are considered in (Lebedev, Pryshchepa, 2007). Taking this into account, let us consider a truncated model |MQ/M/c/N| which has a finite number of sources for repeated calls N. If all the servers are busy and N places of repeated calls are generated, new arrivals are completely lost by the system. The process of service for truncated model is defined by two-dimensional Markov chain    ..., , i N cN N N N c N N cN N c

Optimal control in a set of threshold strategies.
We have analyzed retrial queues with a single retrial attempt in case of the rate of input flow dependent on the number of calls in the orbit. It makes it possible to set and solve optimization problems for such systems. In view of this we consider a class of many-threshold strategies that are set by thresholds where the cost coefficients are: Ci is the income relating to the service of a call in the i-th regime, i=1,…K; CK+1is the penalty relating to service failure; CK+2 is the penalty relating to switching of the input flow rate, CK+3 is the penalty relating to lost call after it's the last failed attempt. The solution of the problem (16) is such many-threshold strategy that maximizes the average system operation profit. The functionals ( ), 1,..., 3, i S H i K   can be written via the stationary probabilities of service process: In the case K=2, formulation of the optimization problem becomes simpler as follows, Thus, the functionals ( ), 1,5 i S H i  can be represented in terms of the stationary probabilities of |MQ/M/c/ꝏ| -queue as follows:

Conclusion
In the present paper, we have investigated retrial queues with one retry to begin service. We have represented formulas for stationary probabilities in terms of systems' parameters, which allow solving optimization problems. In accordance with threshold strategy, the multi-criterion optimization problem of the effective functioning of the system has been formulated and solved. Quality functionals of the optimization problem are represented through the stationary probabilities.