In the classical queueing literature, the server is usually assumed to work at constant speed as long as there is any work present. However, we know that this assumption may not always be appropriate when the system’s workload affects the server’s efficiency in some real world situations. To better understand this fact, we can cite some practical examples to illustrate this point. In a manufacturing system, the decision-maker is responsible for deciding the service speed of the production equipment according to the level of market demand. If the current production capacity is far from meeting market demand, high service rate will be activated to balance the requirements. Nonetheless, once the demand is satisfied and decreases significantly, production will be slowed down to avoid inventory pile up. In addition, the telephone-based directory assistance is another convincing example of service rate depending on the queue length, where as the number of calls increases, the provision of extra attendants is recommended so as to provide better quality of service in terms of reduced waiting time. However, these extra attendants may be removed when the peak time is over and the number of phone calls sharply reduces. Therefore, these real-life applications that mentioned above constitute the main motivation of our study.

Actually, there is a considerable body of queueing literature that deals with workload-dependent service rate. Among some early papers in this area are those by Satty [1] and Gebhard [2] , both of whom considered some fundamental queueing problems such as the stationary queue size distribution and the expected queue length for the M/M/1 queue. Their work spawned research into modeling a queueing system with adaptable service rate, such as by Gross and Harris [3] , Harris and Marchal [4] , William and Wang [5] , Bekker et al. [6] and Zhernovyi [7] . In the past several decades, an important extension to the above model is the multi-server queueing system with queue-dependent servers. Singh [8] respectively analyzed the infinite source M/M/2 queueing systems with two homogeneous and heterogeneous servers. A relationship among the system operating costs, traffic intensity and the queue size is obtained. Later, based on the Singh’s pioneering work, Garg and Singh [9] revisited the same system and established a cost structure to determine the optimal queue length at which the second server was provided, so that the system may gain the maximum profit. With the assumption that the system capacity was limited, Wang and Tai [10] studied queue-dependent servers in the finite buffer M/M/3 queue with three types of service rate. They constructed a relationship among the costs to determine the optimal queue lengths J and K of providing the second server and the third server, respectively. Furthermore, not long ago, Jain [11] investigated the finite capacity M/M/r queueing system with r heterogeneous servers. In particular, the optimal threshold parameters for turning on the servers were obtained in her work. More recently, M/M/r ueueing model with infinite capacity and queue-dependent servers was considered by Lin and Ke [12] . Using the genetic algorithm, they found the best thresholds of queue length in activating servers and their corresponding service rate. These studies greatly enhance the practical value of the multi-server queueing theory since it is realistic to consider the changes in the number of working servers.

However, we may note a common feature existing in the above research works, namely, authors invariably assum that whenever the number of customers or jobs in the system exceeds a certain threshold, the service rate is accelerated to deal with the lengthy queue. Further, if the queue length reduces to less than the threshold, lower service rate is resumed. In fact, such model assumption means that the service rate can be switched countlessly in a regeneration cycle. Here, for the single server queue, the regeneration cycle is the time span between two consecutive starting points of the server’s idle period. Obviously, whenever a server is switched from low service rate to high service rate, or vice-versa, switching cost is incurred. The more the server switches its service rate, the more additional cost it has to face. In other words, if the switch is reiterated over a long period of time, substantial amount of switching cost will be charged to the system. Therefore, the traditional service rate switching policy has some significant drawbacks in the queueing system with a relatively high arrival rate. In order to prevent switches from occurring too frequently, a modified service rate switching policy is proposed in this paper. Under the control of modified switching policy, the high service rate is activated as soon as L customers accumulate in the system and such service rate is preserved until the system becomes completely empty even if the number of customers falls below L. Hence, for the modified switching policy, the change of service rate can only occur at most once in a regeneration cycle. Undoubtedly, this policy will greatly reduce the switch- ing cost of the system. On the other hand, although a lot of continuous-time queueing models with workload- dependent service rate have been studied extensively in the past years, their discrete-time counterparts received very little attention in the literature. Except the studies done by Chaudhry [13] [14] and Parthasarathy and Lenin [15] , no work in this direction has come to our notice. Given that the wide applications of discrete-time queue in digital data networks and flexible manufacturing systems, in this paper, we will develop an analytical model that allows us to extensively analyze and explore the Markovian queueing system with workload-dependent service rate in discrete-time case. Through our work, we wish to develop a computational model that helps decision- makers answer the following important questions: 1) Under a certain cost structure, what is the optimal value of L that minimizes the long-runaverage cost rate function? 2) If the system state information is communicated to the customers upon their arrival, how to evaluate the expected conditional sojourn time of an arriving customer?

The rest of this paper is organized as follows. In Section 2, we describe the mathematical model for the problem under consideration. The steady-state analysis of the model is presented in Section 3 and some important system performance measures are derived in this section. Using the first-step argument, we develop an analytical scheme for the customer’s sojourn time. Furthermore, we also carried out regeneration cycle analysis to find the expected length of two types of busy periods. In Section 4, a long-run average cost rate function is established based on the system characteristics to determine the optimal switching threshold for the service rate. Section 5 concludes the research and suggests some future topics.

We consider a discrete-time queue with single server or machine, whose service rate may be affected by the number of customers or jobs present in the system. In our model, inter-arrival times are independent and identically distributed (i.i.d.) random variables with probability mass function (p.m.f.) and. Arriving customers form a single waiting line based on the order of their arrival. Initially, when the number of customers in the system is less than the given threshold level, the server serves customers with low service rate. The high service rate is activated at the instant when the number of customers in the system becomes equal to L, and it will be preserved until the long queue empties. Here, we assume that the two types of service times, namely and, are independent and geometrically distributed with respective p.m.fs

where, and., and denote the customer arrival rate, low service rate and high service rate, respectively.

In discrete-time queueing system, the time axis is divided into equal intervals called slots and all queueing activities occur at the slot boundaries. Traditionally, there are two types of systems in the discrete-time case (see [16] and [17] ), one is the late arrival with delayed access (LAS-DA) and the other is the early arrival system (EAS). In this paper, we consider the model for the late arrival system with delayed access and therefore, a potential arrival occurs in, and a potential departure takes place in, for. To make it clear, the various time epochs at which events occur are shown in a self-explanatory figure (see Figure 1).

In this section, we first apply the Markov process theory to obtain the steady-state difference equations governing the system. Next, the generating function technique and a recursive method are employed to develop the analytical solutions in a neat close-form. Toward this end, we need to define some commonly used notations to analyze the queueing system as follows:

the number of customers in the system at time,

the server speed state at time,

where

Therefore, is the Markov chain for queueing system with state space

Furthermore, let us define the following stationary probability distributions for the Markov chain:

In manufacturing process management, managers are always interested in minimizing the long-run average cost per unit time of the system. In this section, based on the performance measures that we obtained in the previous section and the renewal reward theorem, we first construct an expected cost rate function for the Geo/Geo/1 queue with switching threshold for the service rate, in which a key decision variable L is considered. Here, our objective is to determine the optimal threshold value under some cost structure, so as to minimize the long-run average cost rate.

Let us consider the following cost elements:

setup cost per cycle;

switching cost for changing the service rate in a regeneration cycle;

holding cost per customer per unit time;

running cost per unit time when the service provides low speed service;

running cost per unit time when the service provides high speed service.

Utilizing the definition of each cost element listed above, the long-run average cost rate minimization problem can be illustrated mathematically as

As shown in Figure 4 and Figure 5, the switching cost is incurred at most only once in a regeneration cycle, and the switching occurs with probability. This is the reason why we multiply by in our cost structure. On the other hand, we also note that it is rather difficult to develop analytic results for the optimal value of L because the long-run average cost rate function is highly non-linear and complex. In spite of that, since L is a discrete variable, the optimal value may be found by using direct substitution of successive values of L into the long-run average cost rate function until the minimum value is achieved.

To illustrate the direct search algorithm described above, a numerical example is provided by considering the following cost parameters:

and other system parameters are taken as, ,. Substituting these values into, we can obtain the results presented in Table 2 and Figure 6. The curve representing the long-run average cost rate function is plotted in Figure 6 for different values of L. As can be seen in Figure 6, we observe that this function is convex and a single relative minimum exists. The optimal value and the corresponding long-run average cost rate are tabulated in Table 2. From Table 2, it appears that the minimum average cost per unit time of 24.2347 is obtained with.

In this paper, we have carried out an analysis of a discrete-time infinite-buffer Geo/Geo/1 queuing system under

a modified service rate switching policy that has potential applications in modeling manufacturing and telecommunication systems. We have developed a recursive method to find the steady-state queue size distribution. The recursive method is powerful and easy to implement. Further, we obtain the analytically explicit expressions for the expected number of customers in the system. Using the first-step argument, a simple algorithm for calculating the customer’s mean sojourn time has been proposed. Moreover, we also performed regeneration cycle analysis of the queue to find the optimal service rate switching threshold L. Our current model is useful and significant to engineers or managers who design an efficient system with economic management. It should be pointed out that the economic importance of this model resides in the multiple applications to manufacturing processes, since most of them operate on a discrete time basis. Furthermore, the optimal control of service rate switching policy is also a main objective from the enterprise point of view. For future studies, the present investigation can be extended by incorporating bulk input or bulk service. Another area of interest may be expanding our model into Geo/G/1 type, because there will be a significant improvement inapplicability to real world system.

The work described in this paper is supported by Sichuan Provincial Department of Education (14ZB0221).