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This paper considers a Geo/Geo/1 queueing system with infinite capacity, in which the service rate changes depending on the workload. Initially, when the number of customers in the system is less than a certain threshold L, low service rate is provided for cost saving. On the other hand, 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. The steady-state probability distribution and the expected number of customers in the system are derived. Through the first-step argument, a recursive algorithm for computing the first moment of the conditional sojourn time is obtained. Furthermore, employing the results of regeneration cycle analysis, the direct search method is also implemented to determine the optimal value of L for minimizing the long-run average cost rate function.

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 [

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 [

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

where

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 [

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:

where

Therefore,

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

From the state-transition-rate diagram for the Geo/Geo/1 queue with service rate switching threshold (see

Remark 1. The Markov chain

Remark 2. Relating the state probabilities at epochs t and

In this subsection, we first derive two important relationships between

Multiplying Equations (1)-(4) by

If we add the right hand sides and left hand sides of the Equations (1)-(4) and cancel the common terms, the following equality holds:

Remark 3. As shown in

Substituting Equation (10) into Equation (9) and after some algebraic manipulation, we have

Using a method similar to the derivation of Equation (11), with Equations (5)-(8), we get

Thus, we can rewrite

Since

Based on Equation (14) the first relationship between

On the other hand, with the help of Equations (2)-(4), another relationship between

Substituting Equation (16) into Equation (15), it follows that:

Remark 4. Obviously, the queueing system for

(17) can further be simplified as follows:

the one given by Hunter [

Having computed the stationary probabilities

Once the explicit expressions for

Begin algorithm |
---|

Input: |

Using L’Hospital’s Rule twice while taking limits

Remark 5. As a matter of fact, the explicit expression for the expected number of customers in the system has been given by Equation (18). Just because the explicit expressions

In this subsection, we deal with the customer’s sojourn time W, defined as the time between the arrival epoch of a customer till the instant at which his service request is satisfied. Here, our aim is to determine the first order moment of the sojourn time. To achieve this goal, we need to introduce some auxiliary random variables.

We also denote the corresponding z-transforms of W,

By differentiating Equation (19) with respect to z, and evaluating at

For determining the unknowns

Assume that a customer arrival will occur in

For the same reason as mentioned above, when a customer arrives at the system during a busy period with high service rate, we conclude that

Alternatively, we can use the memoryless property of the geometric distribution to find the z-transform of

For

Similarly, for

Differentiating both sides of Equation (21) and Equations (25)-(28) with respect to z and evaluating at

Therefore, from the above results and Equations (22)-(24), we obtain

Thus, the problem of computing the mean conditional sojourn times

To demonstrate the feasibility and efficiency of the proposed algorithm, a numerical experiment is carried out on a personal computer implementing an Intel Core i5 CPU (2.7 GHz) and 4.0 GB RAM. In this example, we select

Regeneration cycles are models of stochastic phenomena in which an event (or combination of events) occurs repeatedly over time, and the times between occurrences are independent and identically distributed. Models of such phenomena typically focus on determining limiting averages for costs or other system parameters. In this paper, the reason for performing regeneration cycle analysis is to determine the optimal switching threshold value L, where the high service rate is activated.

A regeneration cycle of our current model consists of a server’s idle period and a server’s busy period. As regeneration points, we choose the points at which the system becomes empty. There are two types of cycles depending on whether there is a change in service rate during the server’s busy period. A cycle is called “type-1” if it does not include switching of the service rate; otherwise it is of “type-2” cycle. To better understand the struc- ture of regeneration cycle, examples of the type-1 and type-2 cycles are shown in

We denote

single server follow geometric distribution with parameter

where

where

On the other hand, let

we can get

Comparing the right hand sides of Equations (38) and (41), we see that

Once we have found the expressions of

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

Let us consider the following cost elements:

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

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

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

L | TC(L) | L | TC(L) | L | TC(L) |
---|---|---|---|---|---|

2 | 29.1306 | 11 | 24.6055 | 20 | 25.5883 |

3 | 26.5196 | 12 | 24.7529 | 21 | 25.6414 |

4 | 25.2645 | 13 | 24.8970 | 22 | 25.6858 |

5 | 24.6309 | 14 | 25.0325 | 23 | 25.7228 |

6 | 24.3345 | 15 | 25.1567 | 24 | 25.7535 |

7 | 24.2347 | 16 | 25.2680 | 25 | 25.7787 |

8 | 24.2522 | 17 | 25.3663 | 26 | 25.7993 |

9 | 24.3386 | 18 | 25.4519 | 27 | 25.8162 |

10 | 24.4630 | 19 | 25.5256 | 28 | 25.8298 |

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).