Parametric programming approach for batch arrival queues with vacation policies and fuzzy parameters
Introduction
Vacation queuing systems in which the server is unavailable during non-deterministic intervals of time have received considerable attention in literature. Levy and Yechiali [1] first proposed an M/G/1 queuing system under a single vacation policy—i.e., under the assumption that the server takes exactly one vacation when the system is empty. A study of the variations and extensions of this single vacation model was presented by Doshi [2] and Takagi [3] and included numerous applications in the study of maintenance problems in production/inventory schedules, computer networks and digital communication systems. The generality and flexibility of these vacation models are useful in modeling many real-life situations (see [4]). For example, in most computer systems, the processor is shared among various types of jobs and hence is not available to each job type at all times. From the perspective of one job type, the processor alternates between handling its job type and other job types. To reflect the occasional unavailability of the processor in queuing systems, the server is modeled as taking vacations (see [5]).
For queuing systems with batch inputs, Choudhury [6] successfully modeled a M[x]/G/1 queuing system (where [x] represents random batch arrival size) with a single vacation policy, extending the results of Levy and Yechiali [1] and Doshi [2]. The M[x]/G/1 queuing system with multiple vacations, where the server goes on vacations repeatedly until it finds at least one waiting customer at the end of a vacation, was first studied by Baba [7]. He derived the expected queue length, waiting time and busy period distributions through a supplementary variable technique. Lee and Srinivasan [8] examined the control operating policy of Baba’s [7] model using a general approach and presented applications in production/inventory systems and other areas. Later, Lee et al. [9], [10] analyzed in detail Lee and Srinivasan’s [8] system with single and multiple vacation policies. They also provided a probabilistic interpretation of the single (multiple) vacation system with a threshold policy.
In the literature described above, the batch inter-arrival times, customer service times and server vacation times are required to follow certain probability distributions. However, in many real-world applications, the parameter distributions may only be characterized subjectively; that is, the arrival, service and vacation patterns are typically described in everyday language summaries of central tendency, such as “the mean arrival rate is around 5 per day”, “the mean service rate is about 10 per hour” or “the mean vacation rate is approximately 2 per week”, rather than with complete probability distributions. In other words, these system parameters are both possibilistic and probabilistic. Thus, fuzzy queues are potentially much more useful and realistic than the commonly used crisp queues (see [11], [12]). By extending the usual crisp queues to fuzzy queues in the context of a vacationing server, these queuing models become appropriate for a wider range of applications.
Li and Lee [11] investigated the analytical results for two typical fuzzy queues (denoted M/F/1/∞ and FM/FM/1/∞, where F represents fuzzy time and FM represents fuzzified exponential distributions) using a general approach based on Zadeh’s extension principle (see also [13], [14]), the possibility concept and fuzzy Markov chains (see [15]). A useful modeling and inferential technique would be to apply their approach to general fuzzy queuing problems. However, their approach is complicated and not suitable for computational purposes; moreover, it cannot easily be used to derive analytic results for other complicated queuing systems (see [16]). In particular, it is very difficult to apply this approach to fuzzy queues with a vacationing server. Negi and Lee [16] proposed a procedure using α-cuts and two-variable simulation to analyze fuzzy queues (see also [17]). Unfortunately, their approach provides only crisp solutions; i.e., it does not fully describe the membership functions of the system characteristics. Using parametric programming, Kao et al. [18] constructed the membership functions of the system characteristics for fuzzy queues and successfully applied them to four simple fuzzy queue models: M/F/1/∞, F/M/1/∞, F/F/1/∞ and FM/FM/1/∞. Recently, Chen [19], [20] developed FM/FM/1/L and FM/FM[K]/1/∞ fuzzy systems using the same approach.
All previous research on fuzzy queuing models is focused on ordinary queues with one or two fuzzy variables. In this paper, we develop an approach that provides system characteristics for queues with a vacationing server and three fuzzy variables: fuzzified exponential arrival, service and vacation rates. Through α-cuts and Zadeh’s extension principle, we transform the fuzzy queues to a family of crisp queues. As α varies, the family of crisp queues is described and solved using parametric nonlinear programming (NLP). The NLP solutions completely and successfully yield the membership functions of the system characteristics, including the expected waiting time in the queue, the expected number of customers in the system and the expected lengths of time the server is idle and busy. Although an explicit closed-form expression for the membership function is very difficult to obtain in the case of three fuzzy variables, we develop a characterization that yields closed-form expressions when interval limits are invertible. Furthermore, this paper extends the analysis of system characteristics to encompass other system indices that are useful in more realistic systems.
The remainder of this paper is organized as follows. Section 2 presents the system characteristics of standard and fuzzy queuing models with a vacationing server. In Section 3, a mathematical programming approach is developed to derive the membership functions of these system characteristics. To demonstrate the validity of the proposed approach, two realistic numerical examples are described and solved. Discussion is provided in Section 4, and conclusions are drawn in Section 5. For notational convenience, our model in this paper is hereafter denoted FM[x]/FM/1/FV, where FV represents the fuzzified exponential vacation rate.
Section snippets
M[x]/M/1/V queues
We consider a queuing system with two different vacation policies. Under Policy I (multiple vacations), the server takes consecutive vacations when there are no customers queued for service, while under Policy II (single vacation), the server takes only a single vacation. It is assumed that customers arrive in batches according to a compound Poisson process with group arrival rate λ. Each batch size, A, has a probability mass function {σk : σk = Pr(A = k), k ⩾ 1}, and denotes the
The solution procedure
To re-express the membership function of (for Policy I) in an understandable and usable form, we adopt Zadeh’s approach, which relies on α-cuts of . Definitions for the α-cuts of , and as crisp intervals are as follows:The constant arrival, service and vacation rates are shown as intervals when the membership functions
Numerical examples
We consider two examples motivated by real-life systems to demonstrate the practical use of the proposed approach. Example 1 The system under Policy I Consider a local postal route in a central mail handling system. A postal worker collects mail from mailboxes with fixed pick-up times and delivers the mail to the local post office. The workers at the local post office collect mail up to a certain point and then send batches to the central mail handling office. The number of parcels sent each time follows a geometric distribution
Conclusions
This paper applies the concepts of α-cuts and Zadeh’s extension principle to a batch arrival queuing model with a vacationing server and constructs membership functions of the expected waiting time in the queue, the expected number of customers in the system and the expected lengths of time the server is idle and busy using paired NLP models. Following the proposed approach, α-cuts of the membership functions are found and their interval limits inverted to attain explicit closed-form
Acknowledgement
The authors wish to thank R. Gulati for polishing the paper which improved its readability.
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