A Batch-Arrival Queue with Multiple Servers and Fuzzy Parameters: Parametric Programming Approach

: This work constructs the membership functions of the system characteristics of a batch-arrival queuing system with multiple servers, in which the batch-arrival rate and customer service rate are all fuzzy numbers. The  -cut approach is used to transform a fuzzy queue into a family of conventional crisp queues in this context. By means of the membership functions of the system characteristics, a set of parametric nonlinear programs is developed to describe the family of crisp batch-arrival queues with multiple servers. A numerical example is solved successfully to illustrate the validity of the proposed approach. Because the system characteristics are expressed and governed by the membership functions, the fuzzy batch-arrival queues with multiple servers are represented more accurately and the analytic results are more useful for system designers and practitioners.


Introduction
Queuing models with multiple servers are effective methods for performance analysis of computer and telecommunication systems, manufacturing/production systems and inventory control (Kleinrock [11], Buzacott and Shanthi Kumar [1], Gross and Harris [7], Trivedi [19]).In general, these analyses consider a queuing system where requests for service arrive in units, one at a time (single-unit arrival).In many practical situations, however, requests for service usually arrive in batches.For example, in manufacturing systems of the job-shop type, each job order often requires the manufacture of more than one unit; in computer communication systems, messages which are to be transmitted could consist of a random number of packets.If the usual crisp batch-arrival queues with multiple servers can be extended to fuzzy batch-arrival queues, such queuing models would have wider applications.
For queuing models with multiple servers under various considerations, the M/M/c vacation systems with a singleunit arrival have attracted much attention from numerous researchers since Levy and Yechiali [12].The extensions of this model can be referred to Vinod [20], Igaki [8], Tian et al. [17], Tian and Xu [18], and Zhang and Tian [23,24].Zhang and Tian [23,24] studied the M/M/c vacation systems with a single-unit arrival and a "partial server vacation policy".They proved several conditional stochastic decomposition results for the queue length and waiting time.Chao and Zhao [3] investigated the GI/M/c vacation models with a single-unit arrival and provided iterative algorithms for computing the stationary probability distributions.
In the literature described above, customer inter-arrival times and customer service times are required to follow certain probability distributions with fixed parameters.However, in many real-world applications, the parameter distributions may only be characterized subjectively; that is, the arrival and service are typically described in everyday language summaries of central tendency, such as "the mean arrival rate is around 5 per day", or "the mean service rate is about 10 per hour", 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 Li and Lee [13] and Zadeh [22]).By extending the usual crisp batch-arrival queues to fuzzy batch-arrival queues in the context of multiple servers, these queuing models become appropriate for a wider range of applications.
Li and Lee [13] 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 Prade [15] and Yager [21]), the possibility concept and fuzzy Markov chains (see Stanford [16]).A useful modeling and inferential technique would be applied their approach to general fuzzy queuing problems (see Stanford [16]).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 Negi and Lee [14]).In particular, it is very difficult to apply this approach to fuzzy queues with more fuzzy variables or multiple servers.Negi and Lee [14] proposed a procedure using αcuts and two-variable simulation to analyze fuzzy queues (see also Chanas and Nowakowski [2]).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. [9] 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 [4,5] developed FM/FM/1/L and FM/FM [K] /1/  fuzzy systems using the same approach.
All previous researches on fuzzy queuing models are focused on ordinary queues with a single server.In this Paper ID: 02091301 www.ijsr.netpaper, we develop an approach that provides system characteristics for batch-arrival queues with multiple servers and fuzzy parameters: fuzzified exponential batch-arrival and service rates.Through  -cuts and Zadeh's extension principle, we transform the fuzzy queues to a family of crisp queues.As  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 number of customers in the system and the expected waiting time in the queue.
The remainder of this paper is organized as follows.Section 2 presents the system characteristics of standard and fuzzy batch-arrival queuing models with multiple servers.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, one realistic numerical example is 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/c.

Fuzzy Batch Queue With Multiple Servers
We consider a batch-arrival queuing system with c servers where the customers arrive in batches to occur according to a compound Poisson process with batch-arrival rate  .Let k A denote the number of customers belonging to the k th arrival batch, where , , and . Customers arriving at the service facility (servers) form a single-file queue and are served in order.The service time for each of all c servers is exponentially distributed with rate  and each server can serve only one customer at a time.Customers who upon entry the service facility find that all servers are busy have to wait in the queue until any one server is available.Let s N and q W represents the expected number of customers in the system and the expected waiting time in the queue, respectively.Through a Markov process, we can easily obtain s N and q W in terms of system parameters ]) [ ( 2) , ( Where ) , (   n P represents the probability that there are n customers in the system.And the probability depends on  and  .In steady-state, it is necessary that we where X and Y are the crisp universal sets of the batcharrival and service rates.Let ) , (   f is also a fuzzy number.Following Zadeh's extension principle (see Yager [21] and Zadeh [22]), the membership function of the system characteristic ) , , (4) Assume that the system characteristic of interest is the expected number of customers in the system.It follows from (1) that the expected number of customers in the system is: The membership function for the expected number of customers in the system is: Unfortunately, the membership function is not expressed in the usual form, making it very difficult to imagine its shape.
In this paper we approach the representation problem using a mathematical programming technique.Parametric NLPs are developed to find the  -cuts of ) , (   f based on the extension principle.

Parametric Nonlinear Programming
This can be accomplished using parametric NLP techniques.
The NLP to find the lower and upper bounds of the  -cut of ) ( and for Case (ii) are:
The crisp interval [ ] obtained from (9) represents the  -cuts of s N ~.
Again, by applying the results of Kaufmann [10] and Zimmermann [25] and convexity properties to , where 1 0 shows the shape of ) ( ~z s N  , although the exact function is not known explicitly.Note that the membership functions for the expected waiting time in the queue can be expressed in a similar manner.

Numerical Example
This section we present one example motivated by real-life systems to demonstrate the practical use of the proposed approach, which is based on http://www.macaudata.com/macauweb/book175/html/19301.htm Paper ID: 02091301 www.ijsr.netExample: Considering one sewerage treatment system collects sewage from the urban areas and sends them to the sewerage treatment plant.The sewerage treatment plant has three supply pipes (referred to 3-servers).Each pipe can settle the larger solids and put the settled into the chemical process tank.After the chemical process, the treated water is discharged to the sea.We assume that the number of arriving sewage solids each time follows a geometric distribution with parameter 0.5 p = ; i.e., the size of arriving sewage solids A is Clearly, this problem can be described by FM [x] /FM/3 system.For efficiency, the management wants to get the system characteristics such as the expected number of sewage solids in the system and the expected waiting time in the queue.Next, we perform  -cuts of batch-arrival and service rates and fuzzy expected number of sewage solids in the system at eleven distinct  values: 0, 0.1, …, 1. Crisp intervals for fuzzy expected number of sewage solids in the system at different possibilistic  levels are presented in Table 1.

Conclusions
This paper applies the concepts of  -cuts and Zadeh's extension principle to a batch-arrival queuing system with multiple servers and constructs membership functions of the expected number of customers and the expected waiting time using paired NLP models.Following the proposed approach,  -cuts of the membership functions are found and their interval limits inverted to attain explicit closed-form expressions for the system characteristics.Even when the membership function intervals cannot be inverted, system designers or managers can specify the system characteristics of interest perform numerical experiments to examine the corresponding  -cuts and then use this information to develop or improve system processes.
applicability of the batch-arrival queuing model with multiple servers, we allow for fuzzy specification of system parameters.Suppose the batcharrival rate  for customers and service rate  for each server are approximately known and can be represented by the fuzzy sets  of  ~ and  ~.We then have the following fuzzy sets: the system characteristic of interest.Since  ~ and  ~ are fuzzy numbers, To re-express the membership function ) ( ~z s N  of s N ~ in an understandable and usable form, we adopt Zadeh's approach, which relies on  -cuts of s N ~.Definitions for the  -cuts of  ~ and  ~ as crisp intervals are as follows: The constant batch-arrival and service rates are shown as intervals when the membership functions are no less than a Paper ID: 02091301 www.ijsr.netgiven possibility level for  .As a result, the bounds of these intervals can be described as functions of  and can can use the  -cuts of s N ~ to construct its membership function since the membership function defined in (6) is parameterized by  .Using Zadeh's extension principle, at least one of the following cases to hold

From
8a) and (8c) have the same smallest element and (8b) and (8d) have the same largest element.To find the membership the  -cuts of s N ~, which can be rewritten as: min of x and y must hit the boundaries of their  -cuts to satisfy ) ( ~z s N  = .This model is a set of mathematical programs with boundary constraints and lends itself to the systematic study of how the optimal solutions change with L x  , U expected number of sewage solids in the system attains its maximum value, and when number of sewage solids in the system attains its minimum value.According to (9), the  -cuts of s Fig.1.The overall shape turns out as expected.of these two functions have no influence on the computational results and can be disregarded.
characteristics to be noted.First, the support of s N ~ ranges from 0.2742 to 1.4443; this indicates that, though the expected number of sewage solids in the system is fuzzy, it is impossible for its values to fall below 0.2742 or exceed 1.4443.Second, the  -cut at 1   contains the values from 0.5933 to 0.9739, which are the most possible values for the fuzzy expected number of sewage solids in the system.

Figure 1 :
Figure 1: The membership function for fuzzy expected number of sewage solids in the system

Figure 2 :
Figure 2: The membership function for fuzzy expected waiting time in the queue 10 3.90 11.10 13.90 0.0662 0.0884 0.20 1.20 3.80 11.20 13.80 0.0667 0.0872 0.30 1.30 3.70 11.30 13.70 0.0672 0.0860 0.40 1.40 3.60 11.40 13.60 0.0678 0.0849 0.50 1.50 3.50 11.50 13.50 0.0684 0.0838 0.60 1.60 3.40 11.60 13.40 0.0689 0.0828 0.70 1.70 3.30 11.70 13.30 0.0695 0.0818 0.80 1.80 3.20 11.80 13.20 0.0701 0.0808 0.90 1.90 3.10 11.90 13.10 0.0708 0.0799 1.00 2.00 3.00 12.00 13.00 0.0714 0.0790 For example, in Example, a designer (manager) can set the range of the number of sewage solids to be [0.5252,1.0596] to reflect the desired service and find that the corresponding  level is 0.8 with  L y  11.80 and  U y  13.20.In other words, the designer can determine that the service rate is between 11.80 and 13.20.Similarly, a designer can also set the expected waiting time with "rounder" numbers like [0.0678, 0.0849] to reflect the desired service, and the corresponding  level is 0.4 with  L y  11.40 and  U y  13.60.As this example demonstrates, the approach proposed in this paper provides practical information for system designers and practitioners.
Suppose the batch-arrival rate and service rate are

Table 2 :
 -cuts of batch-arrival and service rates and