A single server retrial fuzzy queue with priority and unequal service rates

Abstract

This paper develops a latest mathematical model to derive the membership function of fuzzy retrial queue with single server queue model FM₁, FM₂/FM₁, FM₂/1 with priority and unequal service rates. The vital aim of this paper is to combine the parametric non linear programming technique and Yager’s ranking method. Using α - cut approach and Zadeh’s principle the fuzzy queues are changed into crisp queues in this paper. The membership function of the system characteristics is derived for different values of α. The numerical example is given to check the validity of the model.

Keywords

Fuzzy sets membership function nonlinear programming priority retrial queue

1 Introduction

The retrial queues have the characteristic feature that arriving call who discovers all servers occupied join the retrial group to attempt for their requests again. Retrial queues are extensively used in many problems in computer, telephone switching system and communication systems. Many authors have studied on the retrial queue with priority and single server. Choi and Park [8] studied an M₁, M₂/G/1 retrial queue with two types of calls, Type I calls in priority queue and Type II calls in retrial group. Falin, Artalejo and Martin [10] extended Choi and Park [8] to a model where two types of calls have different service times. The retrial queues with priority are useful for evaluation of performance in production, manufacturing systems, inventory control, computer and telecommunication systems.

The general retrial queues with single server and priority with two types of calls are described as follows. In the event that a call upon the arrival finds the server free, he instantly gets the service and leaves the system after service. At the point when Type I customers find the server occupied, he enters the priority queue and waits until the server is free and then gets service. In the event that Type II customers finds the server busy upon the arrival he enters the retrial queue keeping in mind the end goal to receive the service again after a random amount of time. Type II customers are served just when there is no Type I customers in the system. In this manner Type I customers are considered to have non preemptive priority over the Type II customers.

Because of few factors, parameters in the non preemptive priority queues might be fuzzy. In the literature, arrival times and service times are necessary to follow probability distributions. However, in some real-world applications, the parameter distributions are characterized subjectively. Hence fuzzy queues are more practical and useful when compared to the commonly used crisp queues (see Zadeh [27] and Li and Lee [20]. Hence the retrial queues with priority and fuzzy have a extensive range of applications. Bellman and Zadeh [1] presented the idea of fuzziness where inexact information could be solved by decision making problems. The researchers like Li and Lee [20], Buckley [2], Negi and Lee [22], Kaufmann [17], Kao et al. [16], Chen [3] has examined fuzzy queues by Zadeh’s extension principle. The analytical results of fuzzy queues (denoted M/F/1/∞ and FM/FM/1/∞ are studied by Li and Lee [20] where F denotes fuzzy time and FM denotes fuzzified exponential distributions) using Zadeh’s extension principle.

Parametric linear programming approach to derive the membership functions of the system in fuzzy queues has been studied by Kao et al. [16]. F. Choobinesh [5] discussed on different methods for ordering fuzzy numbers. R. Jain [14] made a study on Decision Making in fuzzy variables. L.A. Zadeh [27] has presented the idea of fuzzy probabilities. Buckley [2] studies multi-server queues with finite and infinite capacity queuing models where arrivals and departures follow possibilistic pattern. Chen [3] suggests a strategy of parametric programming in order to derive membership functions of the fuzzy queues. Further, the conversion of fuzzy queues to crisp queues has also been widely discussed in works and many methods and approaches have been used. M.J. Pardo and David de la Fuente [24] optimized a fuzzy priority discipline queuing model. B. Palpandi, and G. Geetharamani [23] computed performance measures of fuzzy non-preemptive priority queues by Robust ranking technique. W. Ritha and L. Robert [25] have analysed priority queuing discipline using fuzzy set theory. Many earlier researches on fuzzy queueing models have shown attention on simple queues with one or two fuzzy variables. In this paper, we discuss an approach that derives system characteristics for retrial queues with priority fuzzy parameters. Using α- cuts and Zadeh’s extension principle, we change the fuzzy queues to a family of crisp queues. The family of crisp queues is derived for different values of α and are solved using parametric nonlinear programming (NLP). The NLP solution wholly and effectively gives the membership functions of the system characteristics, the expected number of jobs and the expected waiting time in the queue.

This paper is structured as follows. Section 2 presents the system characteristics of single server retrial fuzzy queueing models with priority. In Section 3, a mathematical programming approach is developed to derive the membership functions of these system characteristics. In Section 4, realistic numerical examples are solved to exhibit the validity of the given approach. In Section 5, results and discussions are done. Conclusions are given in Section 6.

2 Retrial queue with priority

2.1 M₁, M₂/M₁, M₂/1 queues

Consider a single server retrial queuing system in which Type I and Type II customers arrive according to poisson process with rate λ₁ and λ₂ respectively. The unequal service time is exponentially distributed with μ₁, μ₂ respectively. Type I can be identified as high priority customer. On the other hand, low priority customers who finds the server busy upon arrivals leaves the system and enters the retrial group for the service again after a random exponentially amount of time. The retrial times are assumed to be independent and exponentially distributed with parameter θ.

From the results in Falin, Artalejo and Martin [10] we can obtain the system characteristics in terms of the system parameters.

Mean number of Type I customers in the priority queue is given by $E [N_{1}] = L_{q 1} = \frac{λ_{1} (λ_{1} b_{1}^{(2)} + λ_{2} b_{2}^{(2)})}{2 (1 - ρ_{1})}$

Mean number of Type II customers in the retrial queue is given by $E [N_{2}] = L_{q 2} = \frac{λ_{2} (λ_{1} b_{1}^{(2)} + λ_{2} b_{2}^{(2)})}{2 (1 - ρ_{1}) (1 - ρ)} + \frac{λ_{2} ρ}{θ (1 - ρ)}$ $ρ = ρ_{1} + ρ_{2}, ρ_{1} = \frac{λ_{1}}{μ_{1}}, ρ_{2} = \frac{λ_{2}}{μ_{2}}$ In the steady state it is necessary that we have $ρ = ρ_{1} + ρ_{2} < 1$

2.2 FM₁, FM₂/FM₁, FM₂/1 queues

The retrial queueing model with priority when extended with fuzzy parameters has a wider range of application. Let the arrival rate of high priority customers, the arrival rate of low priority customers, service rate of high priority customers, service rate of low priority customers and retrial rate are approximately known and are represented by the fuzzy numbers, $\tilde{λ_{1}}, \tilde{λ_{2}}, \tilde{μ_{1}}, \tilde{μ_{2}}, \tilde{θ}$ respectively. Let $η_{{\tilde{λ}}_{1}} (x_{1})$ , $η_{{\tilde{λ}}_{2}} (x_{2}), η {\tilde{μ}}_{1} (y_{1}), η_{{\tilde{μ}}_{2}} (y_{2}), η_{\tilde{θ}} (v)$ denote the membership function of $\tilde{λ_{1}}, \tilde{λ_{2}}, \tilde{μ_{1}}, \tilde{μ_{2}}, \tilde{θ}$ , respectively. We then have the following fuzzy sets $\tilde{λ_{1}} = {(x_{1}, η_{{\tilde{λ}}_{1}} (x_{1})) | x_{1} \in X_{1}}$ $\tilde{λ_{2}} = {(x_{2}, η_{{\tilde{λ}}_{2}} (x_{2})) | x_{2} \in X_{2}}$ $\tilde{μ_{1}} = {(y_{1}, η_{{\tilde{μ}}_{1}} (y_{1})) | y_{1} \in Y_{1}}$ $\tilde{μ_{2}} = {(y_{2}, η_{{\tilde{μ}}_{2}} (y_{2})) | y_{2} \in Y_{2}}$ $\tilde{θ} = {(v, η_{\tilde{θ}} (v)) | v \in V}$

Where X₁, X₂, Y₁, Y₂, V are the crisp universal sets of the arrival rate of high priority, low priority, service rate of high priority, low priority and retrial rate respectively.

Let p (x₁, x₂, y₁, y₂, v) denote the system characteristic of interest. Since $\tilde{λ_{1}}, \tilde{λ_{2}}, \tilde{μ_{1}}, \tilde{μ_{2}}, \tilde{θ}$ are fuzzy numbers, $p (\tilde{λ_{1}}, \tilde{λ_{2}}, \tilde{μ_{1}}, \tilde{μ_{2}}, \tilde{θ})$ is also a fuzzy number. By Zadeh’s extension principle the membership function of the system characteristic $p (\tilde{λ_{1}}, \tilde{λ_{2}}, \tilde{μ_{1}}, \tilde{μ_{2}}, \tilde{θ})$ is defined as $\begin{matrix} η_{p (\tilde{λ_{1}}, \tilde{λ_{2}}, \tilde{μ_{1}}, \tilde{μ_{2}}, \tilde{θ})} (z) = sup_{Ω} min {η_{{\tilde{λ}}_{1}} (x_{1}), η_{{\tilde{λ}}_{2}} (x_{2}), \\ η_{{\tilde{μ}}_{1}} (y_{1}), η_{{\tilde{μ}}_{2}} (y_{2}), η_{\tilde{θ}} (v) | z = p (x_{1}, x_{2}, y_{1}, y_{2}, v)} \end{matrix}$ where the supremum is taken over the set $\begin{matrix} Ω = {x_{1} \in X_{1}, x_{2} \in X_{2}, y_{1} \in Y_{1}, y_{2} \in Y_{2}, v \in V | 0 \\ < \frac{x_{1}}{y_{1}} + \frac{x_{2}}{y_{2}} < 1} \end{matrix}$

Assume that the system characteristic of interest is the expected number of customers in the high priority and low priority queue.

The membership function for the expected number of customers in priority queue is $\begin{matrix} η_{\tilde{E} (N_{1})} (z) = {Sup}_{Ω} min {η_{{\tilde{λ}}_{1}} (x_{1}), η_{{\tilde{λ}}_{2}} (x_{2}), η_{{\tilde{μ}}_{1}} (y_{1}), \\ η_{{\tilde{μ}}_{2}} (y_{2}), η_{\tilde{θ}} (v) | z = \frac{x_{1} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}})}} \end{matrix}$

The membership function for the expected number of customers in retrial queue is $\begin{matrix} η_{\tilde{E} (N_{2})} (z) = {Sup}_{Ω} min \\ {\begin{matrix} η_{{\tilde{λ}}_{1}} (x_{1}), η_{{\tilde{λ}}_{2}} (x_{2}), η_{{\tilde{μ}}_{1}} (y_{1}), η_{{\tilde{μ}}_{2}} (y_{2}), η_{\tilde{θ}} (v) | z \\ = \frac{x_{2} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}}) (1 - \frac{x}{y})} + \frac{x_{2} \frac{x}{y}}{v (1 - \frac{x}{y})} \end{matrix}} \end{matrix}$

The membership function is not expressed in the standard 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 α -cuts of $p (\tilde{λ_{1}}, \tilde{λ_{2}}, \tilde{μ_{1}}, \tilde{μ_{2}}, \tilde{θ})$ based on the extension principle.

3 Parametric nonlinear programming

To express the membership function in an understandable form we implement Zadeh’s approach which deals on α–cuts of ${\tilde{L}}_{q 1}$ and ${\tilde{L}}_{q 2}$

Definitions for the α-cuts of ${\tilde{λ}}_{1}$ and ${\tilde{μ}}_{1}$ as crisp intervals are $\begin{matrix} λ_{1} (α) = [x_{1 α}^{L}, x_{1 α}^{U}] = [min_{x_{1} \in X_{1}} {x_{1} / η_{{\tilde{λ}}_{1}} (x_{1}) \geq α}, \\ max_{x_{1} \in X_{1}} {x_{1} / η_{{\tilde{λ}}_{1}} (x_{1}) \geq α}] \end{matrix}$ $\begin{matrix} μ_{1} (α) = [y_{1 α}^{L}, y_{1 α}^{U}] = [min_{y_{1} \in Y_{1}} {y_{1} / η_{{\tilde{μ}}_{1}} (y_{1}) \geq α}, \\ max_{y_{1} \in Y_{1}} {y_{1} / η_{{\tilde{μ}}_{1}} (y_{1}) \geq α}] \end{matrix}$

The constant arrival rates and service rates are given as intervals when the membership functions are no less than a given possibility level for α The bounds of these intervals can be described as functions of α and can be obtained as $x_{1 α}^{L} = min η_{{\tilde{λ}}_{1}}^{- 1} (α), x_{1 α}^{L} = max η_{{\tilde{λ}}_{2}}^{- 1} (α)$ $y_{1 α}^{L} = min η_{{\tilde{μ}}_{1}}^{- 1} (α), y_{1 α}^{L} = max η_{{\tilde{μ}}_{2}}^{- 1} (α)$

We can use the α cuts of L_q1 and L_q2 to construct its membership function since the membership function is parametrized by α

Using Zadeh’s extension principle, $η_{{\tilde{L}}_{q 1}} (z)$ is the minimum of $η_{{\tilde{λ}}_{1}} (x_{1}), η_{{\tilde{λ}}_{2}} (x_{2}), η_{{\tilde{μ}}_{1}} (y_{1}), η_{{\tilde{μ}}_{2}} (y_{2})$ , $η_{\tilde{θ}} (v)$ To derive the membership function we need at least one of the following cases to hold such that $z = \frac{x_{1} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}})}$ satisfies $η_{{\tilde{L}}_{q 1}} (z) = α$

Case 1: $(η_{{\tilde{λ}}_{1}} (x_{1}) = α, η_{{\tilde{λ}}_{2}} (x_{2}) \geq α, η_{{\tilde{μ}}_{1}} \geq α, η_{{\tilde{μ}}_{2}} \geq α)$

Case 2: $(η_{{\tilde{λ}}_{1}} (x_{1}) \geq α, η_{{\tilde{λ}}_{2}} (x_{2}) = α, η_{{\tilde{μ}}_{1}} \geq α, η_{{\tilde{μ}}_{2}} \geq α)$

Case 3: $(η_{{\tilde{λ}}_{1}} (x_{1}) \geq α, η_{{\tilde{λ}}_{2}} (x_{2}) \geq α, η_{{\tilde{μ}}_{1}} = α, η_{{\tilde{μ}}_{2}} \geq α)$

Case 4: $(η_{{\tilde{λ}}_{1}} (x_{1}) \geq α, η_{{\tilde{λ}}_{2}} (x_{2}) \geq α, η_{{\tilde{μ}}_{1}} \geq α, η_{{\tilde{μ}}_{2}} = α)$

This can be solved using parametric NLP techniques. The NLP to find the lower and upper bounds of the αcuts of $η_{{\tilde{L}}_{q 1}}$

For case 1 are $(L_{q_{1}})_{α}^{L_{1}} = min_{Ω} (\frac{x_{1} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}})})$ ${(L_{q 1})}_{α}^{U_{1}} = max_{Ω} (\frac{x_{1} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}})})$

For case 2 are ${(L_{q 1})}_{α}^{L_{2}} = min_{Ω} (\frac{x_{1} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}})})$ ${(L_{q 1})}_{α}^{U_{2}} = max_{Ω} (\frac{x_{1} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}})})$

For case 3 are ${(L_{q 1})}_{α}^{L_{3}} = min_{Ω} (\frac{x_{1} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}})})$ ${(L_{q 1})}_{α}^{U_{3}} = max_{Ω} (\frac{x_{1} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}})})$

For case 4 are ${(L_{q 1})}_{α}^{L_{4}} = min_{Ω} (\frac{x_{1} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}})})$ ${(L_{q 1})}_{α}^{U_{4}} = max_{Ω} (\frac{x_{1} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}})})$

From the definitions of ${\tilde{λ}}_{1}$ and ${\tilde{μ}}_{1}$ , x₁ ∈ λ₁ (α) and y₁ ∈ μ₁ (α) can be replaced by $x_{1} \in [x_{1 α}^{L}, x_{1 α}^{U}]$ , $x_{2} \in [x_{2 α}^{L}, x_{2 α}^{U}]$ and $y_{1} \in [y_{1 α}^{L}, y_{1 α}^{U}]$ , $y_{2} \in [y_{2 α}^{L}, y_{2 α}^{U}]$

The α - cuts form a nested structure with respect to α (see Kaufmann [17] and Zimmermann [28]). Given 0 < α₂ < α₁ ≤ 1, we have $\begin{matrix} [x_{1 α_{1}}^{L}, x_{1 α_{1}}^{U}] \subseteq [x_{1 α_{2}}^{L}, x_{1 α_{2}}^{U}], [x_{2 α_{1}}^{L}, x_{2 α_{1}}^{U}] \subseteq \\ [x_{2 α_{2}}^{L}, x_{2 α_{2}}^{U}] [y_{1 α_{1}}^{L}, y_{1 α_{1}}^{U}] \subseteq [y_{1 α_{2}}^{L}, y_{1 α_{2}}^{U}] \\ [y_{2 α_{1}}^{L}, y_{2 α_{1}}^{U}] \subseteq [y_{2 α_{2}}^{L}, y_{2 α_{2}}^{U}] \end{matrix}$

To find the membership function $η_{{\tilde{L}}_{q 1}} (z)$ it is enough to find the left and right shape functions of $η_{{\tilde{L}}_{q 1}} (z)$ which is equivalent in finding the lower bound ${(L_{q 1})}_{α}^{L}$ and the upper bound ${(L_{q 1})}_{α}^{U}$ of the α- cuts of $({\tilde{L}}_{q 1})$ can be written as ${(L_{q 1})}_{α}^{L} = min_{Ω} (\frac{x_{1}^{2} y_{2}^{2} + x_{1} x_{2} y_{1}^{2}}{y_{1} y_{2}^{2} (y_{1} - x_{1})})$ such that $\begin{matrix} x_{1 α}^{L} \leq x_{1} \leq x_{1 α}^{U}, x_{2 α}^{L} \leq x_{2} \leq x_{2 α}^{U}, y_{1 α}^{L} \leq y_{1} \leq y_{1 α}^{U}, \\ y_{2 α}^{L} \leq y_{2} \leq y_{2 α}^{U} \end{matrix}$ ${(L_{q 1})}_{α}^{U} = max_{Ω} (\frac{x_{1}^{2} y_{2}^{2} + x_{1} x_{2} y_{1}^{2}}{y_{1} y_{2}^{2} (y_{1} - x_{1})})$ such that $\begin{matrix} x_{1 α}^{L} \leq x_{1} \leq x_{1 α}^{U}, x_{2 α}^{L} \leq x_{2} \leq x_{2 α}^{U}, y_{1 α}^{L} \leq y_{1} \leq y_{1 α}^{U}, \\ y_{2 α}^{L} \leq y_{2} \leq y_{2 α}^{U} \end{matrix}$

At least one of x₁ and y₁ must hit the boundaries of their α -cuts to satisfy $η_{{\tilde{L}}_{q 1}} (z) = α$

This model is a set of mathematical programs with boundary constraints is a special case of parametric NLPs (see [11]) where how the optimal solution change with $x_{1 α}^{L}, x_{1 α}^{U}, x_{2 α}^{L}, x_{2 α}^{U}, y_{1 α}^{L}, y_{1 α}^{U}, y_{2 α}^{L}, y_{2 α}^{U}$ as α varies over [0, 1] is studied.

Yet again, by applying the results of Kaufmann [17] and Zimmermann [28] and convexity properties to ${\tilde{L}}_{q 1}$ we have ${(L_{q 1})}_{α 1}^{L} \geq {(L_{q 1})}_{α 2}^{L}$ and ${(L_{q 1})}_{α 1}^{U} \leq {(L_{q 1})}_{α 2}^{U}$ where

0 < α₂ < α₁ ≤ 1. In other words, ${(L_{q 1})}_{α}^{L}$ increases and ${(L_{q 1})}_{α}^{U}$ decreases as α increases.

Similarly the lower bound ${(L_{q 1})}_{α}^{L}$ and the upper bound ${(L_{q 1})}_{α}^{U}$ of the α- cuts of $({\tilde{L}}_{q 2})$ can be written as ${(L_{q 2})}_{α}^{L} = min_{Ω} (\frac{x_{2} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}}) (1 - \frac{x}{y})} + \frac{λ_{2} \frac{x}{y}}{θ (1 - \frac{x}{y})})$ such that $\begin{matrix} x_{1 α}^{L} \leq x_{1} \leq x_{1 α}^{U}, x_{2 α}^{L} \leq x_{2} \leq x_{2 α}^{U}, y_{1 α}^{L} \leq y_{1} \leq y_{1 α}^{U}, \\ y_{2 α}^{L} \leq y_{2} \leq y_{2 α}^{U} v^{L} \leq v \leq v^{U} \end{matrix}$ ${(L_{q 2})}_{α}^{U} = max_{Ω} (\frac{x_{2} (\frac{2 x_{1}}{y_{1}^{2}} + \frac{2 x_{2}}{y_{2}^{2}})}{2 (1 - \frac{x_{1}}{y_{1}}) (1 - \frac{x}{y})} + \frac{λ_{2} \frac{x}{y}}{θ (1 - \frac{x}{y})})$ $\begin{matrix} x_{1 α}^{L} \leq x_{1} \leq x_{1 α}^{U}, x_{2 α}^{L} \leq x_{2} \leq x_{2 α}^{U} y_{1 α}^{L} \leq y_{1} \leq y_{1 α}^{U}, \\ y_{2 α}^{L} \leq y_{2} \leq y_{2 α}^{U} v^{L} \leq v \leq v^{U} \end{matrix}$

If both ${(L_{q 1})}_{α}^{L}$ and ${(L_{q 1})}_{α}^{U}$ are invertible with respect to α then a left shape function ${[{(L_{q 1})}_{α}^{L}]}^{- 1}$ and right shape function ${[{(L_{q 1})}_{α}^{U}]}^{- 1}$ can be derived from which the membership function $η_{{\tilde{L}}_{q 1}}$ is constructed. $η_{{\tilde{L}}_{q 1}} (z) = {\begin{matrix} L (z), {(L_{q 1})}_{α = 0}^{L} \leq z \leq {(L_{q 1})}_{α = 1}^{L} \\ R (z), {(L_{q 1})}_{α = 1}^{U} \leq z \leq {(L_{q 1})}_{α = 0}^{U} \end{matrix}$

Similarly If both $(L_{q_{2}})_{α}^{L}$ and $(L_{q_{2}})_{α}^{U}$ are invertible with respect to αthen $η_{{\tilde{L}}_{q_{2}}} (z) = {\begin{matrix} L (z), (L_{q_{2}})_{α = 0}^{L} \leq z \leq (L_{q_{2}})_{α = 1}^{L} \\ R (z), (L_{q_{2}})_{α = 1}^{L} \leq z \leq (L_{q_{2}})_{α = 0}^{U} \end{matrix}$

In general, the values of ${(L_{q 1})}_{α}^{L}$ and ${(L_{q 1})}_{α}^{U}$ cannot be solved analytically. Hence a closed form membership function for L_q1 cannot be obtained. However the numerical solutions for ${(L_{q 1})}_{α}^{L}$ and ${(L_{q 1})}_{α}^{U}$ at different possibility levels are collected to approximate the shape of L (z) and R (z)

Similarly the membership function of the expected number of customers in retrial queue can be derived.

The values preserve completely all the fuzziness of the arrival and service rate because the system characteristics are described by the membership function. However if we need a single crisp value for a system characteristic rather than a fuzzy set, the fuzzy values are defuzzified using Yager’s ranking index method which is given as $o (\tilde{\land}) = \int_{0}^{1} \frac{\land_{α}^{L} + \land_{α}^{U}}{2} d α$

Where $\tilde{\land}$ is a convex fuzzy number and $(\land_{α}^{L}, \land_{α}^{U})$ is the α- cuts of $\tilde{\land}$ This method possesses the properties of compensation, linearity and additivity.

4 Numerical example

Mostly retrial queues are dealt with one kind of call. Though there are some realistic models which deal with more than a few types of calls. Let us consider an example of telephone switching system (Falin et al. [10]). In recent telephone exchanges, subscriber lines are generally linked to the subscriber line modules. These modules deal both incoming and outgoing calls. A fundamental contrast between these two kinds of calls is in the way that on account of blocking due to all channels busy in the module, outgoing calls can be queued, while incoming calls get busy signal and retried so as to begin the connection. When the channel is free, an active call, if exhibit, occupies the channel quickly. Hence incoming calls may not be connected provided that there are outgoing calls waiting. This shows that there is non-preemptive priorty of outgoing calls over incoming calls.

4.1 The fuzzy expected number of customers in priority queue ${\tilde{L}}_{q 1}$

Let the arrival, service rates and retrial ratesare triangular fuzzy numbers given by ${\tilde{λ}}_{1} = [1, 2, 3], {\tilde{λ}}_{2} = [5, 6, 7], {\tilde{μ}}_{1} = [9, 10, 11], {\tilde{μ}}_{2} = [14,$ $15, 16], \tilde{θ} = [4, 5, 6]$ per minute respectively. It is easy to find that $\begin{matrix} [x_{1 α}^{L}, x_{1 α}^{U}] = [1 + α, 3 - α], [x_{2 α}^{L}, x_{2 α}^{U}] \\ = [5 + α, 7 - α], [y_{1 α}^{L}, y_{1 α}^{U}] = [9 + α, 11 - α] \\ [y_{2 α}^{L}, y_{2 α}^{U}] = [14 + α, 16 - α], [v_{α}^{L}, v_{α}^{U}] \\ = [4 + α, 6 - α] \end{matrix}$

The α–cuts of ${\tilde{L}}_{q 1}$ $(L_{q 1})_{α}^{L} = \frac{601 + 996 α + 447 α^{2} + 54 α^{3} + 2 α^{4}}{28160 - 11712 α + 1646 α^{2} - 96 α^{3} + 2 α^{4}}$ $(L_{q 1})_{α}^{U} = \frac{4845 - 3496 α + 819 α^{2} - 70 α^{3} + 2 α^{4}}{10584 + 6216 α + 1118 α^{2} + 80 α^{3} + 2 α^{4}}$

With the help of MATLAB 8.6, the inverse functions of ${(L_{q 1})}_{α}^{L}$ and ${(L_{q 1})}_{α}^{U}$ exist, yield the membership function $η_{{\tilde{L}}_{q 1}} (z) = {\begin{matrix} L (z); \frac{601}{28160} \leq z \leq \frac{7}{60} \\ R (z); \frac{7}{60} \leq z \leq \frac{1615}{3528} \end{matrix}$

Where L (z) = (48 * x + 27)/(4 * (x - 1)) + (6^(1/2)* ((r))/(12 * ((q)/(x - 1) ³) ^(1/6)) + (6^(1/2) * ((3 * (28037992 * x + 40302864 * x² + 26838656 * x³ - 28672 * x⁴ - 228965) * ((675840 * x)/(x - 1)-(2184192 * x)/(x - 1) ² + (61965 * x)/(x - 1) ³-(2125764 * x)/(x - 1) ⁴ - 14424/(x - 1) -161352/(x - 1) ² - 977589/(2 * (x - 1) ³) -4782969/(16 * (x - 1) ⁴) + (16148 * x + 328 * x² + 399) ²/(16 * (x - 1) ⁴) - (3373056 * x²)/(x - 1) ² + (4854816 * x²)/(x - 1) ³ - (5668704 * x²)/(x - 1) ⁴ + (5688576*x³)/(x - 1) ³ - (6718464 * x³)/(x - 1) ⁴ - (2985984 * x⁴)/(x - 1) ⁴ + ((q)/(x - 1) ³) ^(2/3) + (((q)/(x - 1) ³) ^(1/3) * (16148 * x + 328 * x² + 399))/(2 * (x - 1) ²)) ^(1/2))/(16 * (x - 1) ⁴) - ((q)/(x - 1) ³) ^(2/3) * ((r) - ((16148 * x + 328 * x² + 399) ² * ((675840 * x)/(x - 1) - (2184192 * x)/(x - 1) ² + (61965 * x)/(x - 1) ³ - (2125764 * x)/(x - 1) ⁴ - 14424/(x - 1) -161352/(x - 1) ² - 977589/(2 * (x - 1) ³) -4782969/(16 * (x - 1) ⁴) + (16148 * x + 328 * x² + 399) ²/(16 * (x - 1) ⁴) - (3373056 * x²)/(x - 1) ² + (4854816 * x²)/(x - 1) ³ - (5668704*x²)/(x - 1) ⁴ + (5688576 * x³)/(x - 1) ³ - (6718464 * x³)/(x - 1) ⁴ - (2985984 * x⁴)/(x - 1) ⁴+ ((q)/(x - 1) ³) ^(2/3) + (((q)/(x - 1) ³) ^(1/3) * (16148*x + 328 * x² + 399))/(2 * (x - 1) ²)) ^(1/2))/(16 * (x - 1) ⁴) + (9 * 6^(1/2) * ((q)/(x - 1) ³) ^(1/2) * (71322*x + 69652 * x² - 192 * x³ - 157))/(2 * (x - 1) ³)+ (((q)/(x - 1) ³) ^(1/3) * (16148 * x + 328 * x² + 399)* ((675840* x)/(x - 1) - (2184192 * x)/(x - 1) ² +(61965 * x)/(x - 1) ³ - (2125764 * x)/(x - 1) ⁴ - 14424/(x - 1) -161352/(x - 1) ² - 977589/(2 * (x - 1) ³) -4782969/(16 * (x - 1) ⁴) + (16148 * x + 328 * x² + 399) ²/(16 * (x - 1) ⁴) - (3373056 * x²)/(x - 1) ² + (4854816 * x²)/(x - 1) ³ - (5668704*x²)/(x - 1) ⁴ + (5688576 * x³)/(x - 1) ³ - (6718464 * x³)/(x - 1) ⁴ - (2985984 * x⁴)/(x - 1) ⁴+ ((q)/(x - 1) ³) ^(2/3) + (((q)/(x - 1) ³) ^(1/3) * (16148*x + 328 * x² + 399))/(2 * (x - 1) ²)) ^(1/2))/(x - 1) ²) ^(1/2))/(12 * ((q)/(x - 1) ³) ^(1/6) * ((675840 * x)/(x - 1) - (2184192 * x)/(x - 1) ² + (61965 * x)/(x - 1) ³ - (2125764 * x)/(x - 1) ⁴ - 14424/(x - 1)-161352/(x - 1) ² - 977589/(2* (x - 1) ³) -4782969/(16 * (x - 1) ⁴) + (16148 * x + 328 * x² +399) ²/(16 * (x - 1) ⁴) - (3373056 * x²)/(x - 1) ² + (4854816 * x²)/(x - 1) ³ - (5668704 * x²)/(x - 1) ⁴+(5688576 * x³)/(x - 1) ³ - (6718464 * x³)/(x - 1) ⁴ - (2985984 * x⁴)/(x - 1) ⁴ + ((q)/(x - 1) ³) ^(2/3)+ (((q)/(x - 1) ³) ^(1/3) * (16148 * x + 328 * x² + 399))/(2 * (x - 1) ²)) ^(1/4))

R (z) = (6^(1/2) * ((3 * (28037992 * x + 40302864*x² + 26838656 * x³ - 28672 * x⁴ - 228965) * ((254016 * x)/(x - 1) - (2144400 * x)/(x - 1) ² - (1385475 * x)/(x - 1) ³ - (3858750 * x)/(x - 1) ⁴-116280/(x - 1) -734160/(x - 1) ² - 3009825/(2 *(x - 1) ³) -13505625/(16 * (x - 1) ⁴) + (16148 * x + 328 * x² + 399) ²/(16 * (x - 1) ⁴) - (1491840 * x²)/(x - 1) ² + (2730000 * x²)/(x - 1) ³ - (6615000*x²)/(x - 1) ⁴ + (2683200 * x³)/(x - 1) ³ - (5040000 * x³)/(x - 1) ⁴ - (1440000 * x⁴)/(x - 1) ⁴+ ((q)/(x - 1) ³) ^(2/3) + (((q)/(x - 1) ³) ^(1/3) * (16148*x+ 328 * x² + 399))/(2 * (x - 1) ²)) ^(1/2))/(16 * (x - 1) ⁴) - ((q)/(x - 1) ³) ^(2/3) * ((r - ((16148 * x +328 * x² + 399) ² * ((254016 * x)/(x - 1) - (2144400 * x)/(x - 1) ² - (1385475 * x)/(x - 1) ³-(3858750 * x)/(x - 1) ⁴ - 116280/(x - 1) - 734160/(x - 1) ² - 3009825/(2 * (x - 1) ³) -13505625/(16 * (x - 1) ⁴) + (16148 * x + 328 * x² + 399) ²/(16 * (x - 1) ⁴) - (1491840 * x²)/(x - 1) ² + (2730000 * x²)/(x - 1) ³ - (6615000 * x²)/(x - 1) ⁴ + (2683200 * x³)/(x - 1) ³ - (5040000 * x³)/(x - 1) ⁴ - (1440000* x⁴)/(x - 1) ⁴ + ((q)/(x - 1) ³) ^(2/3) + (((q)/(x - 1) ³) ^(1/3) * (16148 * x + 328 *x²+ 399))/(2 * (x - 1) ²)) ^(1/2))/(16 * (x - 1) ⁴) + (9 *6^(1/2)* ((q)/(x - 1) ³) ^(1/2) * (71322 * x + 69652 *x² - 192 * x³ - 157))/(2 * (x - 1) ³) + (((q)/(x - 1) ³) ^(1/3) * (16148 * x + 328 * x² + 399) * ((254016*x)/(x - 1) - (2144400* x)/(x - 1) ² - (1385475 *x)/(x - 1) ³ - (3858750 * x)/(x - 1) ⁴ - 116280/(x-1) -734160/(x - 1) ² - 3009825/(2 * (x - 1) ³) -13505625/(16 * (x - 1) ⁴) + (16148 * x + 328 * x²+399) ²/(16 * (x - 1) ⁴) - (1491840 * x²)/³ - (5040000 * x³)/(x - 1) ⁴ - (1440000 * x⁴)/(x - 1) ⁴+ ((q)/(x - 1) ³) ^(2/3)+ (((1477201770 * x - 12 *3^(1/2) * ((p) ^(1/2) - 1656476100 * x² + 1331000 * x³+36* 3^(1/2) * x * ((p) ^(1/2) - 36 * 3^(1/2) * x² *((p) ^(1/2) + 12 * 3^(1/2) * x³ * ((p) ^(1/2) - 12228921)/(x - 1) ³) ^(1/3)* (16148 * x + 328 * x² + 399))/(2 *(x - 1) ²)) ^(1/2))/(x - 1) ²) ^(1/2))/(12 * ((q)/(x - 1) ³) ^(1/6) * ((254016 * x)/(x - 1) - (2144400 * x)/(x - 1) ² - (1385475 * x)/(x - 1) ³ - (3858750 * x)/(x - 1) ⁴ - 116280/(x - 1) -734160/(x - 1) ² - 3009825/(2 * (x - 1) ³) -13505625/(16 * (x - 1) ⁴)+(16148 * x + 328 * x² + 399) ²/(16 * (x - 1) ⁴)-(1491840 * x²)/(x - 1) ² + (2730000 * x²)/(x - 1) ³ -(6615000 * x²)/(x - 1) ⁴ + (2683200 * x³)/(x - 1) ³-(5040000 * x³)/(x - 1) ⁴ - (1440000 * x⁴)/(x - 1) ⁴+ ((q)/(x - 1) ³) ^(2/3) + (((q)/(x - 1) ³) ^(1/3) * (16148*x + 328 * x² + 399))/(2 * (x - 1) ²)) ^(1/4)) - (6^(1/2)* ((8074 * x * ((q)/(x - 1) ³) ^(1/3) - 4345980 * x - 2*x * ((q)/(x - 1) ³) ^(2/3) + (399 * ((q)/(x - 1) ³) ^(1/3))/2 + ((q)/(x - 1) ³) ^(2/3) + 12100 * x² + 164 * x² * ((q)/(x - 1) ³) ^(1/3) + x² * ((1477201770 * x - 12 * 3^(1/2)* ((p) ^(1/2) - 1656476100 * x² + 1331000 * x³ + 36 *3^(1/2) * x * ((p) ^(1/2) - 36 * 3^(1/2) * x² * ((764189866500 * x - 1791293889482575 * x² + 178721374968255750 * x³ + 4773621235275000 * x⁴ - 5788552275000 * x⁵ + 3865907200)/(x - 1) ⁶) ^(1/2) + 12 * 3^(1/2) * x³ * ((p) ^(1/2) - 12228921)/(x - 1) ³) ^(2/3) + 52881)/(x - 1) ²) ^(1/2))/(12 * ((q)/(x - 1) ³) ^(1/6)) - (40 * x + 35)/(4 * (x - 1))

Where p = (764189866500 * x - 1791293889482575 * x² + 178721374968255750 * x³ + 4773621235275000 * x⁴ - 5788552275000 * x⁵ + 3865907200)/(x - 1) ⁶q = 1477201770 * x - 12 * 3^(1/2) * (p) ^(1/2) - 1656476100 * x² + 1331000 * x³ + 36 * 3^(1/2) * x * (p) ^(1/2) - 36 * 3¹/2 * x² * (p) ^(1/2) + 12 * 3^(1/2) * x³ * (p) ^(1/2) - 12228921r = (8074 * x * ((q)/(x - 1) ³) ^(1/3) - 4345980 * x - 2 * x * ((q)/(x - 1) ³) ^(2/3) + (399 * ((q)/(x - 1) ³) ^(1/3))/2 + ((q)/(x - 1) ³) ^(2/3) + 12100 * x² + 164 * x² * ((q)/(x - 1) ³) ^(1/3) + x² * ((q)/(x - 1) ³) ^(2/3) + 52881)/(x - 1) ²) ^(1/2)

The overall shape turns out as expected. The membership functions L (z) and R (z) have complex values with their imaginary parts approaching zero when $\frac{601}{28160} \leq z \leq \frac{7}{60}$ for L (z) and $\frac{7}{60} \leq z \leq \frac{1615}{3528}$ for R (z). The imaginary parts of these two functions can be ignored since they do not have influence on the computational parts.

Crisp intervals for fuzzy expected number of customers in priority queue at different possiblistic α- levels are presented in Table 1. The support of $({\tilde{L}}_{q 1})$ ranges from 0.0213 to 0.4578, this indicates that the expected number of customers in priority queue is fuzzy; it is not possible for its values to fall below 0.0213 or exceed 0.4578. The α- cut at α = 1 is 0.1166 which is the most possible value for the fuzzy expected number of customers in priority queue.

Fig.1

The membership function for fuzzy expected number of customers in priority queue.

Table 1

The α–cuts of arrival and service rates and fuzzy expected number of customers in priority queue

α	$x_{1 α}^{L}$	$x_{1 α}^{U}$	$x_{2 α}^{L}$	$x_{2 α}^{U}$	$y_{1 α}^{L}$	$y_{1 α}^{U}$	$y_{2 α}^{L}$
0.0	1	3.0	5	7	9	11	14
0.1	1.1	2.9	5.1	6.9	9.1	11.9	14.1
0.2	1.2	2.8	5.2	6.8	9.2	11.8	14.2
0.3	1.3	2.7	5.3	6.7	9.3	11.7	14.3
0.4	1.4	2.6	5.4	6.6	9.4	1.6	14.4
0.5	1.5	2.5	5.5	6.5	9.5	11.5	14.5
0.6	1.6	2.4	5.6	6.4	9.6	11.4	14.6
0.7	1.7	2.3	5.7	6.3	9.7	11.3	14.7
0.8	1.8	2.2	5.8	6.2	9.8	11.2	14.8
0.9	1.9	2.1	5.9	6.1	9.9	11.1	14.9
1.0	2.0	2.0	6.0	6.0	10	10	15

$y_{2 α}^{U}$	$v_{α}^{L}$	$v_{α}^{U}$	${(L_{q 1})}_{α}^{L}$	${(L_{q 1})}_{α}^{U}$
16	4	6.0	0.0213	0.4578
15.9	4.1	5.9	0.0261	0.4015
15.8	4.2	5.8	0.0316	0.3519
15.7	4.3	5.7	0.0379	0.3082
15.6	4.4	5.6	0.0452	0.2696
15.5	4.5	5.5	0.0536	0.2355
15.4	4.6	5.4	0.0632	0.2055
15.3	4.7	5.3	0.0741	0.1789
15.2	4.8	5.2	0.0865	0.1555
15.1	4.9	5.1	0.1006	0.1349
15	5	5	0.1166	0.1166

4.2 The fuzzy expected number of customers in retrial queue

{\tilde{L}}_{q 2}

The α- cuts of ${\tilde{L}}_{q 2}$ are ${(L_{q 2})}_{α}^{L} = \frac{593980 + 685971 α + 300290 α^{2} + 63881 α^{3} + 6992 α^{4} + 378 α^{5} + 8 α^{6}}{3548160 - 27429112 α + 847076 α^{2} - 133014 α^{3} + 11303 α^{4} - 474 α^{5} + 8 α^{6}}$ ${(L_{q 2})}_{α}^{U} = \frac{3802610 - 2909223 α + 883544 α^{2} - 136217 α^{3} + 11258 α^{4} - 474 α^{5} + 8 α^{6}}{550368 + 630168 α + 280736 α^{2} + 61446 α^{3} + 6896 α^{4} + 378 α^{5} + 8 α^{6}}$

With the help of MATLAB 8.6, the inverse functions of $(L_{q_{2}})_{α}^{L}$ and $(L_{q_{2}})_{α}^{U}$ , yield the membership function $η_{{\tilde{L}}_{q_{2}}} (z) = {\begin{matrix} L (z); \frac{593980}{3548160} \leq z \leq \frac{1651500}{1530147} \\ R (z); \frac{1651506}{1530000} \leq z \leq \frac{3802610}{550368} \end{matrix}$

Where L (z) = root (8 * x * z⁶ - 8 * z⁶ - 474 * x * z⁵ - 378 * z⁵ + 11303 * x * z⁴ - 6992 * z⁴ - 133014 * x * z³ - 63881 * z³ + 725424 * x * z² - 300290 * z² - 2742912 * x * z - 685971 * z + 3548160 * x - 593980, z, 1)

R (z) = root (8 * x * z⁶ - 8 * z⁶ - 474 * x * z⁵ - 378 * z⁵ + 11303 * x * z⁴ - 6992 * z⁴ - 133014 * x * z³ - 63881 * z³ + 847076 * x * z² - 300290 * z² - 2742912 * x * z - 685971 * z + 3548160 * x - 593980, z, 1)

The overall shape turns out as expected. The membership functions L (z) and R (z) have complex values with their imaginary parts approaching zero when $\frac{593980}{3548160} \leq z \leq \frac{1651500}{1530147}$ for L (z) and $\frac{1651506}{15330000} \leq z \leq \frac{3802610}{550368}$ for R (z). Hence, the imaginary parts of these two functions have no influence on the computational results and can be disregarded.

Crisp intervals for fuzzy expected number of customers in retrial queue at different possiblistic α- levels are presented in Table 2. The support of $({\tilde{L}}_{q 1})$ ranges from 0.1674 to 6.9092, this indicates that the expected number of customers in retrial queue is fuzzy; it is not possible for its values to fall below 0.1674 or exceed 6.9092. The α- cut at α = 1 is 1.0793 which is the most possible value for the fuzzy expected number of customers in retrial queue.

Table 2
The α- cuts of arrival and service rates and fuzzy queue length of retrial customers

α $x_{1 α}^{L}$ $x_{1 α}^{U}$ $x_{2 α}^{L}$ $x_{2 α}^{U}$ $y_{1 α}^{L}$ $y_{1 α}^{U}$ $y_{2 α}^{L}$

0.0 1 3.0 5 7 9 11 14

0.1 1.1 2.9 5.1 6.9 9.1 11.9 14.1

0.2 1.2 2.8 5.2 6.8 9.2 11.8 14.2

0.3 1.3 2.7 5.3 6.7 9.3 11.7 14.3

0.4 1.4 2.6 5.4 6.6 9.4 1.6 14.4

0.5 1.5 2.5 5.5 6.5 9.5 11.5 14.5

0.6 1.6 2.4 5.6 6.4 9.6 11.4 14.6

0.7 1.7 2.3 5.7 6.3 9.7 11.3 14.7

0.8 1.8 2.2 5.8 6.2 9.8 11.2 14.8

0.9 1.9 2.1 5.9 6.1 9.9 11.1 14.9

1.0 2.0 2.0 6.0 6.0 10 10 15

α	$x_{1 α}^{L}$	$x_{1 α}^{U}$	$x_{2 α}^{L}$	$x_{2 α}^{U}$	$y_{1 α}^{L}$	$y_{1 α}^{U}$	$y_{2 α}^{L}$
0.0	1	3.0	5	7	9	11	14
0.1	1.1	2.9	5.1	6.9	9.1	11.9	14.1
0.2	1.2	2.8	5.2	6.8	9.2	11.8	14.2
0.3	1.3	2.7	5.3	6.7	9.3	11.7	14.3
0.4	1.4	2.6	5.4	6.6	9.4	1.6	14.4
0.5	1.5	2.5	5.5	6.5	9.5	11.5	14.5
0.6	1.6	2.4	5.6	6.4	9.6	11.4	14.6
0.7	1.7	2.3	5.7	6.3	9.7	11.3	14.7
0.8	1.8	2.2	5.8	6.2	9.8	11.2	14.8
0.9	1.9	2.1	5.9	6.1	9.9	11.1	14.9
1.0	2.0	2.0	6.0	6.0	10	10	15

$y_{2 α}^{U}$	$v_{α}^{L}$	$v_{α}^{U}$	${(L_{q 1})}_{α}^{L}$	${(L_{q 1})}_{α}^{U}$
16	4	6.0	0.1674	6.9092
15.9	4.1	5.9	0.2028	5.7126
15.8	4.2	5.8	0.2453	4.7302
15.7	4.3	5.7	0.2961	3.9219
15.6	4.4	5.6	0.3571	3.2554
15.5	4.5	5.5	0.4301	2.7047
15.4	4.6	5.4	0.5176	2.2488
15.3	4.7	5.3	0.6224	1.8709
15.2	4.8	5.2	0.7480	1.5572
15.1	4.9	5.1	0.8986	1.2964
15	5	5	1.0793	1.0793

5 Results and discussions

Since the performance measures are described by the membership function, the values conserve completely all of fuzziness of arrival rate, service rate and retrial rate If we prefer a suitable single crisp value for the system characteristics, we defuzzify the fuzzy values using Yager’s ranking index method. The system characteristics are calculated by

$o (\tilde{\land}) = \int_{0}^{1} \frac{\land_{α}^{L} + \land_{α}^{U}}{2} d α$

Based on the example the suitable number of customers in priority queue is $\begin{matrix} o ({\tilde{L}}_{q 1}) = \int_{0}^{1} \frac{1}{2} \\ [\frac{601 + 996 α + 447 α^{2} + 54 α^{3} + 2 α^{4}}{28160 - 11712 α + 1646 α^{2} - 96 α^{3} + 2 α^{4}} \\ + \frac{4845 - 3496 α + 819 α^{2} - 70 α^{3} + 2 α^{4}}{10584 + 6216 α + 1118 α^{2} + 80 α^{3} + 2 α^{4}}] \end{matrix}$ $d α = 0.1558$

Fig.2

The membership functions for fuzzy expected no of retrial customers in the queue.

The suitable number of customers in retrial queue is $\begin{matrix} o ({\tilde{L}}_{q 2}) = \int_{0}^{1} \frac{1}{2} [\frac{593980 + 685971 α + 300290 α^{2} + 63881 α^{3} + 6992 α^{4} + 378 α^{5} + 8 α^{6}}{3548160 - 27429112 α + 847076 α^{2} - 133014 α^{3} + 11303 α^{4} - 474 α^{5} + 8 α^{6}} \\ + \frac{3802610 - 2909223 α + 883544 α^{2} - 136217 α^{3} + 11258 α^{4} - 474 α^{5} + 8 α^{6}}{550368 + 630168 α + 280736 α^{2} + 61446 α^{3} + 6896 α^{4} + 378 α^{5} + 8 α^{6}}] d α = 1.8116 \end{matrix}$

6 Conclusion

This paper applies the concepts of α -cuts and Zadeh’s extension principle to retrial queue with single server queue model with priority and unequal service rates and constructs membership functions of the fuzzy expected number of customers in high priority queue and fuzzy expected number of customers in low priority queue using paired NLP models. Using the proposed approach, α-cuts of the membership functions are set up and their interval limits inverted to attain explicit closed-form expressions for the system characteristics. The system characteristics of interest can be specified and numerical experiments can be performed when the membership function intervals cannot be inverted, to examine the corresponding a -cuts and then use this information to improve system processes. Since the system characteristics are given by the membership functions, the fuzzy queues with priority are represented more precisely and the results are more useful.

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A single server retrial fuzzy queue with priority and unequal service rates

Abstract

Keywords

1 Introduction

2 Retrial queue with priority

2.1 M1, M2/M1, M2/1 queues

2.2 FM1, FM2/FM1, FM2/1 queues

3 Parametric nonlinear programming

4 Numerical example

4.1 The fuzzy expected number of customers in priority queue L ˜ q 1

References

2.1 M₁, M₂/M₁, M₂/1 queues

2.2 FM₁, FM₂/FM₁, FM₂/1 queues

4.1 The fuzzy expected number of customers in priority queue ${\tilde{L}}_{q 1}$