Optimal Control Policy for a Two-Phase M/M/1 Unreliable Gated Queue under N-Policy with a Fuzzy Environment

Controllable operating policy is useful in several domains, including computer network management, communication systems, manufacturing, and inventory control operations. Instead, let's use an example: computer-controlled manufacturing, where raw materials are processed at the initial stage into finished products like silicon chips, bearings, and electronic components and then undergo quality checks before packaging into batches. N-policy is a well-known queue control operating policy. Under this N-policy, the server is off as soon as the system is empty and on when the waiting queue size is ‘N'.

Yadin and Naor [1] were the first to propose the concept of N-policy. Many researchers, including Teghem [2], Lee and Srinivasan [3], Medhi and Templeton [4], Takagi [5], Wang and Hsich [6], Lee and Park [7], Hur and Paik [8], Ke [9], Tadj and Ke [10], Wang [11], Wang et al. [12]. Tadj and Choudhury [13], Tadj et al. [14], Wang et al. [15], and others have studied individual or batch arrival, general service, exponential or general startup, and unreliable server queueing models.

In the study of fuzzy queueing models, the basic idea is to translate a fuzzy queue to a family of conventional crisp queues by applying the α-cut approach. Kao et al. [16] proposed a general procedure to construct the membership functions of the performance measures in queueing systems when the interarrival time and service time are fuzzy numbers. Subsequently, Chen [17] proposed a procedure for constructing the membership functions of the performance measures in finite capacity queueing system with arrival rate and service rate being fuzzy numbers in which he formulated a pair of nonlinear programs to describe the family of finite capacity queues, using the α-cut approach. Thereafter, Lin and Ke [18] constructed the membership function of the fuzzy objective values of a controllable queueing model in which cost elements, arrival rate and service rate are fuzzy numbers. Based on the Zadeh’s extension principle they developed, a set of nonlinear programs to find the upper and lower bounds of the minimal average total cost per unit time at the possibility level. Of late, Das and Baruah [19] applied the parametric programming approach to construct the membership functions of the interarrival time and service time. Bagherinejad and Pishkenari [20] proposed an FM/FM/C system based on fuzzy approach. They developed performance measures using the concept of parametric nonlinear programming and characteristics of the fuzzy numbers. Ashok Kumar [21] used the mathematical programming to construct the membership function of the system performance measures. Varadarajan and Susmitha [22] extended the queueing model of priority classes to the fuzzy environment by constructing the parametric programming problem where the trapezoidal fuzzy numbers are introduced for the queue parameters. Using the α-cut approach, Kalpana and Anusheela [23] derived the membership functions of the system characteristics in non-preemptive priority queueing model with priority subscribers in which the arrival rate and service rates are fuzzy numbers. George et al. [24] described a method to find variety of performance measures for fuzzy queueing model with an unreliable server. Aarthi et al. [25] discussed the characteristics of N-policy fuzzy queue under vague data. Vonodhini and Anitha [26] proposed a mathematical programming technique to find the membership function of the performance measures of a fuzzy single server queue in steady state.

In the process of two-phase service queueing with two services per customer, which consists of first phase batch service followed by second phase single service for all customers in the batch were initiated by Krishna and Lee [27]. Subsequently, the two-phase M/G/1 queue with N-policy was analyzed by Doshi [28]. Later, the two-phase queueing system with N-policy were explored by Kim and Chae [29]. In continuation of the above studies, the two stages M/M/1 queueing system with N-policy and gating, Vasanta Kumar & Chandan [30] presented an optimal operational policy. The detailed and performance examination of the two-phase M/E_k/1 and M^X/M/1 server breakdown and gating N-policy systems were extensively and exhaustively explored by Vasanta Kumar et al. [31, 32]. As a consequence of the above studies, batch arrival and Erlangian service two-phase controllable M^X/M/1 and M^X/E_k/1 queueing systems with server startup and breakdowns were discussed by Vasanta Kumar et al. [33, 34]. Hari Prasad [35] presented an extensive study of Markovian server start-up queueing system with two service phases, N policy and an unreliable server. He determined the optimum threshold N*, which reduced the overall anticipated expenses per unit time, including six distinct costs. He assumed that the arrival rate, batch service rate, individual service rate, vacation rate, breakdown rate, repair rate and various cost variables are exactly known. These parameters and cost aspects cannot, however, be accurately approximated in many real-world applications. For example, the costs of maintaining a customer per unit time and the costs per unit time to keep the server in operation are more suitably described in linguistic terms high, modest, or low as well as installation costs, breakdown cost and revenue generated during secondary vacation activities. These system and cost characteristics are both possibilistic and probabilistic. There are so potentially much more practical and practical adjustable operation strategy for queueing models with fuzzy parameters than the usual crisp queues.

It has been observed from the review of literature that existing research works, including those mentioned above have not considered the controllable two-phase, N-policy Markovian queues with server breakdowns under fuzzy environment. Hence, the authors propose to consider a controllable M/M/1 queue with two service phases and an unreliable server with a Fuzzy environment. By means of the membership functions of the system characteristics, we develop the membership functions of the minimal total expected cost using the parametric nonlinear programming approach with exponential arrival rate, batch service rate, individual service rate, startup rate, breakdown rate, repair rates and fuzzy cost elements as fuzzy numbers. Through α-cuts and Zadeh’s extension principle [36] we transform the fuzzy minimal total expected cost function into a family of crisp functions. As the nonlinear programs developed are complicated to derive the closed form solution, utilizing the nonlinear programming problem solver available in MATLAB, numerical solutions are computed for assumed values of the system parameters and cost elements. These numerical solutions are helpful for the system designers and practitioners to improve the system process. The MATLAB code developed for the numerical computations is presented in APPENDIX.

The α-cuts of $\tilde{\lambda}, \tilde{\beta}, \tilde{\mu}, \tilde{\theta}, \tilde{\delta}, \tilde{\gamma}, \tilde{\mathrm{C}}_{\mathrm{h}}, \tilde{\mathrm{C}}_{0}, \tilde{\mathrm{C}}_{m}, \tilde{\mathrm{C}}_{\mathrm{s}}, \tilde{C}_{b}$ and $\tilde{\mathrm{C}}_{\mathrm{r}}$ can be defined by:

$\lambda \left( \alpha  \right)=\left[ a_{\alpha }^{L},a_{\alpha }^{U} \right]\,=\left[ _{a\in A}^{\min }\left\{ a:{{\xi }_{{\tilde{\lambda }}}}(a)\ge \alpha  \right\},\,_{a\in A}^{\max }\,\left\{ a:{{\xi }_{{\tilde{\lambda }}}}(a)\ge \alpha  \right\} \right]$                          (9a)

$\beta \left( \alpha  \right)=\left[ b_{\alpha }^{L},b_{\alpha }^{U} \right]=\,\left[ _{b\in B}^{\min }\left\{ b:{{\xi }_{{\tilde{\beta }}}}(b)\ge \alpha  \right\},\,_{b\in B}^{\max }\,\left\{ b:{{\xi }_{{\tilde{\beta }}}}(b)\ge \alpha  \right\} \right]$                           (9b)

$\mu \left( \alpha  \right)=\left[ c_{\alpha }^{L},c_{\alpha }^{U} \right]=\,\left[ _{c\in C}^{\min }\left\{ c:{{\xi }_{{\tilde{\mu }}}}(c)\ge \alpha  \right\},\,_{c\in C}^{\max }\,\left\{ c:{{\xi }_{{\tilde{\mu }}}}(c)\ge \alpha  \right\} \right]$                            (9c)

$\theta \,(\alpha )=\left[ d\,_{^{\alpha }}^{L},\,\,d\,_{^{\alpha }}^{U} \right]=\left[ _{d\in D}^{\min }\left\{ d:{{\xi }_{{\tilde{\theta }}}}(d)\ge \alpha  \right\},\,_{d\in D}^{\max }\,\left\{ d:{{\xi }_{{\tilde{\theta }}}}(d)\ge \alpha  \right\} \right]$                          (9d)

$\delta (\alpha )=\left[ e\,_{^{\alpha }}^{L},\,\,e\,_{^{\alpha }}^{U} \right]=\left[ _{e\in E}^{\min }\left\{ e:{{\xi }_{{\tilde{\delta }}}}(e)\ge \alpha  \right\},\,_{e\in E}^{\max }\,\left\{ e:{{\xi }_{{\tilde{\delta }}}}(e)\ge \alpha  \right\} \right]$                              (9e)

$\gamma (\alpha )=\left[ f\,_{^{\alpha }}^{L},\,\,f\,_{^{\alpha }}^{U} \right]=\left[ _{f\in F}^{\min }\left\{ f:{{\xi }_{{\tilde{\gamma }}}}(f)\ge \alpha  \right\},\,_{f\in F}^{\max }\,\left\{ f:{{\xi }_{{\tilde{\gamma }}}}(f)\ge \alpha  \right\} \right]$                          (9f)

${{C}_{h}}\left( \alpha  \right)=\left[ g\,_{^{\alpha }}^{L},\,\,g_{^{\alpha }}^{U} \right]=\left[ _{g\in G}^{\min }\left\{ g:{{\xi }_{{{{\tilde{C}}}_{h}}}}(g)\ge \alpha  \right\},\,_{g\in G}^{\max }\,\left\{ g:{{\xi }_{{{{\tilde{C}}}_{h}}}}(g)\ge \alpha  \right\} \right]$                            (9g)

${{C}_{o}}\left( \alpha  \right)=\left[ h\,_{^{\alpha }}^{L},\,\,h_{^{\alpha }}^{U} \right]=\left[ _{h\in H}^{\min }\left\{ h:{{\xi }_{{{{\tilde{C}}}_{o}}}}(h)\ge \alpha  \right\},\,_{h\in H}^{\max }\,\left\{ h:{{\xi }_{{{{\tilde{C}}}_{o}}}}(h)\ge \alpha  \right\} \right]$                       (9h)

${{C}_{m}}\left( \alpha  \right)=\left[ i\,_{^{\alpha }}^{L},\,\,i_{^{\alpha }}^{U} \right]=\left[ _{i\in I}^{\min }\left\{ i:{{\xi }_{{{{\tilde{C}}}_{m}}}}(i)\ge \alpha  \right\},\,_{i\in I}^{\max }\,\left\{ i:{{\xi }_{{{{\tilde{C}}}_{m}}}}(i)\ge \alpha  \right\} \right]$                             (9i)

${{C}_{s}}\left( \alpha  \right)=\left[ j\,_{^{\alpha }}^{L},\,\,j_{^{\alpha }}^{U} \right]=\left[ _{j\in J}^{\min }\left\{ j:{{\xi }_{{{{\tilde{C}}}_{s}}}}(j)\ge \alpha  \right\},\,_{j\in J}^{\max }\,\left\{ j:{{\xi }_{{{{\tilde{C}}}_{s}}}}(j)\ge \alpha  \right\} \right]$                          (9j)

${{C}_{b}}\left( \alpha  \right)=\left[ k\,_{^{\alpha }}^{L},\,\,k_{^{\alpha }}^{U} \right]=\left[ _{k\in K}^{\min }\left\{ k:{{\xi }_{{{{\tilde{C}}}_{b}}}}(k)\ge \alpha  \right\},\,_{k\in K}^{\max }\,\left\{ k:{{\xi }_{{{{\tilde{C}}}_{b}}}}(k)\ge \alpha  \right\} \right]$                             (9k)

${{C}_{r}}\left( \alpha  \right)=\left[ l_{^{\alpha }}^{L},\,\,l_{^{\alpha }}^{U} \right]=\left[ _{l\in L}^{\min }\left\{ l:{{\xi }_{{{{\tilde{C}}}_{r}}}}(l)\ge \alpha  \right\},\,_{l\in L}^{\max }\,\left\{ l:{{\xi }_{{{{\tilde{C}}}_{r}}}}(l)\ge \alpha  \right\} \right]$                            (9l)

The queue parameters and cost elements are shown as intervals when the membership functions are not less than a given possibility level α.

Let$H\left( a,b,c,d,e,f,g,h,i,j,k,l \right)$

$=g\left[ \begin{align}  & \left( \frac{N(N-1)}{2(1-{{\rho }_{1}})}+\frac{aN(a+Nd)}{{{d}^{2}}(1-{{\rho }_{1}})} \right){{P}_{0,0,0}} \\ & +\frac{a}{b(1-{{\rho }_{1}})}+\frac{{{\rho }_{1}}}{(1-{{\rho }_{1}})}+\frac{ae\rho }{{{f}^{2}}(1-{{\rho }_{1}})}+{{\left( \frac{a}{b} \right)}^{2}}\frac{{{\rho }_{1}}}{(1-{{\rho }_{1}})} \\\end{align} \right]$

$+h\left( \frac{a}{b}+\rho  \right)+i\left( \frac{a}{d}{{P}_{0,0,0}} \right)+j\left( a{{P}_{0,0,0}} \right)+k\left( \frac{\rho e}{f} \right)-l\left( N{{P}_{0,0,0}} \right)$                  (10)

The lower and upper bounds of the α-cuts of ${{\xi }_{{\tilde{\psi }}}}(z)$ are:

$\psi _{\alpha }^{L}=\underset{a,b,c,d,e,f,g,h,i,j,k,l\in {{R}^{+}}}{\mathop{\min }}\quad \underset{N>0}{\mathop{\min }}\,\,\,H\left( a,b,c,d,e,f,g,h,i,j,k,l \right)$                     (11a)

such that

$a_{\alpha}^{L} \leq a \leq a_{\alpha}^{U}, b_{\alpha}^{L} \leq b \leq b_{\alpha}^{U}, c_{\alpha}^{L} \leq c \leq c_{\alpha}^{U}, d_{\alpha}^{L} \leq d \leq d_{\alpha}^{U}$

$e_{\alpha}^{L} \leq e \leq e_{\alpha}^{U}, f_{\alpha}^{L} \leq f \leq f_{\alpha}^{U}, g_{\alpha}^{L} \leq g \leq g_{\alpha}^{U}, h_{\alpha}^{L} \leq h \leq h_{\alpha}^{U}$

$i_{\alpha}^{L} \leq i \leq i_{\alpha}^{U}, j_{\alpha}^{L} \leq j \leq j_{\alpha}^{U}, k_{\alpha}^{L} \leq k \leq k_{\alpha}^{U}, l_{\alpha}^{L} \leq l \leq l_{\alpha}^{U}$.

$\psi _{\alpha }^{U}=\underset{a,b,c,d,e,f,g,h,i,j,k,l\in {{R}^{+}}}{\mathop{\max }}\quad\underset{N>0}{\mathop{\min }}\,\,\,H\left( a,b,c,d,e,f,g,h,i,j,k,l \right)$                  (11b)

such that

$a_{\alpha}^{L} \leq a \leq a_{\alpha}^{U}, b_{\alpha}^{L} \leq b \leq b_{\alpha}^{U}, c_{\alpha}^{L} \leq c \leq c_{\alpha}^{U}, d_{\alpha}^{L} \leq d \leq d_{\alpha}^{U}$

$e_{\alpha}^{L} \leq e \leq e_{\alpha}^{U}, f_{\alpha}^{L} \leq f \leq f_{\alpha}^{U}, g_{\alpha}^{L} \leq g \leq g_{\alpha}^{U}, h_{\alpha}^{L} \leq h \leq h_{\alpha}^{U}$

$i_{\alpha}^{L} \leq i \leq i_{\alpha}^{U}, j_{\alpha}^{L} \leq j \leq j_{\alpha}^{U}, k_{\alpha}^{L} \leq k \leq k_{\alpha}^{U}, l_{\alpha}^{L} \leq l \leq l_{\alpha}^{U} .$

Since Eq. 11(a) is to find the minimum of all the minimum objective values, the two-level mathematical programming problem can be simplified to the following one level mathematical programming problem.

$\psi _{\alpha }^{L}=_{N>0}^{\min }T(N,a,b,c,d,e,f,g,h,i,j,k,l)$                     (12)

such that

$a_{\alpha}^{L} \leq a \leq a_{\alpha}^{U}, b_{\alpha}^{L} \leq b \leq b_{\alpha}^{U}, c_{\alpha}^{L} \leq c \leq c_{\alpha}^{U}, d_{\alpha}^{L} \leq d \leq d_{\alpha}^{U}$

$e_{\alpha}^{L} \leq e \leq e_{\alpha}^{U}, f_{\alpha}^{L} \leq f \leq f_{\alpha}^{U}, g_{\alpha}^{L} \leq g \leq g_{\alpha}^{U}, h_{\alpha}^{L} \leq h \leq h_{\alpha}^{U}$

$i_{\alpha}^{L} \leq i \leq i_{\alpha}^{U}, j_{\alpha}^{L} \leq j \leq j_{\alpha}^{U}, k_{\alpha}^{L} \leq k \leq k_{\alpha}^{U}, l_{\alpha}^{L} \leq l \leq l_{\alpha}^{U}$.

To solve 11(b), we treat the inner $\underset{N}{\mathop{\min }}\,$ problem, in that a,b,c,d,e,f,g,h,i,j,k,l could be viewed as constants. The only unknown in the objective function of the inner $\underset{N}{\mathop{\min }}\,$ problem is N.

Let $T(N;a,b,c,d,e,f,g,h,i,j,k,l)\triangleq G(N)\text{ }$                        (13)

Then the inner $\underset{N}{\mathop{\min }}\,$ problem of Eq. 11(b) becomes $\min G(N),$ which is a local minimization problem. From the second derivative test, if ${G}'({{N}_{0}})=0$ and ${G}''({{N}_{0}})>0$ then G(N) has a local minimum at N=N₀ Hence, Eq. 11(b) can be reformulated as the following mathematical programming problem.

$\psi _{\alpha }^{U}=\underset{{{N}_{0}}>0}{\mathop{\min }}\,G(N)$                       (14)

such that

$a_{\alpha}^{L} \leq a \leq a_{\alpha}^{U}, b_{\alpha}^{L} \leq b \leq b_{\alpha}^{U}, c_{\alpha}^{L} \leq c \leq c_{\alpha}^{U}, d_{\alpha}^{L} \leq d \leq d_{\alpha}^{U}$

$e_{\alpha}^{L} \leq e \leq e_{\alpha}^{U}, f_{\alpha}^{L} \leq f \leq f_{\alpha}^{U}, g_{\alpha}^{L} \leq g \leq g_{\alpha}^{U}, h_{\alpha}^{L} \leq h \leq h_{\alpha}^{U}$

$i_{\alpha}^{L} \leq i \leq i_{\alpha}^{U}, j_{\alpha}^{L} \leq j \leq j_{\alpha}^{U}, k_{\alpha}^{L} \leq k \leq k_{\alpha}^{U}, l_{\alpha}^{L} \leq l \leq l_{\alpha}^{U}$.

At least one a,b,c,d,e,f,g,h,i,j,k and l must hit the boundary of their α–cuts to satisfy ${{\xi }_{{\tilde{\psi }}}}=\alpha .$

The lower bound $\psi _{\alpha }^{L}$ and upper bound $\psi _{\alpha }^{U}$ of the α–cuts of $\tilde{\psi }$ can be found by solving Eqns. (12) and (14), respectively.

This article applies the ideas of α-cut and Zadeh's extension principle to an N-policy controlled Markovian queue with an unreliable server to obtain the lower and upper bound for the optimal threshold N* as well as the minimal total expected cost per unit time. When the classical queues with unreliable server can be extended to queueing models with fuzzy parameters, the fuzzy queue can be represented more accurately by using the proposed approach and will have realistic applications. The results obtained from the analysis for the queueing model studied in this paper are useful and significant for system designers and practitioners in designing production systems and communication systems. The fuzzy approach presented in this paper can be extended to study other crisp two-phase, N-policy queues with batch arrival, server breakdowns and delayed repair.