A queueing-inventory model with skeptical and trusting customers

Abstract

Performing a part of the service without the customer's presence is becoming an increasingly common practice in many real-life service systems. This practice can generate a two-phase service system composed of (i) an opening service (OS), which can be provided only when the customer is present, and (ii) a complementary service (CS), which can be conducted without the customer. However, in practice, not all customers favor being absent when their service is provided. It follows that customers are generally of two kinds: those who insist that the whole service is provided in their presence and those who are willing to leave the system whenever their presence is not required. Moreover, after providing the OS, the server can postpone the execution of the CS, store the ready OS in the designated storage facility and handle it once the system becomes empty of customers. By adopting such a policy, the server’s idle time is efficiently used, reducing customers’ mean sojourn time in the system. In contrast to classical queueing-inventory models, where each customer's service may require a unit from inventory, in our model a customer's service may create a unit in the inventory. We formulate and analyze this novel queueing-inventory system and derive its steady-state probabilities using matrix geometric methods. We subsequently consider the spoilage of OSs in the inventory, and carry out an economic analysis to determine the optimal OS capacity and optimal level of investment in preservation technologies preventing the spoilage.

Appendices

Appendix

See Fig. 

Fig. 15

The system's states and transition rate diagram for $n = 3$

15

The matrices A ₀ , A ₁ , A ₂ and B for n  = 3

$$ A_{0} = \left( {\begin{array}{*{20}c} \lambda & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \lambda & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \lambda & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\lambda + \beta } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \lambda & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \lambda & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\lambda + \beta } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \lambda & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\lambda + \beta } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\lambda + \beta } \\ \end{array} } \right), $$

$$\begin{aligned} A_{1} &= \left( {\begin{array}{*{20}c} { - (\lambda + \beta + \mu )} & \beta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & { - (\lambda + \beta + \alpha )} & \beta & 0 & \alpha & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - (\lambda + \beta + \alpha )} & \beta & 0 & \alpha & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - (\lambda + \beta + \alpha )} & 0 & 0 & \alpha & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - (\lambda + \beta + \mu )} & \beta & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - (\lambda + \beta + \alpha )} & \beta & \alpha & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - (\lambda + \beta + \alpha )} & 0 & \alpha & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - (\lambda + \beta + \mu )} & \beta & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - (\lambda + \beta + \alpha )} & \alpha \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - (\lambda + \beta + \mu )} \\ \end{array} } \right),\\ A_{2} &= \left( {\begin{array}{*{20}c} \mu & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \mu \\ \end{array} } \right).\end{aligned}$$

$B = \left( {\begin{array}{*{20}c} { - (\lambda + \beta )} & \beta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & { - (\lambda + \beta + \alpha )} & \beta & 0 & \alpha & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - (\lambda + \beta + \alpha )} & \beta & 0 & \alpha & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - (\lambda + \beta + \alpha )} & 0 & 0 & \alpha & 0 & 0 & 0 \\ \delta & 0 & 0 & 0 & { - (\lambda + \beta + \delta )} & \beta & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - (\lambda + \beta + \alpha )} & \beta & \alpha & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - (\lambda + \beta + \alpha )} & 0 & \alpha & 0 \\ 0 & 0 & 0 & 0 & \delta & 0 & 0 & { - (\lambda + \beta + \delta )} & \beta & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - (\lambda + \beta + \alpha )} & \alpha \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \delta & 0 & { - (\lambda + \beta + \delta )} \\ \end{array} } \right),$ Formulae for the rate matrix R

$$ \begin{array}{*{20}c} {r^{v,t} = 0} & {{\text{for all}}\;v > t} \\ \end{array} , $$

(10)

$$ r^{v,v} = \left\{ {\begin{array}{*{20}c} {\frac{{ - a_{1}^{v,v} - \sqrt {(a_{1}^{v,v} )^{2} - 4\,a_{0}^{v,v} \,a_{2}^{v,v} } }}{{2\,a_{2}^{v,v} }},} & {\,\,\,\,\,a_{2}^{v,v} > 0,\,\,\,a_{0}^{v,v} > 0} \\ {0,} & {\,\,\,\,\,a_{2}^{v,v} > 0,\,\,\,a_{0}^{v,v} = 0} \\ {\frac{{ - a_{0}^{v,v} }}{{a_{1}^{v,v} }},} & {\,\,\,\,\,a_{2}^{v,v} = 0} \\ \end{array} } \right.,\,\,\,\,\,\,\forall v, $$

(11)

$$ \begin{array}{*{20}c} {r^{v,t} = - \frac{{a_{0}^{v,t} + \sum\limits_{k = v}^{t - 1} {r^{v,k} a_{1}^{k,t} } + \sum\limits_{\tau = v + 1}^{t - 1} {r^{v,\tau } r^{\tau ,t} a_{2}^{t,t} } + \sum\limits_{k = v}^{t - 1} {\sum\limits_{\tau = v}^{k} {r^{v,\tau } r^{\tau ,k} a_{2}^{k,t} } } }}{{a_{1}^{t,t} + a_{2}^{t,t} (r^{t,t} + r^{v,v} )}}} & {{\text{for all}}.} \\ \end{array} $$

(12)