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Continuous review (s, Q) inventory system at a service facility with positive order lead times

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Abstract

In this study, a continuous review (s,Q) inventory system with a service facility is examined. There is only one server and a limited number of customers waiting rooms in this facility. The demands arrive to the queueing-inventory system according to the Poisson process. Every customer needs a single product with a service period that is unpredictable and distributed arbitrarily. An external supplier replenishes the inventory, and the lead time for the reorder is predicated on an independent exponential distribution. Demands that arise during a stock out period must wait in the waiting area, and when the ordered items arrive, they are served using the first-come-first-serve queueing discipline. With the help of the imbedded Markov chain technique, we are able to compute the joint probability distribution of the number of customers in the system and the number of items in inventory at post-departure epoch. With the remaining service time of a customer in service as the supplementary variable, we are able to relate the system length distributions at post-departure and random epochs in order to determine the joint probability distribution at random epoch. The analysis of waiting time of an accepted customer in the queue is also examined. Several stationary system performance measures are computed and the total expected cost is determined under an appropriate cost structure to determine the optimal values for waiting space (N), reorder level (s), and order quantity (Q). In order to explain the important performance indicators of the system, some numerical findings are given for a variety of model parameters.

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Acknowledgements

The first and third authors acknowledge the National Board for Higher Mathematics (NBHM), DAE, Mumbai, India, for financial support from the project grant 02012/1/2019 NBHM(R.P)/R &D II/1231.

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Author notes

  1. K. P. Sapna Isotupa and A. Verma are contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Raipur, Raipur, Chhattisgarh, 492010, India

    S. K. Samanta & A. Verma

  2. School of Business and Economics, Wilfrid Laurier University, Waterloo, ON, N2L 3C5, Canada

    K. P. Sapna Isotupa

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  1. S. K. Samanta
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  2. K. P. Sapna Isotupa
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Correspondence to S. K. Samanta.

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Appendices

Appendix A: Develop system of differential-difference equations

Relating the states of the system at time epochs t and \(t+dt\), we easily derive the following steady-state system of differential-difference equations as

$$\begin{aligned} 0= & {} -(\lambda +\theta )\pi (0,0)+\pi (1,1,0), \end{aligned}$$
(A1)
$$\begin{aligned} 0= & {} -(\lambda +\theta )\pi (0,j)+\lambda \pi (0,j-1)+\pi (1,j+1,0),\quad 1\le j\le N-1,\end{aligned}$$
(A2)
$$\begin{aligned} 0= & {} -\theta \pi (0,N)+\lambda \pi (0,N-1)+\pi (1,N+1,0),\end{aligned}$$
(A3)
$$\begin{aligned} 0= & {} -(\lambda +\theta )\pi (i,0)+\pi (i+1,1,0),\quad 1\le i\le s,\end{aligned}$$
(A4)
$$\begin{aligned} 0= & {} -\lambda \pi (i,0)+\pi (i+1,1,0),\quad s+1\le i\le Q-1,\end{aligned}$$
(A5)
$$\begin{aligned} 0= & {} -\lambda \pi (i,0)+\pi (i+1,1,0)+\theta \pi (i-Q,0),\quad Q\le i\le Q+s-1,\end{aligned}$$
(A6)
$$\begin{aligned} 0= & {} -\lambda \pi (Q+s,0)+\theta \pi (s,0), \end{aligned}$$
(A7)
$$\begin{aligned} -\frac{d}{dx}\pi (i,1,x)= & {} -(\lambda +\theta )\pi (i,1,x)+\pi (i+1,2,0)g(x)+\lambda \pi (i,0)g(x),\quad 1\le i\le s,\end{aligned}$$
(A8)
$$\begin{aligned} -\frac{d}{dx}\pi (i,j,x)= & {} -(\lambda +\theta )\pi (i,j,x)+\pi (i+1,j+1,0)g(x)+\lambda \pi (i,j-1,x),\nonumber \\{} & {} 1\le i\le s,~~ 2\le j\le N,\end{aligned}$$
(A9)
$$\begin{aligned} -\frac{d}{dx}\pi (i,N+1,x)= & {} -\theta \pi (i,N+1,x)+\lambda \pi (i,N,x),\quad 1\le i\le s,\end{aligned}$$
(A10)
$$\begin{aligned} -\frac{d}{dx}\pi (i,1,x)= & {} -\lambda \pi (i,1,x)+\pi (i+1,2,0)g(x)+\lambda \pi (i,0)g(x),\nonumber \\{} & {} \quad s+1\le i\le Q-1,\end{aligned}$$
(A11)
$$\begin{aligned} -\frac{d}{dx}\pi (i,j,x)= & {} -\lambda \pi (i,j,x)+\pi (i+1,j+1,0)g(x)+\lambda \pi (i,j-1,x),\nonumber \\{} & {} s+1\le i\le Q-1,~~ 2\le j\le N,\end{aligned}$$
(A12)
$$\begin{aligned} -\frac{d}{dx}\pi (i,N+1,x)= & {} \lambda \pi (i,N,x),\quad s+1\le i\le Q-1,\end{aligned}$$
(A13)
$$\begin{aligned} -\frac{d}{dx}\pi (Q,1,x)= & {} -\lambda \pi (Q,1,x)\nonumber \\{} & {} +\pi (Q+1,2,0)g(x)+\lambda \pi (Q,0)g(x)+\theta \pi (0,1)g(x),\end{aligned}$$
(A14)
$$\begin{aligned} -\frac{d}{dx}\pi (Q,j,x)= & {} -\lambda \pi (Q,j,x)+\pi (Q+1,j+1,0)g(x)\nonumber \\{} & {} +\lambda \pi (Q,j-1,x)+\theta \pi (0,j)g(x),2\le j\le N, \end{aligned}$$
(A15)
$$\begin{aligned} -\frac{d}{dx}\pi (Q,N+1,x)= & {} \lambda \pi (Q,N,x),\end{aligned}$$
(A16)
$$\begin{aligned} -\frac{d}{dx}\pi (i,1,x)= & {} -\lambda \pi (i,1,x)+\pi (i+1,2,0)g(x)+\lambda \pi (i,0)g(x)\nonumber \\{} & {} +\theta \pi (i-Q,1,x), Q+1\le i\le Q+s-1,\end{aligned}$$
(A17)
$$\begin{aligned} -\frac{d}{dx}\pi (i,j,x)= & {} -\lambda \pi (i,j,x)+\pi (i+1,j+1,0)g(x) +\lambda \pi (i,j-1,x)\nonumber \\{} & {} +\theta \pi (i-Q,j,x),\nonumber \\{} & {} Q+1\le i\le Q+s-1,~~ 2\le j\le N,\end{aligned}$$
(A18)
$$\begin{aligned} -\frac{d}{dx}\pi (i,N+1,x)= & {} \lambda \pi (i,N,x)+\theta \pi (i-Q,N+1,x),\nonumber \\{} & {} \quad Q+1\le i\le Q+s-1,\end{aligned}$$
(A19)
$$\begin{aligned} -\frac{d}{dx}\pi (Q+s,1,x)= & {} -\lambda \pi (Q+s,1,x)+\lambda \pi (Q+s,0)g(x) +\theta \pi (s,1,x),\end{aligned}$$
(A20)
$$\begin{aligned} -\frac{d}{dx}\pi (Q+s,j,x)= & {} -\lambda \pi (Q+s,j,x)+\lambda \pi (Q+s,j-1,x)\nonumber \\{} & {} +\theta \pi (s,j,x),~2\le j\le N,\end{aligned}$$
(A21)
$$\begin{aligned} -\frac{d}{dx}\pi (Q+s,N+1,x)= & {} \lambda \pi (Q+s,N,x)+\theta \pi (s,N+1,x). \end{aligned}$$
(A22)

Multiplying (A8) and (A22) by \(e^{-\alpha x}\) and integrating them w.r.t. x over 0 to \(\infty \), we obtain

$$\begin{aligned} (\lambda +\theta -\alpha )\pi ^{*}(i,1,\alpha )= & {} -\pi (i,1,0)+\pi (i+1,2,0)G^{*}(\alpha )+\lambda \pi (i,0)G^{*}(\alpha ),\quad 1\le i\le s,\nonumber \\ \end{aligned}$$
(A23)
$$\begin{aligned} (\lambda +\theta -\alpha )\pi ^{*}(i,j,\alpha )= & {} -\pi (i,j,0)+\pi (i+1,j+1,0)G^{*}(\alpha )+\lambda \pi ^{*}(i,j-1,\alpha ),\nonumber \\ {}{} & {} \quad 1\le i\le s,~~ 2\le j\le N,\end{aligned}$$
(A24)
$$\begin{aligned} (\theta -\alpha )\pi ^{*}(i,N+1,\alpha )= & {} -\pi (i,N+1,0)+\lambda \pi ^{*}(i,N,\alpha ),\quad 1\le i\le s,\end{aligned}$$
(A25)
$$\begin{aligned} (\lambda -\alpha )\pi ^{*}(i,1,\alpha )= & {} -\pi (i,1,0)+\pi (i+1,2,0)G^{*}(\alpha )+\lambda \pi (i,0)G^{*}(\alpha ),\nonumber \\{} & {} s+1\le i\le Q-1,\end{aligned}$$
(A26)
$$\begin{aligned} (\lambda -\alpha )\pi ^{*}(i,j,\alpha )= & {} -\pi (i,j,0)+\pi (i+1,j+1,0)G^{*}(\alpha )+\lambda \pi ^{*}(i,j-1,\alpha ),\nonumber \\ {}{} & {} s+1\le i\le Q-1,~~ 2\le j\le N,\end{aligned}$$
(A27)
$$\begin{aligned} -\alpha \pi ^{*}(i,N+1,\alpha )= & {} -\pi (i,N+1,0)+\lambda \pi ^{*}(i,N,\alpha ),\quad s+1\le i\le Q-1,\end{aligned}$$
(A28)
$$\begin{aligned} (\lambda -\alpha )\pi ^{*}(Q,1,\alpha )= & {} -\pi (Q,1,0)+\pi (Q+1,2,0)G^{*}(\alpha )+\lambda \pi (Q,0)G^{*}(\alpha )\nonumber \\{} & {} +\theta \pi (0,1)G^{*}(\alpha ),\end{aligned}$$
(A29)
$$\begin{aligned} (\lambda -\alpha )\pi ^{*}(Q,j,\alpha )= & {} -\pi (Q,j,0)+\pi (Q+1,j+1,0)G^{*}(\alpha )+\lambda \pi ^{*}(Q,j-1,\alpha )\nonumber \\{} & {} +\theta \pi (0,j)G^{*}(\alpha ), 2\le j\le N,\end{aligned}$$
(A30)
$$\begin{aligned} -\alpha \pi ^{*}(Q,N+1,\alpha )= & {} -\pi (Q,N+1,0)+\lambda \pi ^{*}(Q,N,\alpha ),\end{aligned}$$
(A31)
$$\begin{aligned} (\lambda -\alpha )\pi ^{*}(i,1,\alpha )= & {} -\pi (i,1,0)+\pi (i+1,2,0)G^{*}(\alpha )+\lambda \pi (i,0)G^{*}(\alpha )\nonumber \\{} & {} +\theta \pi ^{*}(i-Q,1,\alpha ), Q+1\le i\le Q+s-1,\end{aligned}$$
(A32)
$$\begin{aligned} (\lambda -\alpha )\pi ^{*}(i,j,\alpha )= & {} -\pi (i,j,0)+\pi (i+1,j+1,0)G^{*}(\alpha )+\lambda \pi ^{*}(i,j-1,\alpha )\nonumber \\{} & {} +\theta \pi ^{*}(i-Q,j,\alpha ),Q+1\le i\le Q+s-1,\quad 2\le j\le N,\end{aligned}$$
(A33)
$$\begin{aligned} -\alpha \pi ^{*}(i,N+1,\alpha )= & {} -\pi (i,N+1,0)+\lambda \pi ^{*}(i,N,\alpha )+\theta \pi ^{*}(i-Q,N+1,\alpha ),\nonumber \\{} & {} Q+1\le i\le Q+s-1,\end{aligned}$$
(A34)
$$\begin{aligned} (\lambda -\alpha )\pi ^{*}(Q+s,1,\alpha )= & {} -\pi (Q+s,1,0)+\lambda \pi (Q+s,0)G^{*}(\alpha )+\theta \pi ^{*}(s,1,\alpha ),\end{aligned}$$
(A35)
$$\begin{aligned} (\lambda -\alpha )\pi ^{*}(Q+s,j,\alpha )= & {} -\pi (Q+s,j,0)+\lambda \pi ^{*}(Q+s,j-1,\alpha )+\theta \pi ^{*}(s,j,\alpha ),\nonumber \\{} & {} 2\le j\le N,\end{aligned}$$
(A36)
$$\begin{aligned} -\alpha \pi ^{*}(Q+s,N+1,\alpha )= & {} -\pi (Q+s,N+1,0)+\lambda \pi ^{*}(Q+s,N,\alpha )\nonumber \\{} & {} +\theta \pi ^{*}(s,N+1,\alpha ). \end{aligned}$$
(A37)

Appendix B: Derivation of \(\pi ^{*(1)}(i,j,0)\)

The first order derivative \(\pi ^{*(1)}(i,j,0)\) of \(\pi ^{*}(i,j,\alpha )\) at \(\alpha =0\) are obtained from (A23), (A24), (A25), (A26), (A27), (A29), (A30), (A32), (A33), (A35) and (A36) as

$$\begin{aligned} \pi ^{*(1)}(i,1,0)= & {} \frac{1}{\lambda +\theta }\left[ \pi (i,1)-\frac{E^{\star }}{\mu }\pi ^+(i,1)-\frac{\lambda }{\mu }\pi (i,0)\right] ,\; 1\le i\le s, \end{aligned}$$
(B38)
$$\begin{aligned} \pi ^{*(1)}(i,j,0)= & {} \frac{1}{\lambda +\theta }\left[ \pi (i,j)-\frac{E^{\star }}{\mu }\pi ^+(i,j)+\lambda \pi ^{*(1)}(i,j-1,0)\right] ,\nonumber \\{} & {} 1\le i\le s,\; 2\le j\le N,\end{aligned}$$
(B39)
$$\begin{aligned} \pi ^{*(1)}(i,N+1,0)= & {} \frac{1}{\theta }\left[ \pi (i,N+1)+\lambda \pi ^{*(1)}(i,N,0)\right] ,\nonumber \\{} & {} 1\le i\le s,\end{aligned}$$
(B40)
$$\begin{aligned} \pi ^{*(1)}(i,1,0)= & {} \frac{1}{\lambda }\left[ \pi (i,1)-\frac{E^{\star }}{\mu }\pi ^+(i,1) -\frac{\lambda }{\mu }\pi (i,0)\right] ,\; \nonumber \\{} & {} s+1\le i\le Q-1,\end{aligned}$$
(B41)
$$\begin{aligned} \pi ^{*(1)}(i,j,0)= & {} \frac{1}{\lambda }\left[ \pi (i,j)-\frac{E^{\star }}{\mu }\pi ^+(i,j) +\lambda \pi ^{*(1)}(i,j-1,0)\right] ,\nonumber \\{} & {} s+1\le i\le Q-1,\; 2\le j\le N,\end{aligned}$$
(B42)
$$\begin{aligned} \pi ^{*(1)}(Q,1,0)= & {} \frac{1}{\lambda }\left[ \pi (Q,1)-\frac{1}{\mu }\left[ E^{\star } \pi ^+(Q,1)+\lambda \pi (Q,0)+\theta \pi (0,1)\right] \right] ,\end{aligned}$$
(B43)
$$\begin{aligned} \pi ^{*(1)}(Q,j,0)= & {} \frac{1}{\lambda }\left[ \pi (Q,j)-\frac{E^{\star }}{\mu }\pi ^+(Q,j) +\lambda \pi ^{*(1)}(Q,j-1,0)-\frac{\theta }{\mu }\pi (0,j)\right] ,\nonumber \\{} & {} 2\le j\le N,\end{aligned}$$
(B44)
$$\begin{aligned} \pi ^{*(1)}(i,1,0)= & {} \frac{1}{\lambda }\left[ \pi (i,1)-\frac{E^{\star }}{\mu }\pi ^+(i,1) -\frac{\lambda }{\mu }\pi (i,0)+\theta \pi ^{*(1)}(i-Q,1,0)\right] ,\nonumber \\{} & {} Q+1\le i\le Q+s-1,\end{aligned}$$
(B45)
$$\begin{aligned} \pi ^{*(1)}(i,j,0)= & {} \frac{1}{\lambda }\left[ \pi (i,j)-\frac{E^{\star }}{\mu }\pi ^+(i,j) +\lambda \pi ^{*(1)}(i,j-1,0)+\theta \pi ^{*(1)}(i-Q,j,0)\right] ,\nonumber \\{} & {} Q+1\le i\le Q+s-1,\; 2\le j\le N,\end{aligned}$$
(B46)
$$\begin{aligned} \pi ^{*(1)}(Q+s,1,0)= & {} \frac{1}{\lambda }\left[ \pi (Q+s,1) -\frac{\lambda }{\mu }\pi (Q+s,0)+\theta \pi ^{*(1)}(s,1,0)\right] ,\end{aligned}$$
(B47)
$$\begin{aligned} \pi ^{*(1)}(Q+s,j,0)= & {} \frac{1}{\lambda }\left[ \pi (Q+s,j) +\lambda \pi ^{*(1)}(Q+s,j-1,0)+\theta \pi ^{*(1)}(s,j,0)\right] ,\nonumber \\{} & {} \quad 2\le j\le N. \end{aligned}$$
(B48)

Appendix C: Derivation of \(\pi ^{*}(i,j,\theta )\) and \(\pi ^{*(1)}(i,j,\theta )\)

The \(\pi ^{*}(i,j,\theta )\) and \(\pi ^{*(1)}(i,j,\theta )\) are obtained from (A23), (A24), (A26), (A27), (A29), (A30), (A32), (A33), (A35) and (A36) as

$$\begin{aligned} \pi ^{*}(i,1,\theta )= & {} \frac{1}{\lambda }\left[ E^{\star }\pi ^+(i,1)G^{*}(\theta )+\lambda \pi (i,0)G^{*}(\theta )-E^{\star }\pi ^+(i-1,0)\right] ,\quad 1\le i\le s,\\ \pi ^{*}(i,j,\theta )= & {} \frac{1}{\lambda }\left[ E^{\star }\pi ^+(i,j)G^{*}(\theta )+\lambda \pi ^{*}(i,j-1,\theta )-E^{\star }\pi ^+(i-1,j-1)\right] , \\{} & {} \quad 1\le i\le s,~~ 2\le j\le N,\\ \pi ^{*}(i,1,\theta )= & {} \frac{1}{\lambda -\theta }\left[ E^{\star }\pi ^+(i,1)G^{*}(\theta )+\lambda \pi (i,0)G^{*}(\theta )-E^{\star }\pi ^+(i-1,0)\right] ,\\{} & {} s+1\le i\le Q-1,\\ \pi ^{*}(i,j,\theta )= & {} \frac{1}{\lambda -\theta }\left[ E^{\star }\pi ^+(i,j)G^{*}(\theta )+\lambda \pi ^{*}(i,j-1,\theta )-E^{\star }\pi ^+(i-1,j-1)\right] ,\\{} & {} s+1\le i\le Q-1,~~ 2\le j\le N,\\ \pi ^{*}(Q,1,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \left[ E^{\star }\pi ^+(Q,1)+\lambda \pi (Q,0)+\theta \pi (0,1)\right] G^{*}(\theta )-E^{\star }\pi ^+(Q-1,0)\right] ,\\ \pi ^{*}(Q,j,\theta )= & {} \frac{1}{\lambda -\theta }\left[ E^{\star }\pi ^+(Q,j)G^{*}(\theta )+\lambda \pi ^{*}(Q,j-1,\theta )+\theta \pi (0,j)G^{*}(\theta )\right. \\{} & {} \left. -E^{\star }\pi ^+(Q-1,j-1)\right] ,\quad 2\le j\le N,\\ \pi ^{*}(i,1,\theta )= & {} \frac{1}{\lambda -\theta }\left[ E^{\star }\pi ^+(i,1)G^{*}(\theta )+\lambda \pi (i,0)G^{*}(\theta )+\theta \pi ^{*}(i-Q,1,\theta )\right. \\{} & {} \left. -E^{\star }\pi ^+(i-1,0)\right] ,\quad Q+1\le i\le Q+s-1,\\ \pi ^{*}(i,j,\theta )= & {} \frac{1}{\lambda -\theta }\left[ E^{\star }\pi ^+(i,j)G^{*}(\theta )+\lambda \pi ^{*}(i,j-1,\theta )+\theta \pi ^{*}(i-Q,j,\theta )\right. \\{} & {} \left. -E^{\star }\pi ^+(i-1,j-1)\right] ,\quad Q+1\le i\le Q+s-1,\quad 2\le j\le N,\\ \pi ^{*}(Q+s,1,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \lambda \pi (Q+s,0)G^{*}(\theta )+\theta \pi ^{*}(s,1,\theta )-E^{\star }\pi ^+(Q+s-1,0)\right] ,\\ \pi ^{*}(Q+s,j,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \lambda \pi ^{*}(Q+s,j-1,\theta )+\theta \pi ^{*}(s,j,\theta )-E^{\star }\pi ^+(Q+s-1,j-1)\right] ,\\{} & {} 2\le j\le N,\\ \pi ^{*(1)}(i,1,\theta )= & {} \frac{1}{\lambda }\left[ \pi ^{*}(i,1,\theta )+E^{\star }\pi ^+(i,1)G^{*(1)}(\theta )+\lambda \pi (i,0)G^{*(1)}(\theta )\right] ,\quad 1\le i\le s,\\ \pi ^{*(1)}(i,j,\theta )= & {} \frac{1}{\lambda }\left[ \pi ^{*}(i,j,\theta )+E^{\star }\pi ^+(i,j)G^{*(1)}(\theta )+\lambda \pi ^{*(1)}(i,j-1,\theta )\right] , \\{} & {} 1\le i\le s,~~ 2\le j\le N, \end{aligned}$$
$$\begin{aligned} \pi ^{*(1)}(i,1,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \pi ^{*}(i,1,\theta )+E^{\star }\pi ^+(i,1)G^{*(1)}(\theta )+\lambda \pi (i,0)G^{*(1)}(\theta )\right] ,\\{} & {} s+1\le i\le Q-1,\\ \pi ^{*(1)}(i,j,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \pi ^{*}(i,j,\theta )+E^{\star }\pi ^+(i,j)G^{*(1)}(\theta )+\lambda \pi ^{*(1)}(i,j-1,\theta )\right] ,\\{} & {} s+1\le i\le Q-1,~~ 2\le j\le N,\\ \pi ^{*(1)}(Q,1,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \pi ^{*}(Q,1,\theta )+\left[ E^{\star }\pi ^+(Q,1)+\lambda \pi (Q,0)+\theta \pi (0,1)\right] G^{*(1)}(\theta )\right] ,\\ \pi ^{*(1)}(Q,j,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \pi ^{*}(Q,j,\theta )+E^{\star }\pi ^+(Q,j)G^{*(1)}(\theta )+\lambda \pi ^{*(1)}(Q,j-1,\theta )\right. \\{} & {} \left. +\theta \pi (0,j)G^{*(1)}(\theta )\right] ,\quad 2\le j\le N,\\ \pi ^{*(1)}(i,1,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \pi ^{*}(i,1,\theta )+E^{\star }\pi ^+(i,1)G^{*(1)}(\theta )+\lambda \pi (i,0)G^{*(1)}(\theta )\right. \\{} & {} \left. +\theta \pi ^{*(1)}(i-Q,1,\theta )\right] ,\quad Q+1\le i\le Q+s-1,\\ \pi ^{*(1)}(i,j,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \pi ^{*}(i,j,\theta )+E^{\star }\pi ^+(i,j)G^{*(1)}(\theta )+\lambda \pi ^{*(1)}(i,j-1,\theta )\right. \\{} & {} \left. +\theta \pi ^{*(1)}(i-Q,j,\theta )\right] ,\quad Q+1\le i\le Q+s-1,\quad 2\le j\le N,\\ \pi ^{*(1)}(Q+s,1,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \pi ^{*}(Q+s,1,\theta )+\lambda \pi (Q+s,0)G^{*(1)}(\theta )+\theta \pi ^{*(1)}(s,1,\theta )\right] ,\\ \pi ^{*(1)}(Q+s,j,\theta )= & {} \frac{1}{\lambda -\theta }\left[ \pi ^{*}(Q+s,j,\theta )+\lambda \pi ^{*(1)}(Q+s,j-1,\theta )+\theta \pi ^{*(1)}(s,j,\theta )\right] ,\\{} & {} 2\le j\le N. \end{aligned}$$

Appendix D: Derivation of \(G^{*(n)}(\alpha )\) at \(\alpha =\theta \) for deterministic, exponential and Erlang service time distributions

Deterministic distribution

Let \(g(t)=1\) for \(t=1/\mu \) and 0 in all other cases. Thus, \(g^{*(n)}(\alpha )=\frac{(-1)^n}{\mu ^n}e^{-\alpha /\mu }\).

Then \(G(t)=1\) for \(0\le t\le 1/\mu \) and 0 in all other cases. Thus, \(G^{*(n)}(\alpha )=\frac{(-1)^{n}n!}{\alpha ^{n+1}}\Bigg [1-e^{-\alpha /\mu }\sum _{i=0}^n\frac{1}{i!}\bigg (\frac{\alpha }{\mu }\bigg )^i\Bigg ]\).

Exponential distribution

Let \(g(t)=\mu e^{-\mu t}\). Thus, \(g^{*(n)}(\alpha )=\frac{(-1)^{n}n!\mu }{(\alpha +\mu )^{n+1}}\). Then \(G(t)=e^{-\mu t}\). Thus, \(G^{*(n)}(\alpha )=\frac{(-1)^{n}n!}{(\alpha +\mu )^{n+1}}\).

Erlang (of order p) distribution

Let \(g(t)=\frac{\mu ^pt^{p-1}e^{-\mu t}}{(p-1)!}\). Thus, \(g^{*(n)}(\alpha )=\frac{(-1)^{n}\mu ^p(n+p-1)!}{(p-1)!(\alpha +\mu )^{n+p}}\). Then \(G(t)=\sum _{i=0}^{p-1}\frac{(\mu t)^ie^{-\mu t}}{i!}\).

Thus, \(G^{*(n)}(\alpha )=\frac{(-1)^{n}n!}{\alpha ^{n+1}}\Bigg [1-\bigg (\frac{\mu }{\alpha +\mu }\bigg )^p\sum _{i=0}^n{i+p-1\atopwithdelims ()i}\bigg (\frac{\alpha }{\alpha +\mu }\bigg )^i\Bigg ]\).

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Samanta, S.K., Isotupa, K.P.S. & Verma, A. Continuous review (s, Q) inventory system at a service facility with positive order lead times. Ann Oper Res 331, 1007–1028 (2023). https://doi.org/10.1007/s10479-023-05171-2

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