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Strategic customer behavior in a two-stage batch processing system

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Abstract

We consider a two-stage service system with batch processing. More specifically, customers arrive according to a Poisson process at the first stage of the system, where they do not receive any service, but wait until a number K of them are accumulated. Then, these K customers form a batch which is instantaneously transferred to the second stage where the batches are served sequentially, as single units, by a single server. We derive customer equilibrium strategies, regarding the joining/balking dilemma, for the (partially) observable case where the customers are informed upon arrival about the number of present customers at the first stage. We prove the existence and study the uniqueness of equilibrium strategies and give a simple procedure for their computation. Moreover, we juxtapose the customer strategic behavior in this system with the corresponding behavior in the unobservable system and study the effect of information on the resulting social welfare per time unit under equilibrium.

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Acknowledgements

We would like to cordially thank two anonymous referees and the guest editors for their constructive comments and feedback that help us to improve considerably the present paper.

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Correspondence to Olga Bountali.

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Appendices

Appendix I: Performance evaluation under an MRT strategy using a matrix-analytic approach

We begin with the derivation of the stationary distribution of the process \(\{(M(t),J(t))\}\) when the customers follow an x-MRT strategy.

Proposition 8.1

The stationary distribution \((\pi (m,j):m\ge 0, 0\le j\le K-1)\) of the process \(\{(M(t),J(t))\}\), when the customers follow an x-MRT strategy with \(x\in (K-1,K]\), exists if and only if (5.4) holds. In that case, it is given by

$$\begin{aligned} \pi (0,j)= & {} \alpha _j \left( 1- \frac{\lambda q_x}{\lambda q_x+\mu (1-\eta )} \rho ^{j}\right) ,\; 0\le j\le K-1, \end{aligned}$$
(8.1)
$$\begin{aligned} \pi (m,j)= & {} \alpha _j \frac{\lambda q_x}{\lambda q_x+\mu (1-\eta )} \rho ^{j} (1-\eta ) \eta ^{m-1},\; m\ge 1, 0\le j\le K-1, \end{aligned}$$
(8.2)

where \(\eta =\eta (q_x)\) is the unique solution in (0, 1) of equation (5.6), \((\alpha _j: 0\le j\le K-1)\) is the corresponding marginal stationary distribution of \(\{J(t)\}\) given by (5.2)–(5.3) and \(\rho =\frac{\lambda }{\lambda +\mu (1-\eta )}\).

Proof

Under the x-MRT strategy, the process \(\left\{ (M(t),J(t))\right\} \) is a CTMC with transition rate diagram shown in Fig. 1 with \(q_0=q_x\) and \(q_j=1\) for \(j=1,2,\ldots , K-1\). Arranging the states lexicographically and grouping them according to levels, we have that the transition rate matrix Q of \(\{(M(t),J(t))\}\) assumes the block-partitioned form

$$\begin{aligned} Q= \left( \begin{array}{cccccc} B &{}\quad A_0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots \\ A_2 &{}\quad A_1 &{}\quad A_0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots \\ 0 &{}\quad A_2 &{}\quad A_1 &{}\quad A_0 &{}\quad 0 &{}\quad \cdots \\ 0 &{}\quad 0 &{}\quad A_2 &{}\quad A_1 &{}\quad A_0 &{}\quad \cdots \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad A_2 &{}\quad A_1 &{}\quad \cdots \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots \\ \end{array} \right) . \end{aligned}$$

Let \(A=A_0+A_1+A_2\) be the transition matrix of the phase process and \({\alpha }=(\alpha _0,\alpha _1,\ldots ,\alpha _{K-1})^T\) be the column vector of its stationary distribution, i.e., \({\alpha }^T A={\mathbf {0}}^T\), \({\alpha }^T{\mathbf {e}}=1\), with \({\mathbf {0}}\) and \({\mathbf {e}}\) being the K-dimensional vectors of zeros and ones, respectively. Then, the positive recurrence (stability) condition for the QBD is \({\alpha }^T A_0 {\mathbf {e}}<{\alpha }^T A_2 {\mathbf {e}}\) (see Latouche and Ramaswami [36], Theorem 7.2.4, page 158). Due to the special structure of the submatrices \(A_0\) and \(A_2\), the stability condition assumes the form \(\lambda \alpha _{K-1}<\mu \sum _{i=0}^{K-1}\alpha _i\), which implies the condition (5.4).

We now proceed to the computation of the stationary distribution of \(\left\{ \left( M(t),J(t)\right) \right\} \). We first interpret the problem in discrete time by applying the uniformization technique. We choose the uniformization rate \(\Lambda =\lambda +\mu \), and let \(P=I+\frac{1}{\Lambda }Q\) be the transition probability matrix of the corresponding discrete-time Markov chain which has the same stationary distribution \({\pi }=(\pi (m,j;x):m\ge 0, 0\le j\le K-1)\) with \(\left\{ \left( M(t),J(t)\right) \right\} \). We set \({\pi }_m=(\pi (m,0),\pi (m,1),\ldots ,\pi (m,K-1))^T\), \(m\ge 0\). Then, P corresponds to a discrete-time QBD with block-partitioned form P identical to the matrix Q and we mark the corresponding blocks by a tilde, i.e., \({\tilde{B}}= I+\Lambda ^{-1}B\), \({\tilde{A}}_0=\Lambda ^{-1}A_{0}\), \({\tilde{A}}_1=I+\Lambda ^{-1}A_{1}\), and \({\tilde{A}}_2=\Lambda ^{-1}A_{2}\). Following Latouche and Ramaswami [36], Theorem 6.2.1, page 131, we have that the subvectors \({\pi }_m\) of the stationary distribution have a matrix-geometric form: \({\pi }_m^T={\pi }_0^T R^m\), \(m\ge 0\), where R is the minimal nonnegative solution of the matrix equation \(R={\tilde{A}}_{0}+R{\tilde{A}}_{1}+R^2 {\tilde{A}}_{2}\) (see Latouche and Ramaswami [36], Theorem 8.1.4, page 172) and \({\pi }_0\) is the unique solution of the system \({\pi }_0^T\left( {\tilde{B}}+R{\tilde{A}}_{2}\right) ={\pi }_0^T\) and \({\pi }_0^T \left( I-R\right) ^{-1}{\mathbf {e}}=1\) (see Latouche and Ramaswami [36], Lemma 6.3.1, page 139).

However, in the present case, a more explicit form for the stationary distribution is possible, because of the special form of \({\tilde{B}}\). Indeed, we observe that \({\tilde{B}}={\tilde{A}}_{1}+{\tilde{A}}_{2}\), so Latouche and Ramaswami [36], Lemma 6.3.2, page 140 is applicable. We conclude that

$$\begin{aligned} {\pi }_m^T={\alpha }^T\left( I-R\right) R^m, m\ge 0, \end{aligned}$$
(8.3)

where \({\alpha }\) is the unique solution of \({\alpha }^T={\alpha }^T {\tilde{A}}\), \({\alpha }^T {\mathbf {e}}=1\), with \({\tilde{A}}=\tilde{A_0}+\tilde{A_1}+\tilde{A_2}\), which coincides with the marginal stationary distribution of \(\{J(t)\}\) with coordinates \(\alpha _j\) given by (5.2) and (5.3).

Note, moreover, that the matrix \({\tilde{A}}_0\) has rank 1 and in particular \({\tilde{A}}_0={\mathbf {c}} {\mathbf {r}}^T\), with \({\mathbf {c}}=(0,0,\ldots ,0,\frac{\lambda }{\lambda +\mu })^T\) and \({\mathbf {r}}=(1,0,0,\ldots ,0)^T\). Therefore, results of Ramaswami and Latouche [41] are applicable (see also Latouche and Ramaswami [36], Theorem 8.5.2, page 198) and we have that

$$\begin{aligned} R={\mathbf {c}}{\xi }^T, \end{aligned}$$
(8.4)

where

$$\begin{aligned} {\xi }^T={\mathbf {r}}^T\left( I-{\tilde{A}}_1-\eta {\tilde{A}}_2\right) ^{-1} \text{ and } \eta ={\xi }^T{\mathbf {c}}. \end{aligned}$$
(8.5)

In particular, \(\eta \) is the spectral radius of R. We now proceed to the computation of \({\xi }=(\xi _0,\xi _1,\ldots ,\xi _{K-1})^T\). Equation (8.5) yields \({\xi }^T \left( I-{\tilde{A}}_1-\eta {\tilde{A}}_2\right) ={\mathbf {r}}^T\). This system is bidiagonal, and therefore the components \(\xi _j\) of \({\xi }\) can be computed recursively. We then obtain

$$\begin{aligned} {\xi }=\frac{\lambda +\mu }{\lambda q_x+\mu (1-\eta )}(1,q_x \rho ,q_x \rho ^2, q_x \rho ^3,\ldots ,q_x \rho ^{K-1})^T, \end{aligned}$$
(8.6)

with \(\rho =\frac{\lambda }{\lambda +\mu (1-\eta )}\). For \(m\ge 1\), iteration of (8.4), using the definition of \(\eta \) in (8.5), yields \(R^m={\mathbf {c}}\left( {\xi }^T{\mathbf {c}}\right) ^{m-1} {\xi }^T=\eta ^{m-1} {\mathbf {c}} {\xi }^T\), so

$$\begin{aligned} R^m= & {} \frac{\lambda \eta ^{m-1}}{\lambda q_x+\mu (1-\eta )} \left( \begin{array}{cccccc} 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad &{}\quad \vdots \\ 0 &{}\quad 0&{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \\ 1 &{}\quad q_x \rho &{}\quad q_x \rho ^2 &{}\quad q_x \rho ^3 &{}\quad \cdots &{}\quad q_x \rho ^{K-1} \\ \end{array} \right) , m\ge 1. \end{aligned}$$

Having obtained \({\alpha }\) and \(R^m\) in closed form, we can now compute \({\pi }_m\), for \(m\ge 0\), using (8.3), and we deduce (8.1)–(8.2). It remains to compute \(\eta \) to complete the proof. Recall that the matrix \({\tilde{A}}_0\) has rank 1, \({\tilde{A}}_0={\mathbf {c}}{\mathbf {r}}^T\), and therefore Latouche and Ramaswami [36], Theorem 8.5.2, page 198 is applicable. Then, the quantity \(\eta \) and the spectral radius of R (also known as the caudal characteristic) are characterized as the unique root \(z\in (0,1)\) of the equation \(z={\mathbf {r}}^T\left( I-{\tilde{A}}_1-z {\tilde{A}}_2\right) ^{-1}{\mathbf {c}}\), which is seen to be equivalent to (5.6) after a bit of algebra. \(\square \)

We can now use Proposition 8.1 to compute S(jx), by conditioning on the number of batches, m, at the service node for a tagged customer who arrives and sees j customers in the assembly station. We have that

$$\begin{aligned} S(j;x)=\sum _{m=0}^{\infty } S(m,j) \pi _{M|J}(m|j), \end{aligned}$$
(8.7)

where S(mj) is the conditional mean sojourn time of a customer that arrives when the system is in state (mj) and decides to join, and \(\pi _{M|J}(m|j)\) is the conditional probability of m batches at the service node given j customers at the assembly node, when the customers follow the x-MRT strategy (the Poisson Arrivals See Time Averages (PASTA) property has been used). Therefore, we have that \(\pi _{M|J}(m|j)=\frac{\pi (m,j)}{\alpha _j}\). Using (8.7), we can now proceed to the computation of \(S(K-1;x)\). Note that \(S(m,K-1)=\frac{1+m}{\mu }\), because a joining customer that sees \(K-1\) customers in the assembly node will complete the consolidation process taking place there and so the corresponding batch will move immediately to the service node, where it will occupy the \((1+m)\)th position. Using (8.1)–(8.2), for \(j=K-1\), yields

$$\begin{aligned} S(K-1;x)= & {} S(0,K-1)\pi (0|K-1)+\sum _{m=1}^{\infty } S(m,K-1)\pi (m|K-1)\\= & {} \frac{1}{\mu }\left( 1-\frac{\lambda q_x}{\lambda q_x+\mu (1-\eta )}\rho ^{K-1}\right) \\&+\sum _{m=1}^{\infty } \left( \frac{1}{\mu }+\frac{m}{\mu }\right) \frac{\lambda q_x}{\lambda q_x+\mu (1-\eta )}\rho ^{K-1}(1-\eta )\eta ^{m-1}. \end{aligned}$$

Summing the geometric series yields

$$\begin{aligned} S(K-1;x)= & {} \frac{1}{\mu }+\frac{\lambda (\lambda +\mu )\lambda q\lambda ^{K-2}}{\mu (\lambda +\mu )(1-\eta )\left( \lambda q+\left( 1-\eta \right) \mu \right) \left( \lambda +\left( 1-\eta \right) \mu \right) ^{K-1}}\nonumber \\= & {} \frac{1}{\mu }+\frac{\lambda ^K q}{\left( \lambda q+\left( 1-\eta \right) \mu \right) \left( \lambda +\left( 1-\eta \right) \mu \right) ^{K-1}}\cdot \frac{1}{\mu (1-\eta )}. \end{aligned}$$
(8.8)

But, because \(\eta \) is a root of (5.6), (8.8) reduces to \(S(K-1;x)=\frac{1}{\mu }+\frac{\eta }{\mu (1-\eta )}=\frac{1}{\mu (1-\eta )}\). This concludes the alternative derivation of Proposition 5.2.

Appendix II: Continuity of the equilibrium social benefit as function of the arrival rate

The equilibrium social benefit for the observable model as function of \(\lambda \) is continuous. Indeed, for \(\lambda <\lambda ^*\) it is identically 0, while for \(\lambda >\lambda ^*\) it is composition of two continuous functions (see the bottom branch of (6.6) and formulas (6.1)–(6.2)). To show the continuity at \(\lambda =\lambda ^*\), it suffices to show that the equilibrium joining probability \(q_x\) tends to 0 as \(\lambda \) tends to \(\lambda ^*\) from the right. For \(\epsilon \in (0,1)\), let

$$\begin{aligned} \lambda (\epsilon )=\frac{K-1}{\frac{R}{C}-\frac{1}{\mu (1-\epsilon )}}. \end{aligned}$$

Then, \(\lambda (\epsilon )\rightarrow \lambda ^*\) as \(\epsilon \rightarrow 0\). Note, now, that for \(\epsilon \) small enough we have that Case IV of Theorem 6.1 occurs. Moreover, in this case we have that \(\eta (\epsilon )\) given by (6.2) for \(\lambda =\lambda (\epsilon )\) yields \(\eta (\epsilon )=\epsilon \), so (6.1) reduces to

$$\begin{aligned} q_x(\epsilon )=\frac{\epsilon (1-\epsilon ) \mu [\lambda (\epsilon )+\mu (1-\epsilon )]^{K-1}}{\lambda (\epsilon )^K-\lambda (\epsilon ) \epsilon [\lambda (\epsilon )+\mu (1-\epsilon )]^{K-1}}, \end{aligned}$$

which obviously tends to 0 as \(\epsilon \rightarrow 0\).

Appendix III: Limiting equilibrium throughput when the arrival rate tends to infinity

As \(\lambda \rightarrow \infty \), the various cases of Theorem 6.1 collapse into just two cases: \(\frac{R}{C}\in [0,\frac{1}{\mu })\) and \(\frac{R}{C}\in [\frac{1}{\mu },\infty )\).

For \(\frac{R}{C}\in [0,\frac{1}{\mu })\), there exists a unique SPE strategy which is the 0-MRT strategy. In this case, the equilibrium throughput is 0.

For \(\frac{R}{C}\in [\frac{1}{\mu },\infty )\), Case IV of Theorem 6.1 is applicable. To emphasize the dependence of \(q_x\) given by (6.1) and \(\eta \) given by (6.2) on \(\lambda \), we denote them by \(q_x(\lambda )\) and \(\eta (\lambda )\), respectively. Then, the throughput of the system (number of served batches per time unit) is \(\lambda q_x(\lambda )\). We are interested in computing \(\lim _{\lambda \rightarrow \infty } \lambda q_x(\lambda )\).

Taking the limit as \(\lambda \rightarrow \infty \) in (6.2) immediately yields

$$\begin{aligned} \lim _{\lambda \rightarrow \infty } \eta (\lambda )=1-\frac{C}{R\mu }. \end{aligned}$$
(8.9)

Multiplying (6.1) by \(\lambda \) and dividing both the numerator and the denominator by \(\lambda ^{K-1}\) yields

$$\begin{aligned} \lambda q_x(\lambda )=\frac{\eta (\lambda ) (1-\eta (\lambda )) \mu [1+\frac{\mu }{\lambda }(1-\eta (\lambda ))]^{K-1}}{1- \eta (\lambda ) [1+\frac{\mu }{\lambda }(1-\eta (\lambda ))]^{K-1}}. \end{aligned}$$
(8.10)

Taking, now, the limit in (8.10) and using (8.9) yields

$$\begin{aligned} \lim _{\lambda \rightarrow \infty } \lambda q_x(\lambda )=\lim _{\lambda \rightarrow \infty } \frac{\eta (\lambda ) (1-\eta (\lambda )) \mu }{1- \eta (\lambda ) }= \lim _{\lambda \rightarrow \infty } \mu \eta (\lambda )=\mu -\frac{C}{R}. \end{aligned}$$
(8.11)

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Bountali, O., Economou, A. Strategic customer behavior in a two-stage batch processing system. Queueing Syst 93, 3–29 (2019). https://doi.org/10.1007/s11134-019-09615-0

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