Strategic joining rules in a single server Markovian queue with Bernoulli vacation

Abstract

The strategic joining behavior of customers in a single-server Markovian queueing system with Bernoulli vacation is studied. It is assumed that the server begins a vacation period if the queue is empty upon completion of a service, and if the queue is not empty, the server will take a Bernoulli type vacation. Assuming that arriving customers can observe various levels of the system information, we study strategic customers’ decision on whether to join or balk the queue based on a linear reward-cost structure. The Nash equilibrium strategies in the fully observable case and the unobservable cases are investigated. The effect of the information level as well as several parameters on the equilibrium behavior is illustrated via numerical examples.

Appendix

1. The functions S(0, q(0), q(1)) and S(1, q(0), q(1)) are both strictly decreasing for $q(1)\in [0,1]$.

Proof

Denote the expected mean sojourn time of an arriving customer who decides to enter at state i by $E[W_i]$. Then we have

$$E[W_0]=\lambda _0\frac{({\lambda _1-\mu })^{2}+\lambda _1 \mu p}{({\mu -\lambda _1})({\mu \theta -\theta \lambda _1-\lambda _0 \mu p})}\left( {\frac{1}{\mu }+\frac{p}{\theta }}\right) +\frac{1}{\mu }+\frac{1}{\theta },$$

(59)

$$E[W_1]=\frac{\lambda _0({\mu -\mu p-\lambda _1})+\mu \theta }{\mu \theta -\theta \lambda _1-\lambda _0 \mu p}\left( {\frac{1}{\mu }+\frac{p}{\theta }}\right) +\frac{1}{\mu }.$$

(60)

It is readily to verify that $E[W_0]$ and $E[W_1]$ are increasing for $\lambda _0 \in [0,1]$, respectively. Next we will prove that $E[W_0]$ and $E[W_1]$ are all increasing for $\lambda _1 \in [0,1]$.

Firstly, we prove that $E[W_0]$ is increasing for $\lambda _1 \in [0,1]$.

Letting

$$\begin{aligned} g(x)= & \frac{({\mu -x})^{2}+\mu p x}{({\mu -x})({\mu \theta -\theta x-\lambda _0 \mu p})} \\= & \frac{\mu -x}{\mu \theta -\theta x-\lambda _0 \mu p}+\frac{\mu p x}{({\mu -x})({\mu \theta -\theta x-\lambda _0 \mu p})}, \end{aligned}$$

(61)

where $x \in [0,1]$.

From the above equation, we can observe that the first part is increasing for $x \in [0,1]$, then we only need to give the evidence of the latter part.

We let

$$g_1(x)=\frac{1}{\mu -x},$$

(62)

$$g_2(x)=\frac{\mu p x}{({\mu \theta -\theta x-\lambda _0 \mu p})}.$$

(63)

Obviously, Eq.  (62) is strictly increasing for $x \in [0,1]$. Now differentiating (63) with respect to x, we obtain

$$g_2^{'}(x)=\mu ^{2} p \frac{\theta -\lambda _0 p}{({\mu \theta -\theta x-\lambda _0 \mu p})^{2}}.$$

(64)

By using the condition $\mu \theta >\lambda (\mu p+\theta )$, we get

$$\theta -\lambda _0 p > \theta -\frac{\mu p \theta }{\mu p+\theta }= \frac{\theta ^{2}}{\mu p+\theta }> 0.$$

That is, we get $g_2^{'}(x)>0$ for $x \in [0,1]$, which follows that Eq.  (63) is strictly increasing for $x \in [0,1]$. From the above discussion, we can get that g(x) is strictly increasing for $x \in [0,1]$, i.e., $E[W_0]$ is increasing for $\lambda _1 \in [0,1]$.

Similarly, we can prove that $E[W_1]$ is increasing for $\lambda _1 \in [0,1]$. $\square$

2. In the fully unobservable M/M/1 queue with Bernoulli vacation, the system is stable if and only if $\mu \theta >\lambda q(\mu p+ \theta ).$

Proof

Firstly, we prove the necessity of the condition, i.e. if the system is stable, then $\mu \theta >\lambda q(\mu p+ \theta ).$

As we have already obtained the probability generating functions $P_0(z)$ and $P_1(z)$ in Proposition 4.1, plugging $z=1$ in Eqs. (21) and (22), we can get

$$\begin{aligned}P_0(1)= & \frac{(\mu -\lambda _1)[\theta +(1-p)\lambda _0]}{\mu \theta -\lambda _1\theta -\lambda _0\mu p}p(0,0), \\ P_1(1)= & \frac{\lambda _0[\theta +(1-p)\lambda _0]}{\mu \theta -\lambda _1\theta -\lambda _0\mu p}p(0,0).\end{aligned}$$

If the system is stable, then the following conditions are established:

$$\begin{aligned}P_0(1)> & 0, \\ P_1(1)> & 0,\end{aligned}$$

where $\lambda _0=\lambda _1=\lambda$.

Through combining the expression of p(0, 0) [i.e. Eq.  (23)] which has been obtained in the proof of Proposition 4.1, we can get the condition $\mu \theta >\lambda q(\mu p+ \theta )$.

In other words, the interpretation of this stability condition could be referred to Theorem 3.1. of Wang et al. (2001), i.e., “the generalized service time of a single customer (i.e., $\frac{\theta +\mu p}{\mu \theta }$) is less than the interarrival time between two customers (i.e., $\frac{1}{\lambda q}$)”.

Next, we prove that the inequality $\mu \theta >\lambda q(\mu p+ \theta )$ is also a sufficient condition for the system to be stable. We use the method in Theorem 3.1. of Wang et al. (2001). Because of the Markovian property of the system in our paper, here $\Psi _k=E(Q_{n+1}-Q_n|Q_n=k)$ equals the mean number of customers arriving during the generalized service time minuses 1, i.e.,

$$\Psi _k=\lambda q \frac{\theta +\mu p}{\mu \theta }-1.$$

Obviously, if $\mu \theta >\lambda q(\mu p+ \theta )$, we can obtain that the system is stable. $\square$