Queues with Simultaneous Arrival of Customers and the Dependence Structure of the Waiting Times

Abstract

We consider a queueing system where some customers decide to simultaneously wait in two queues, rather than in a single queue, to receive their service. In practice, when there exists a number of queues rendering the same service, the customers may tend to simultaneously wait in more than one queue in order to receive the service sooner and thus scale down their waiting time. In this framework, the customers may abandon one of the queues when they are called to receive the service from the other. We treat this situation as customer reneging or abandonment. We study the customer’s waiting time under this model for the cases of independence and dependence of the waiting time random variables of the queues. In conducting this study, a Copula approach is applied to take into account the dependence structure of the waiting time random variables. We compare the numerical results of the dependence with that of the independence.

Appendices

Appendix I

Proof of Proposition 2:

The first term in the right-hand side of the above relation can be written as follows.

The last equation was concluded due to the point that T₁ and T₂ are assumed to be independent random variables. Thus, one can write

where $A=\frac {\theta \cdot \omega \left (-\beta ,\sqrt {\frac {\mu }{\theta }}\right )h\left (\beta \sqrt {\frac {\mu }{\theta }}\right )}{\phi \left (\beta \sqrt {\frac {\mu }{\theta }} \right )}$ and $B=\frac {\sqrt {N\mu \theta }}{\phi \left (\beta \sqrt {\frac {\mu }{\theta }}\right )}\omega \left (-\beta ,\sqrt {\frac {\mu }{\theta }}\right )h\left (\beta \sqrt {\frac {\mu }{\theta }}\right )$.

Likewise, the second term in the right-hand side of the above relation can be expanded.

Appendix II

Let the second term in the right-hand side of Eq. 7 be denoted by J, and let g(t₂) be the integral inside this term. Then, we can write

$$\begin{array}{@{}rcl@{}} J\!&=&\!B \cdot E \left[ g(T_{2}) \right]=B {\int}_{0}^{\infty} g(t_{2})\cdot f(t_{2})dt_{2}\\ &=&\! B \!{\int}_{0}^{\infty} \! g(t_{2}) \frac{\theta^{\prime}\left[ \!1\,-\,{\Phi}\!\left( \! \beta^{\prime}\sqrt{\frac{\mu^{\prime}}{\theta^{\prime}}} \,+\,\sqrt{N\mu^{\prime}\theta^{\prime}}\cdot t_{2} \!\right)\!\right] \,+\, \sqrt{N\mu^{\prime}\theta^{\prime}}\!\cdot\! \phi \! \left( \! \beta^{\prime}\sqrt{\frac{\mu^{\prime}}{\theta^{\prime}}} \,+\, \sqrt{N\mu^{\prime}\theta^{\prime}}\cdot t_{2} \right) }{\phi \!\left( \! \beta^{\prime}\sqrt{\frac{\mu^{\prime}}{\theta^{\prime}}} \right)}\\ && \cdot \omega\left( -\beta^{\prime},\sqrt{\frac{\mu^{\prime}}{\theta^{\prime}}} \right)\cdot h\left( \beta^{\prime}\sqrt{\frac{\mu^{\prime}}{\theta^{\prime}}} \right)\cdot e^{-\theta^{\prime}t_{2}}dt_{2}\\ \end{array} $$

$$\begin{array}{@{}rcl@{}} &&{\kern-9.6pc} = \frac{B\cdot \omega\left( -\beta^{\prime},\sqrt{\frac{\mu^{\prime}}{\theta^{\prime}}} \right)\cdot h \left( \beta^{\prime}\sqrt{\frac{\mu^{\prime}}{\theta^{\prime}}} \right)}{\theta^{\prime}\phi\left( \beta^{\prime}\sqrt{\frac{\mu^{\prime}}{\theta^{\prime}}} \right)}\\ &&{\kern-9.6pc}\quad \cdot E_{\xi xp}\left[ g\left( T_{2}\right)\left( \theta^{\prime}\left[ 1-{\Phi} \left( \beta^{\prime}\sqrt{\frac{\mu^{\prime}}{\theta^{\prime}}} +\sqrt{N\mu^{\prime}\theta^{\prime}}\cdot T_{2} \right) \right] \right.\right.\\ &&{\kern-9.6pc}\quad + \left.\left. \sqrt{N\mu^{\prime}\theta^{\prime}}\cdot \phi\left( \beta^{\prime}\sqrt{\frac{\mu^{\prime}}{\theta^{\prime}}} +\sqrt{N\mu^{\prime}\theta^{\prime}}\cdot T_{2} \right) \right) \right] \end{array} $$

where E_ξxp(⋅) denotes an expectation to be computed under exponential distribution with parameter 𝜃^′. Let the expression inside this expectation which has been multiplied by g(T₂) be denoted by R(T₂).

Now, g(t₂) can be written as follows:

$$\begin{array}{@{}rcl@{}} g(t_{2})&=&{\int}_{0}^{t_{2}}t_{1}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}\left( \beta\sqrt{\frac{\mu}{\theta}} + \sqrt{N\mu\theta}\cdot t_{1} \right)^{2}e^{-\theta t_{1}}}dt_{1} \\ &=&\frac{1}{\sqrt{2\pi}}e^{-\frac{\beta^{2}\mu}{2\theta}} {\int}_{0}^{t_{2}}t_{1}\cdot e^{-\frac{N\mu\theta}{2}{t_{1}^{2}}-\left( \beta\sqrt{N}\cdot\mu+\theta \right)t_{1}} dt_{1} \end{array} $$

Then, by applying some manipulation, we arrive at

$$g(t_{2})=\frac{1}{\sqrt{2\pi}N\mu\theta}e^{-\frac{\beta^{2}\mu}{2\theta}}\left[ 1- e^{-\frac{N\mu\theta}{2}{t_{2}^{2}}-\left( \beta\sqrt{N}\cdot\mu+\theta \right)t_{2}} - \left( \beta\sqrt{N}\cdot\mu+\theta \right) \cdot g^{\prime}(t_{2})\right] $$

where $g^{\prime }(t_{2})={\int }_{0}^{t_{2}} e^{-\frac {N\mu \theta }{2}{t_{1}^{2}}-\left (\beta \sqrt {N}\cdot \mu +\theta \right )t_{1}} dt_{1}$. By considering a uniform distribution for U, i.e. U ∼ U(0,t₂), then g^′(t₂) can be written as follows:

$$g^{\prime}(t_{2})= {\int}_{0}^{t_{2}}t_{2}e^{-\frac{N\mu\theta}{2}u^{2}-\left( \beta\sqrt{N}\cdot\mu+\theta \right)u} \cdot \frac{1}{t_{2}}du=E_{U}\left[K(U) \right] $$

where $K(u)= t_{2} \cdot e^{-\frac {N\mu \theta }{2}u^{2}-\left (\beta \sqrt {N}\cdot \mu +\theta \right )u}$.

We then apply the following algorithm based on the importance sampling method to estimate J.

• i) Generate n random values from E_ξxp(𝜃^′) to be considered as values of t₂.

• ii) For each value of t₂ as obtained above, compute R(t₂).

• iii) For each value of t₂ as obtained above, compute g^′(t₂). To this end, the following steps are taken:

  • a) Generate n random numbers from U(0,t₂) to be considered as values of u.

  • b) For each value of u, compute K(u).

  • c) Calculate $\frac {1}{n}{\sum }_{i = 1}^{n}K(u_{i})$ to be utilized as an estimate for g^′(t₂).

• iv) Compute g(t₂) using g^′(t₂) obtained above.

• v) Take $\widehat {E}=\frac {1}{n}{\sum }_{i = 1}^{n}g(t_{2i})\cdot R(t_{2i})$ as an estimate for E_ξxp[g(T₂) ⋅ R(T₂)].

• vi) Finally, take $\frac {B\cdot w\left (-\beta ^{\prime },\sqrt {\frac {\mu ^{\prime }}{\theta ^{\prime }}}\right )h\left (\beta ^{\prime }\sqrt {\frac {\mu ^{\prime }}{\theta ^{\prime }}}\right )}{\theta ^{\prime }\phi \left (\beta ^{\prime }\sqrt {\frac {\mu ^{\prime }}{\theta ^{\prime }}}\right )}\cdot \widehat {E}$ as an estimate for J.