Two-way communication retrial queues with multiple types of outgoing calls

Abstract

In this paper, we start with single server Markovian retrial queues with multiple types of outgoing calls. Incoming calls arrive at system according to a Poisson process. Service times of incoming calls follow the exponential distribution. Incoming calls that find the server busy upon arrival join an orbit and retry after some exponentially distributed time. On the other hand, the server makes an outgoing call after some exponentially distributed idle time. We assume that there are multiple types of outgoing calls whose durations follow distinct exponential distributions. For this model, we obtain explicit expressions for the joint stationary distribution of the number of calls in the orbit and the state of the server via the generating function approach. We also obtain simple asymptotic and recursive formulae for the joint stationary distribution. We show a stochastic decomposition property where we prove that the number of incoming calls in the system (server and orbit) can be decomposed into the sum of three independent random variables which have a clear physical meaning. We then consider the multiserver model for which we obtain the stability condition and derive some exact formulae by mean value analysis. Finally, we extend the single server model to the case where service time distribution of incoming calls and that of each type of outgoing calls are arbitrary. For this case, we obtain explicit expressions for the partial generating functions and recursive formulae for the joint stationary distribution of the server’s state and the number of calls in the orbit.

Appendix: Special case

We consider only the case $\nu _1 = \lambda + \nu _3$ but the case $\nu _1 = \lambda + \nu _2$ is the same. It should be noted that (15) and (24)–(27) are also established for these special cases.

Theorem 16

For $|z| \le 1$,

$$\begin{aligned} \Pi _0(z)&= \frac{1 - \rho }{1 + \sigma _2 + \sigma _3} \exp \left[ - \frac{\alpha _2 \rho \hat{\rho }}{\mu } \right] \nonumber \\&\times \left( \frac{1 - \rho }{1 - \rho z} \right) ^{\frac{D_1}{\mu }} \exp \left[ \frac{\alpha _2 \rho ^2 z}{\mu (1 - \rho z)} \right] \left( \frac{1 - \theta _3}{1 - \theta _3 z} \right) ^{\frac{D_3}{\mu }} \nonumber \\&= \pi _{0,0} (1 - \rho z)^{- \frac{D_1}{\mu }} \exp \left[ \frac{\alpha _2 \rho ^2 z}{\mu (1 - \rho z)} \right] (1 - \theta _3 z)^{- \frac{D_3}{\mu }}, \qquad \\ \Pi _1(z)&= \left( \frac{\lambda + C_{13}}{\nu _1 - \lambda z} + \frac{\lambda \alpha _2}{(\nu _1 - \lambda z)^2} + \frac{C_3}{\lambda + \nu _3 - \lambda z} \right) \Pi _0(z), \nonumber \\ \Pi _2(z)&= \frac{\alpha _2}{\nu _1 - \lambda z} \Pi _0(z), \nonumber \\ \Pi _3(z)&= \frac{\alpha _3}{\lambda + \nu _3 - \lambda z} \Pi _0(z), \nonumber \end{aligned}$$

(48)

where

$$\begin{aligned} C_{13}&= - \frac{\lambda \alpha _3}{\nu _1 - (\lambda + \nu _3 )},~~~~C_3 = \frac{\lambda \alpha _3}{\nu _1 - (\lambda + \nu _3 )}, \\ D_1&= \lambda + \alpha _2 - \frac{\lambda \alpha _3}{\nu _1 - (\lambda + \nu _3 )},~~~~D_3 = \frac{\alpha _3 (\nu _1 - \nu _3 )}{\nu _1 - (\lambda + \nu _3 )}, \\ \rho&= \frac{\lambda }{\nu _1},~~~~\hat{\rho } = \frac{\rho }{1 - \rho },~~~~\theta _3 = \frac{\lambda }{\lambda + \nu _3}, \\ \sigma _2&= \frac{\alpha _2}{\nu _2},~~~~\sigma _3 = \frac{\alpha _3}{\nu _3}, \\ \pi _{0,0}&= \frac{1 - \rho }{1 + \sigma _2 + \sigma _3} (1 - \rho )^{\frac{D_1}{\mu }} \exp \left[ - \frac{\alpha _2 \rho \hat{\rho }}{\mu } \right] (1 - \theta _3 )^{\frac{D_3}{\mu }}. \end{aligned}$$

Proof

This theorem is obtained after a minor modification of the derivation in Section 3.1. $\square$

Theorem 17

For $j \in {\mathbb Z}_+$,

$$\begin{aligned} \pi _{0,j}&= \sum _{k=0}^{j-1} \sum _{\ell =0}^{j-k-1} \bigg \{ \left( \frac{D_1}{\mu } \right) _k \frac{\rho ^k}{k!} \left( \frac{D_3}{\mu } \right) _\ell \frac{\theta _3^\ell }{\ell !} \nonumber \\&\times \left( \sum _{m=1}^{j-k-\ell } {}_{j-k-\ell -1} \mathrm{C}_{m-1} \frac{1}{m!} \left( \frac{\alpha _2 \rho }{\mu } \right) ^m \right) \rho ^{j-k-\ell } \bigg \} \nonumber \\&+ \sum _{k=0}^j \left( \frac{D_1}{\mu } \right) _k \frac{\rho ^k}{k!} \left( \frac{D_3}{\mu } \right) _{j-k} \frac{\theta _3^{j-k}}{(j-k)!}, \\ \pi _{1,j}&= \frac{1}{\lambda + \nu _1} \sum _{k=0}^j (\lambda \pi _{0,k} + (k + 1) \mu \pi _{0,k+1} ) \left( \frac{\lambda }{\lambda + \nu _1} \right) ^{j-k} \nonumber \\&= \left( \rho + \frac{C_{13}}{\nu _1} \right) \sum _{k=0}^j \pi _{0,k} \rho ^{j-k} \nonumber \\&+ \frac{\lambda \alpha _2}{\nu _1^2} \sum _{k=0}^j \pi _{0,k} (j - k + 1) \rho ^{j-k} + \frac{C_3}{\lambda + \nu _3} \sum _{k=0}^j \pi _{0,k} \theta _3^{j-k}, \qquad \nonumber \\ \pi _{2,j}&= \frac{\alpha _2}{\nu _1} \sum _{k=0}^j \pi _{0,k} \rho ^{j-k}, \nonumber \\ \pi _{3,j}&= \frac{\alpha _3}{\lambda + \nu _3} \sum _{k=0}^j \pi _{0,k} \theta _3^{j-k}. \nonumber \end{aligned}$$

(49)

Proof

We prove (49) from which the expressions for $\pi _{i,j}$ ($i=1,2,3$) are straightforward. First, we derive the Taylor expansion at $z=0$ for the exponential part in (48) as follows.

$$\begin{aligned} \exp \left[ \frac{\alpha _2 \rho ^2 z}{\mu (1 - \rho z)} \right]&= \sum _{k=0}^\infty \frac{1}{k!} \left( \frac{\alpha _2 \rho ^2 z}{\mu (1 - \rho z)} \right) ^k \\&= \sum _{k=0}^\infty \frac{1}{k!} \left( \frac{\alpha _2 \rho }{\mu } \left( \frac{1}{1 -\rho z} - 1 \right) \right) ^k \\&= \sum _{k=0}^\infty \frac{1}{k!} \left( \frac{\alpha _2 \rho }{\mu } \right) ^k \left( \sum _{j=1}^\infty (\rho z)^j \right) ^k \\&= \sum _{k=0}^\infty \frac{1}{k!} \left( \frac{\alpha _2 \rho ^2 z}{\mu } \right) ^k \left( \sum _{j=0}^\infty (\rho z)^j \right) ^k \\&= 1 + \sum _{k=1}^\infty \frac{1}{k!} \left( \frac{\alpha _2 \rho ^2 z}{\mu } \right) ^k \sum _{j=0}^\infty {}_{j+k-1} \mathrm{C}_{k-1} (\rho z)^j \\&= 1 + \sum _{k=1}^\infty \frac{1}{k!} \left( \frac{\alpha _2 \rho }{\mu } \right) ^k \sum _{j=0}^\infty {}_{j+k-1} \mathrm{C}_{k-1} (\rho z)^{j+k} \\&= 1 + \sum _{k=1}^\infty \frac{1}{k!} \left( \frac{\alpha _2 \rho }{\mu } \right) ^k \sum _{j=k}^\infty {}_{j-1} \mathrm{C}_{k-1} (\rho z)^j \\&= 1 + \sum _{j=1}^\infty \left( \sum _{k=1}^j {}_{j-1} \mathrm{C}_{k-1} \frac{1}{k!} \left( \frac{\alpha _2 \rho }{\mu } \right) ^k \rho ^j \right) z^j. \end{aligned}$$

It follows from (48) that

$$\begin{aligned} \Pi _0(z)&= \pi _{0,0} (1 - \rho z)^{- \frac{D_1}{\mu }} \exp \left[ \frac{\alpha _2 \rho ^2 z}{\mu (1 - \rho z)} \right] (1 - \theta _3 z)^{- \frac{D_3}{\mu }} \\&= \pi _{0,0} \left( \sum _{j=0}^\infty \left( \frac{D_1}{\mu } \right) _j \frac{\rho ^j}{j!} z^j \right) \left( \sum _{j=0}^\infty \left( \frac{D_3}{\mu } \right) _j \frac{\theta _3^j}{j!} z^j \right) \\&\times \left\{ 1 + \sum _{j=1}^\infty \left( \sum _{k=1}^j {}_{j-1} \mathrm{C}_{k-1} \frac{1}{k!} \left( \frac{\alpha _2 \rho }{\mu } \right) ^k \rho ^j \right) z^j \right\} , \end{aligned}$$

leading to (49). Other formulae are obtained by the same manner as is presented in Artalejo and Phung-Duc (2012). $\square$