An M/M/1 Queueing Model Subject to Differentiated Working Vacation and Customer Impatience

Abstract

This paper deals with the stationary and transient analysis of a single server queueing model subject to differentiated working vacation and customer impatience. Customers are assumed to arrive according to a Poisson process and the service times are assumed to be exponentially distributed. When the system empties, the single server takes a vacation of some random duration (Type I) and upon his return if the system is still empty, he takes another vacation of shorter duration (Type II). Both the vacation duration are assumed to follow exponential distribution. Further, the impatient behaviour of the waiting customer due to slow service during the period of vacation is also considered. Explicit expressions for the time dependent system size probabilities are obtained in terms of confluent hyper geometric series and modified Bessel’s function of first kind using Laplace transform, continued fractions and generating function methodologies. Numerical illustrations are added to depict the effect of variations in different parameter values on the time dependent probabilities.

Appendix: Derivation of \( \phi_{n} \left( t \right) \,{\text{and}} \,\psi_{n} \left( t \right) \)

The confluent hypergeometric function represented by ${}_{1}{\text{F}}_{1} \left( {{\rm{a}};{\text{c}};{\rm{z}}} \right)$ has a series representation given by

$$ {}_{1}{\text{F}}_{1} \left( {{\rm{a}};{\text{c}};{\rm{z}}} \right) = 1 + \frac{\text{a}}{\rm{c}} \frac{\text{z}}{1!} + \frac{\rm{a}}{\text{c}} \frac{{{\rm{a}}\left( {{\text{a}} + 1} \right)}}{{\left( {{\rm{c}} + 1} \right)}} \frac{{{\text{z}}^{2} }}{2!} + \ldots = 1 + \sum\limits_{k = 1}^{\infty } {\frac{{\mathop \prod \nolimits_{j = 1}^{k - 1} \left( {a + i} \right)}}{{\mathop \prod \nolimits_{i = 0}^{k - 1} \left( {c + i} \right)}} \frac{{z^{k} }}{k!}} $$

Consider the repression for $\hat{\phi }_{n} \left( s \right)$ obtained as

$$ \hat{\phi }_{n} \left( s \right) = \left( {\frac{\lambda }{\xi }} \right)^{n} \frac{1}{{\mathop \prod \nolimits_{i = 1}^{n} \left( {\frac{{s + \gamma_{1} + \mu_{1} }}{\xi } + i} \right)}}\frac{{{}_{1}{\text{F}}_{1} \left( {\frac{{\mu_{1} }}{\xi } + n + 1 ;\frac{{s + \gamma_{1} + \mu_{1} }}{\xi } + n + 1; - \frac{\lambda }{\xi }} \right)}}{{{}_{1}{\text{F}}_{1} \left( {\frac{{\mu_{1} }}{\xi } + 1 ;\frac{{s + \gamma_{1} + \mu_{1} }}{\xi } + 1; - \frac{\lambda }{\xi }} \right)}} . $$

(A.1)

Using the definition of confluent hypergeometric function, we obtain

$$ {}_{1}{\text{F}}_{1} \left( {\frac{{\mu_{1} }}{\xi } + n + 1 ;\frac{{s + \gamma_{1} + \mu_{1} }}{\xi } + n + 1; - \frac{\lambda }{\xi }} \right) = \sum\limits_{k = 0}^{\infty } {\frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + \left( {n + j} \right)\xi } \right)}}{{\mathop \prod \nolimits_{i = 1}^{n + k} \left( {s + \gamma_{1} + \mu_{1} + i\xi } \right)}} \frac{{\left( { - \lambda } \right)^{k} }}{{\xi^{k - n} k!}}} $$

And hence

$$ \frac{{{}_{1}{\text{F}}_{1} \left( {\frac{{\mu_{1} }}{\xi } + n + 1 ;\frac{{s + \gamma_{1} + \mu_{1} }}{\xi } + n + 1; - \frac{\lambda }{\xi }} \right)}}{{\mathop \prod \nolimits_{i = 1}^{n} \left( {\frac{{s + \gamma_{1} + \mu_{1} }}{\xi } + i} \right)}} = \sum\limits_{k = 0}^{\infty } {\frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + \left( {n + j} \right)\xi } \right)}}{{\mathop \prod \nolimits_{i = 1}^{n + k} \left( {s + \gamma_{1} + \mu_{1} + i\xi } \right)}} \frac{{\left( { - \lambda } \right)^{k} }}{{\xi^{k - n} k!}}} $$

Applying partial fraction in the above equation, we get

$$ \begin{aligned} \frac{{{}_{1}{\text{F}}_{1} \left( {\frac{{\mu_{1} }}{\xi } + n + 1 ;\frac{{s + \gamma_{1} + \mu_{1} }}{\xi } + n + 1; - \frac{\lambda }{\xi }} \right)}}{{\mathop \prod \nolimits_{i = 1}^{n} \left( {\frac{{s + \gamma_{1} + \mu_{1} }}{\xi } + i} \right)}} = \mathop \sum \limits_{k = 0}^{\infty } \frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + \left( {n + j} \right)\xi } \right)}}{k!} \frac{{\left( { - \lambda } \right)^{k} }}{{\xi^{2k - 1} }} \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathop \sum \limits_{i = 1}^{ n + k} \left( {\frac{{\left( { - 1} \right)^{i - 1} }}{{\left( {i - 1} \right)!\left( {n + k - i} \right)!}}} \right) \left( {\frac{1}{{s + \gamma_{1} + \mu_{1} + i\xi }}} \right) \hfill \\ \end{aligned} $$

(A.2)

Now, consider the term in the denominator of $\hat{\phi }_{n} \left( s \right)$ as

$$ \begin{aligned} {}_{1}{\text{F}}_{1} \left( {\frac{{\mu_{1} }}{\xi } + 1 ;\frac{{s + \gamma_{1} + \mu_{1} }}{\xi } + 1; - \frac{\lambda }{\xi }} \right) = \mathop \sum \limits_{k = 0}^{\infty } \frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + j\xi } \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} \left( {s + \gamma_{1} + \mu_{1} + i\xi } \right)}} \frac{{\left( { - \lambda } \right)^{k} }}{{\xi^{k} k!}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop \sum \limits_{k = 0}^{\infty } \left( { - \lambda } \right)^{k} \hat{a}_{k} \left( s \right) \hfill \\ \end{aligned} $$

Where $\hat{a}_{k} \left( s \right) = \frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + j\xi } \right)}}{{\mathop \prod \nolimits_{i = 1}^{k} \left( {s + \gamma_{1} + \mu_{1} + i\xi } \right)}}\left( {\frac{1}{{\xi^{k} k!}}} \right)\, {\text{and}} \,\hat{a}_{0} \left( s \right) = 1.$ By resolving into partial fractions, we have

$$ \hat{a}_{k} \left( s \right) = \frac{1}{{\xi^{2k - 1} k!}}\mathop \sum \limits_{r = 1}^{k} \frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + j\xi } \right)\left( { - 1} \right)^{r - 1} }}{{\left( {r - 1} \right)!\left( {k - r} \right)!}}\frac{1}{{s + \gamma_{1} + \mu_{1} + r\xi }} , \,{\text{for}} \, k = 1,2,3 \ldots $$

Using the identity is given by Gradshteyn et al. (2007), it is seen that

$$ \left[ {{}_{1}{\text{F}}_{1} \left( {\frac{{\mu_{1} }}{\xi } + 1 ;\frac{{s + \gamma_{1} + \mu_{1} }}{\xi } + 1; - \frac{\lambda }{\xi }} \right)} \right]^{ - 1} = \left[ {\mathop \sum \limits_{k = 0}^{\infty } \hat{a}_{k} \left( s \right)\left( { - \lambda } \right)^{k} } \right]^{ - 1} = \mathop \sum \limits_{k = 0}^{\infty } \hat{b}_{k} \left( s \right)\lambda^{k} $$

(A.3)

where $\hat{b}_{0} \left( s \right) = 1 \,{\text{and }}\,{\rm{for}}\, k = 1,2,3 \ldots$

$$ \begin{aligned} & \hat{b}_{k} \left( s \right) = \left| {\begin{array}{*{20}c} {\hat{a}_{1} \left( s \right)} & 1 & {} & \ldots & {} & {} \\ {\hat{a}_{2} \left( s \right)} & {\hat{a}_{1} \left( s \right)} & 1 & \ldots & {} & {} \\ { \hat{a}_{3} \left( s \right)} & {\hat{a}_{2} \left( s \right)} & { \hat{a}_{2} \left( s \right)} & \ldots & {} & {} \\ {} & \ldots & \ldots & \ldots & \ldots & {} \\ {\hat{a}_{k - 1} \left( s \right)} & { \hat{a}_{k - 2} \left( s \right)} & { \hat{a}_{k - 3} \left( s \right)} & \ldots & {\hat{a}_{1} \left( s \right)} & 1 \\ {\hat{a}_{k} \left( s \right)} & {\hat{a}_{k - 1} \left( s \right)} & {\hat{a}_{k - 2} \left( s \right)} & \ldots & {\hat{a}_{2} \left( s \right)} & {\hat{a}_{1} \left( s \right)} \\ \end{array} } \right| \\ & \quad \quad \quad = \,\mathop \sum \limits_{i = 1}^{k} \left( { - 1} \right)^{i - 1} \hat{a}_{i} \left( s \right)\hat{b}_{k - i} \left( s \right). \\ \end{aligned} $$

Substituting Eq. (A.3) and Eq. (A.2) in Eq. (A.1), we get

$$ \hat{\phi }_{n} \left( s \right) = \lambda^{n} \xi^{n} \mathop \sum \limits_{k = 0}^{\infty } \left( { - \lambda } \right)^{k} \frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + \left( {n + j} \right)\xi } \right)}}{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + j\xi } \right)}}\hat{a}_{n + k} \left( s \right)\mathop \sum \limits_{j = 1}^{\infty } \lambda^{j} \hat{b}_{j} \left( s \right) . $$

Taking inverse Laplace transform of the above equation leads to

$$ \phi_{n} \left( t \right) = \lambda^{n} \xi^{n} \mathop \sum \limits_{k = 0}^{\infty } \left( { - \lambda } \right)^{k} \frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + \left( {n + j} \right)\xi } \right)}}{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + j\xi } \right)}}a_{n + k} \left( t \right)*\mathop \sum \limits_{j = 1}^{\infty } \lambda^{j} b_{j} \left( t \right) , $$

where

$$ a_{k} \left( t \right) = \frac{1}{{\xi^{2k - 1} k!}}\mathop \sum \limits_{r = 1}^{k} \frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{1} + j\xi } \right)\left( { - 1} \right)^{r - 1} }}{{\left( {r - 1} \right)!\left( {k - r} \right)!}}e^{{ - \left( {\gamma_{1} + \mu_{1} + r\xi } \right)t}} , k = 1,2, \ldots $$

and

$$ b_{k} \left( t \right) = \mathop \sum \limits_{i = 1}^{k} \left( { - 1} \right)^{i - 1} a_{i} \left( t \right)*b_{k - i} \left( t \right), k = 2,3, \ldots , b_{1} \left( t \right) = a_{1} \left( t \right) $$

Similarly equation of $\hat{\psi }_{n} \left( s \right)$ as

$$ \hat{\psi }_{n} \left( s \right) = \lambda^{n} \xi^{n} \mathop \sum \limits_{k = 0}^{\infty } \left( { - \lambda } \right)^{k} \frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{2} + \left( {n + j} \right)\xi } \right)}}{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{2} + j\xi } \right)}}\hat{c}_{n + k} \left( s \right)\mathop \sum \limits_{j = 1}^{\infty } \lambda^{j} \hat{d}_{j} \left( s \right) . $$

Proceeding in the similar manner as that of $\hat{\phi }_{n} \left( s \right)$, it is seen that the Laplace inverse of $\hat{\psi }_{n} \left( s \right)$ is

$$ \psi_{n} \left( t \right) = \lambda^{n} \xi^{n} \mathop \sum \limits_{k = 0}^{\infty } \left( { - \lambda } \right)^{k} \frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{2} + \left( {n + j} \right)\xi } \right)}}{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{2} + j\xi } \right)}}c_{n + k} \left( t \right)*\mathop \sum \limits_{j = 1}^{j} \lambda^{j} d_{j} \left( t \right). $$

where

$$ c_{k} \left( t \right) = \frac{1}{{\xi^{2k - 1} k!}}\mathop \sum \limits_{r = 1}^{k} \frac{{\mathop \prod \nolimits_{j = 1}^{k} \left( {\mu_{2} + j\xi } \right)\left( { - 1} \right)^{k} }}{{\left( {r - 1} \right)!\left( {k - r} \right)!}}e^{{ - \left( {\gamma_{2} + \mu_{2} + r\xi } \right)t}} , k = 1,2, \ldots $$

and

$$ d_{k} \left( t \right) = \mathop \sum \limits_{i = 1}^{k} \left( { - 1} \right)^{i - 1} c_{i} \left( t \right)*d_{k - i} \left( t \right), k = 2,3, \ldots $$