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.
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Appendix: Derivation of \( \phi_{n} \left( t \right) \,{\text{and}} \,\psi_{n} \left( t \right) \)
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
Consider the repression for \( \hat{\phi }_{n} \left( s \right) \) obtained as
Using the definition of confluent hypergeometric function, we obtain
And hence
Applying partial fraction in the above equation, we get
Now, consider the term in the denominator of \( \hat{\phi }_{n} \left( s \right) \) as
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
Using the identity is given by Gradshteyn et al. (2007), it is seen that
where \( \hat{b}_{0} \left( s \right) = 1 \,{\text{and }}\,{\rm{for}}\, k = 1,2,3 \ldots \)
Substituting Eq. (A.3) and Eq. (A.2) in Eq. (A.1), we get
Taking inverse Laplace transform of the above equation leads to
where
and
Similarly equation of \( \hat{\psi }_{n} \left( s \right) \) as
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
where
and
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Vijayashree, K.V., Ambika, K. (2020). An M/M/1 Queueing Model Subject to Differentiated Working Vacation and Customer Impatience. In: Balusamy, S., Dudin, A.N., Graña, M., Mohideen, A.K., Sreelaja, N.K., Malar, B. (eds) Computational Intelligence, Cyber Security and Computational Models. Models and Techniques for Intelligent Systems and Automation. ICC3 2019. Communications in Computer and Information Science, vol 1213. Springer, Singapore. https://doi.org/10.1007/978-981-15-9700-8_9
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