Question

Marshall [44] shows that for an M/G/1 queueing system the Laplace-Stieltjes transform of the interdeparture time is given by $$ D^[\theta]=\frac{(\theta+\mu) \rho W_s^[\theta]}{\theta+\lambda} . $$ Using the above result, show that the interdeparture time distribution is exponential if and only if the service time is exponential. (Disney et al. [15] showed that the only $\mathrm{M} / \mathrm{G} / 1$ queueing system having independent, identically distributed, interdeparture times is the $M / M / 1$ system. Such a stream is called a renewal process. Laslett [39] showed that the only GI/M/1 queueing system having renewal output was the $\mathrm{M} / \mathrm{M} / 1$ system.)

   Marshall [44] shows that for an M/G/1 queueing system the Laplace-Stieltjes transform of the interdeparture time is given by
$$
D^*[\theta]=\frac{(\theta+\mu) \rho W_s^*[\theta]}{\theta+\lambda} .
$$
Using the above result, show that the interdeparture time distribution is exponential if and only if the service time is exponential. (Disney et al. [15] showed that the only $\mathrm{M} / \mathrm{G} / 1$ queueing system having independent, identically distributed, interdeparture times is the $M / M / 1$ system. Such a stream is called a renewal process. Laslett [39] showed that the only GI/M/1 queueing system having renewal output was the $\mathrm{M} / \mathrm{M} / 1$ system.)

Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Arnold O. Allen 2nd Edition

Chapter 5, Problem 58

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The Laplace-Stieltjes transform of the interdeparture time for an M/G/1 queue is given by: \[ D^*[\theta] = \frac{(\theta + \mu) \rho W_s^*[\theta]}{\theta + \lambda} \] where: - $ \theta $ is the Laplace transform variable, - $ \mu $ is the service rate, - \( Show more…

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Marshall [44] shows that for an M/G/1 queueing system the Laplace-Stieltjes transform of the interdeparture time is given by $$ D^*[\theta]=\frac{(\theta+\mu) \rho W_s^*[\theta]}{\theta+\lambda} . $$ Using the above result, show that the interdeparture time distribution is exponential if and only if the service time is exponential. (Disney et al. [15] showed that the only $\mathrm{M} / \mathrm{G} / 1$ queueing system having independent, identically distributed, interdeparture times is the $M / M / 1$ system. Such a stream is called a renewal process. Laslett [39] showed that the only GI/M/1 queueing system having renewal output was the $\mathrm{M} / \mathrm{M} / 1$ system.)

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