Section 11.3

M/G/1

[4/100%]
Show
LaTeX
Problem 11.3.1

Consider M(λ)/D(d)/1\mathrm{M}(\lambda ) / \mathrm{D}(d) / 1 where ρ=λd<1\rho = \lambda d < 1. Show that the mean queue length at moments of departure in equilibrium is 12ρ(2−ρ)/(1−ρ)\frac{1}{2} \rho (2-\rho ) /(1-\rho ).

?
Problem 11.3.2

Consider M(λ)/M(μ)/1\mathrm{M}(\lambda ) / \mathrm{M}(\mu ) / 1, and show that the moment generating function of a typical busy period is given by

MB(s)=(λ+μ−s)−(λ+μ−s)2−4λμ2λ M_{B}(s) = \frac{(\lambda +\mu -s)-\sqrt{(\lambda +\mu -s)^{2}-4 \lambda \mu }}{2 \lambda }

for all sufficiently small but positive values of ss.

?
Problem 11.3.3
?
(a)

Show that, for a M/G/1M / G / 1 queue, the sequence of times at which the server passes from being busy to being free constitutes a renewal process.

(b)

When the above queue is in equilibrium, what is the moment generating function of the total time that an arriving customer spends in the queue including service?

Problem 11.3.4

Consider the M(λ)/G/1M(\lambda ) / G / 1 queue with no waiting room. Customers who arrive while the server is busy are lost. Show that the long run proportion of arrivals lost is 1/(1+ρ−1)1 /\left(1+\rho^{-1}\right) where ρ=λE[S]\rho = \lambda \mathbb {E}\left[S\right] is the traffic intensity.

?