MAST30001 Stochastic Modelling
Tutorial Sheet 9
1. Show that in in M/M/1 queue with arrival rate λ and service rate µ > λ, the
expected lengths of the idle and busy periods are 1/λ and 1/(µ − λ), respectively.
[Hint: the proportion of time the server is idle is equal to the stationary chance the
system is empty.]
2. A rental car washing facility can wash one car at a time. Cars arrive to be washed
according to a Poisson process with rate 3 per day and the service time to wash a car
is exponential with mean 7/24 days. It costs the company $150 per day to operate
the facility and the company loses $10 per day for each car tied up in the washing
facility. The company can upgrade the facility to get down to a mean service time
of 1/4 days at the cost of $C per day. What’s the largest C can be for this upgrade
to make economic sense?
3. (M/G/∞ queue) In a certain communications system, information packets arrive
according to a Poisson process with rate λ per second and each packet is processed in
one second with probability p and in two seconds with probability 1−p, independent
of the arrival times and other service times. Let Nt be the number of packets that
have entered the system up to time t and Xt be the number of packets in the system
(including those being served) at time t.
(a) Is (Xt)t≥0 a Markov chain? (No detailed argument is necessary here, just think
about it heuristically.)
(b) If X0 = 0, what is the distribution of X2?
(c) IfX0 = 0, is there a “stationary” limiting distribution πk = limt→∞ P (Xt = k)?
If so, what is it?
(d) If X0 = N0 = 0, what is the joint distribution of Xt and Nt?