Q1) A document printing machine receives documents according to a Poisson process with an expected interarrival time of 5 minutes from the legal department. When the machine has just one document to print, the expected processing time is 5 minutes. When machine has more than one document, then it does not require initial set-up time and expected processing time for each document to 4 minutes. In both cases, the processing times have an exponential distribution.

(a) Construct the rate diagram for this queueing system.

(b) Check that this is a birth-and-death process and find the steady-state distribution of the number of documents in the system but not yet finished printing. (Hint: Use the sum of Geometric series)

(c) Derive L and Lq for this system. (Hint: Use the sum of Arithmetic-Geometric series)

(d) Use the information of L and Lq to determine W and Wq.

Q2) Jacob runs a shoe repair store by himself. Customers arrive to bring a pair of shoes to be repaired according to a Poisson process at a mean rate of 1 per hour. The time Antonio requires to repair each individual shoe has an exponential distribution with a mean of 15 minutes.

Consider the formulation of this queuing system where the individual shoes (not pairs of shoes) are considered to be customers.

(a) For this formulation, construct the rate diagram.

(b) Explain whether this formulation satisfies the assumption of birth-and-death process.