Question

Problem 2. Jobs sent to a printer are held in a buffer until they can be printed. Jobs are printed sequentially on a first-come, first-served basis. Jobs arrive at the printer at the rate of four per minute. The average time to print a job is 10 seconds. Assume M/M/1 queuing and steady state. (a) Find λA and λS (b) Find the expected value and standard deviation of the number of jobs in this system at any time. (c) On average, how long does a job spend in the system? (d) If a person submits a job to be printed, what is the probability it will begin printing immediately?

Answer 1

Given,

No of servers (s) = 1

This is a M|M|1 problem

a) Arrival rate (l) = 4 jobs/minute

Service rate (m)= 1 job in 10 secs = 6 jobs in 60 secs = 6 jobs/minute

b) Average no of jobs in system L=l/(m-l) = 4/(6-4)= 4/2 = 2 jobs

Average jobs waiting in line Lq = (l*L)/m = (4*2)/6 = 1.33 jobs

c) Average time spent in system W= 1/(m-l) = 1/(6-4) = 1/2 = 0.5 min = 30 secs

Average time spent in line Wq= (l*W)/m = (4*0.5)/6 = 0.333 min = 20 secs

d) Probability that job will begin printing immediately = Probability that 0 customers are in the system Po = 1- (l/m) = 1-(4/6) =1 – 0.6667 = 0.3333 = 33.33%