A library has only one librarian to help borrowers check out their books. Borrow

Question

A library has only one librarian to help borrowers check out their books. Borrowers arrive according to a Poisson process at a rate of 5 per hour. The service time for each customer follows an Exponential distribution with expectation 0.1 hour. If the librarian is serving someone, arriving borrowers will stay in a line.

(a) What is this queuing system?

(b) What is the traffic intensity?

(c) What is the (steady state) probability that there is no one in the queue?

(d) What is the (steady state) expected waiting time to check out (time in the queue plus time in service)?

Explanation / Answer

a) M/M/1 queueing system

b) = 5 per hour and = 1/0.1 = 10 per hour. Thus the trac intensity a =/= 0.5.

c) That meansP(X(t) = 0 or 1) =p0 + p1.p0= 1-a= 1-0.5 = 0.5 and p1= p0a= 0.25

d) That means the averag waiting time for borrower in the library

W = 1/( -) = 1/(10-5) = 1/5 = 0.2

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