Question

he reservation office for Central Airlines has two agents answering incoming phone calls for flight reservations. A caller can be put on hold until one of the agents is available to take the call. If all three phone lines (both agent lines and the hold line) are busy, a potential customer gets a busy signal, in which case the call is lost. All calls occur randomly (i.e., according to a Poisson process) at a mean rate of 15 per hour. The length of a telephone conversation has an exponential distribution with a mean of 4 minutes.

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

(b) Find the steady-state probability that: (Show every calculation)

1. A caller will get to talk to an agent immediately.

2. The caller will be put on hold, and

3. The caller will get a busy signal.

Answer 1

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

Since this queueing system has two servers (the agents) and the maximum number of customers (the incoming calls) allowed in the system is 3, the applicable model is the finite queue variation of the M/M/s model with

K = 3, λ = 15, and µ = 60/4 = 15.

The rate diagram is shown below:

(b) Find the steady-state probability that

(i) A caller will get to talk to an agent immediately,

(ii) The caller will be put on hold, and

(iii) The caller will get a busy signal.

We use the Excel template for the finite queue variation of the M/M/s model to calculate the steady-state probabilities and obtain.

P0 = 0.36364, P1 = 0.36364, P2 = 0.18182, P3 = 0.09091.

Hence,

(i) the steady-state probability that a caller will get to talk to an agent immediately = P0 + P1 = 0.727.

(ii) the steady-state probability that a caller will be put on hold = P2 = 0.182.

(iii) the steady-state probability that a caller will get a busy signal = P3 = 0.091.