Question

The 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:

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

Solution a:

As this queuing system has two agents and the maximum number of customers (the incoming calls) allowed in the system is 3 therefore the applicable model is of the finite queue variation of the M/M/s model having

                                K = 3, l = 15, and m = 60/4 = 15.

The rate diagram is given below:

Solution b:

For finding the steady state probabilities, we will use the Excel template for the finite queue variation of the M/M/s model and we got the following results:

                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 is given as :

    = P0 + P1= 0.727.

(ii) The steady-state probability that a caller will be put on hold is given as:

= P2 = 0.182.

(iii) The steady-state probability that a caller will get a busy signal is given as:

= P3 = 0.091.

The excel template is as shown below: