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: (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

There are 2 agents and 3 lines. Model is finite queue of M/M/s/K model

Arrival rate = = 15 per hr

Service rate = 60/4 = 15 per hr

s = 2

K (Max Customer) = 3

Rate Diagram

Staedy state Probabilities

P0 = 1 / (1+1+0.5+0.5*0.5)= 1/2.75 = 0.363636

P1 = (15/15)1 / 1! * P0 = 1*P0 = 0.3636

P2 = (15/15)2 / 2! * P0 = 0.5*P0 = 0.5*0.363636 = 0.181818

P3 = (15/15)3 * 1/2(3-2) * 1/ 2! *P0 = 1*0.5*0.5*0.3636 = 0.090909 (for n=s , second formula from above to be used)

1) A caller will get to talk to an agent immediately if there are 0 or 1 customer in the system .So Probability is

P0 +P1 = 0.363636+0.363636 = 0.727272

2) The caller will be put on hold if there are 2 customer in the system = P2 = 0.181818

3) The caller will get a busy signal if there are 3 customers

P3 = 0.090909