Question

All airplane passengers at the Lake City Regional airport must pass through security screening before proceeding to the boarding area. The airport has 3 screening stations available, and the facility manager must decide how many to have open at any particular time. The service rate for processing passengers at each screening station is 3 passengers per minute. On Monday morning, the arrival rate is 5.1 passengers per minute. Assume exponential service times and Poisson arrival rates. Compute the probability that an arriving customer will have to wait for service (i.e., probability of waiting) for 2 screening stations and for 3 screening stations. If the service target is that the probability that a customer should have to wait for service should be no more than 45%, would you have 2 or 3 stations open on Monday morning?

Answer 1

Average arrival rate, λ = 5.1 passengers per minute
Average service rate, μ = 3 passengers per minute

Use the following formula, for a given number of servers (s), to compute the probability of idle server (P₀), average waiting time (L_q), and Probability of waiting (P_w).

Implement the above formula for s=2 and s=3 to find P_w in both the cases.

s	P0	Lq	Pw
2	0.0811	4.4261	0.7811 > 0.45
3	0.1657	0.4095	0.3131 < 0.45

So, if the target of P_w is less than 45%, then there must be 3 servers at least.