Telephone calls arrive at the rate of 48 per hour at the res

Telephone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. fund the probability of receiving 3 calls in a 5-minute interval. Find the probability of receiving 10 calls in 15 minutes. Suppose that no calls are currently on hold. If the agent takes 5 minutes to complete processing the current call, how many callers do you expect to be waiting by that time? What is the probability that no one will be waiting? If no calls are currently being processed, what is the probability that the agent can take 3 minutes for personal time without being interrupted?

Solution

Let X be the random variable denotes number of telephone calls received per hour at the reservation desk for Regional Airways.

Then, X ~ Poisson (), where = 48

a. 5 minutes = 5/60 hour = 1/12 hour

P(receiving 3 calls in 5 minutes)

= P(receiving 3 calls in 1/12 hour)

= P(X(t) = 3) where, X(t) ~ Poisson (t), t = 1/12, i.e. X(t) ~ Poisson(4)

= (4³/3!). e^-4   (Answer)

b. 15 minutes = 15/60 hour = ¼ hour

P(receiving 10 calls in 15 minutes)

= P(receiving 10 calls in ¼ hour)

= P(X(t) = 10) where, X(t) ~ Poisson (t), t = ¼, i.e. X(t) ~ Poisson(12)

= (12¹⁰/10!). e^-10   (Answer)

c. The Exponential Distribution is the probability distribution that describes the time between events in a Poisson process. Let T be a random variable denotes the waiting time between events for the poisson process with rate = 48/hour = 0.8/minute

Therefore, T ~ Exp ()

Mean waiting time = E(T) = 1/ = 1/48 hour = 60/48 minutes = 5/4 minutes

The agent takes 5 minutes to complete processing the current call

So, expected number of callers to be waiting by that time = 5/E(T) = 5/(5/4) = 4 (Answer)

Arrival rate = 0.8/minute

Service rate = 0.2/minute (as the agent takes 5 minutes to complete processing the current call, on average 1/5 = 0.2 call is taken per minute)

Prob (No one will be waiting) = 0 as >

d. Prob (the agent can take 3 minutes for personal time without being interrupted)

= P(T > 3)

= 1 – P( 3)

= 1 – (1 – e^-3)

= e^-3 = e^-2.4 ( = 0.8/minute)

= 0.0907     (Answer)