Telephone calls arrive at a rate of 48 per hour at a reservation desk for Region

Question

Telephone calls arrive at a rate of 48 per hour at a reservation desk for Regional Airways.

a. Find the probability of receiving 3 calls in a 5 minute interval.

b. Find the probability of receiving 10 calls in 15 minutes.

c. 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 nobody will be waiting?

d. If no calls are currently being processed, what is the probability that the agent can take 3 minutes for personal time without being interrupted?

Explanation / Answer

Arrival rate = Lambda = 48 calls per hour

Service rate = Miu = 60 calls per hour

Utilization Factor = Chi = Lambda / Miu = 48/60 = 0.8

Ls = Chi / (1-Chi) = 0.8 / (1-0.8) = 4

Lq = Ls – Chi = 4 – 0.8 = 3.2

Ws = Ls / Lambda = 4 / 48 = 0.0833

Wq = Lq / Lambda = 3.2 / 48 = 0.0666

Number of servers = C = 1

a)

Pn = Probability of n = (1-Chi) * Chi ^ n

P3 = (1-0.8) * (0.8)^3 = 0.1024

3 calls in 5 minute interval = 3 calls once in every 5 minutes means = 12*3 = 36 calls per hour

New value of Lambda = 36 calls per hour

Chi = 36/60 = 0.6

P36 = (1-0.6) * (0.6)^36 = 4.1257*e-9

b)

10 calls in 15 minutes = 4*10 = 40 calls per hour

P10 = (1-0.8) * (0.8)^10 = 0.0215

Lambda = 40

Chi = 40/60 = 4/6=2/3 = 0.6667

P40 = (1-0.667)*(0.667)^40 = 3.07 e -8

c) currently no calls are on hold, Ls = 4 callers will be there in the system (both waiting and being served) and 3.2 callers will be waiting in the system

d) Probability of having a 3 minute break between the calls = 1/12 because calls arrive only at the rate of 3 calls once in every 5 minutes

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