Arrivals of customers at a telephone booth follow poisson distribution, with an average time of 10 minutes between one arrival and the next. the length of the phone call is assumed to be distributed exponentially with a mean of 3 minutes. find
a. the average number of persons waiting and making telephone calls
b. the average length of the queue that is formed from time to time.
c. probability that a customer arrive and find telephone booth is busy.
d. probability that a customer arrive and find telephone booth is empty.
Answers
a. the average number of persons waiting and making telephone calls
b. the average length of the queue that is formed from time to time.
c. probability that a customer arrive and find telephone booth is busy.
d. probability that a customer arrive and find telephone booth is empty.
Answer:
(i) 0.43
(ii) 0.43
(iii) 0.3
(iv) 0.7
Step-by-step explanation:
a. the average number of persons waiting and making telephone calls
Average arrival rate = λ = 60/10 = 6 customers/hour
Average service rate = μ = 60/3 = 20 customers/hour
Utilization parameter = ρ = 6 /20 = 0.3
=/ − =6/20 −6 = 6/14= 0.43 customers
b. the average length of the queue that is formed from time to time.
Ls= ρ/1- ρ
= 0.3/1-0.3
= 0.43 customers
c. probability that a customer arrive and find telephone booth is busy.
P (w > 0) = 1 – P0= 1 – (1 - λ / μ) = λ / μ= 0.10/0.33 = 0.3
d. probability that a customer arrive and find telephone booth is empty.
λ=0.1 per minute,
Service rate, μ=0.33 per minute
ρ= 0.1 / 0.33 =0.3
Probability that an arrival does not have to wait before service or probability that a system is idle,
p0 = 1- p = 1-0.3 =0.7
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