Scenario One: Automobiles arrive at the drive-through window of the Elm Street branch of the local bank at the rate of 1 every 12 minutes. The average service time is 7.5 minutes. The Poisson distribution is appropriate for the arrival rate and service times are exponentially distributed.
The table given below shows the workings of arrival rate and service rate times of local bank at Elm street branch
|
Parameter
|
Value
|
Minutes
|
Seconds
|
|
M/M/1 (exponential service times)
|
|
|
|
|
Arrival rate(lambda)
|
5
|
|
|
|
Service rate(mu)
|
8
|
|
|
|
Number of servers
|
1
|
|
|
|
Average server utilization
|
0.63
|
|
|
|
Average number in the queue(Lq)
|
1.04
|
|
|
|
Average number in the system(L)
|
1.67
|
|
|
|
Average time in the queue(Wq)
|
0.21
|
12.5
|
750
|
|
Average time in the system(W)
|
0.33
|
20
|
1200
|
The probability table is given below
|
k
|
Prob (num in sys = k)
|
Prob (num in sys <= k)
|
Prob (num in sys >k)
|
|
0
|
0.38
|
0.38
|
0.63
|
|
1
|
0.23
|
0.61
|
0.39
|
|
2
|
0.15
|
0.76
|
0.24
|
|
3
|
0.09
|
0.85
|
0.15
|
|
4
|
0.06
|
0.9
|
0.1
|
|
5
|
0.04
|
0.94
|
0.06
|
|
6
|
0.02
|
0.96
|
0.04
|
|
7
|
0.01
|
0.98
|
0.02
|
|
8
|
0.01
|
0.99
|
0.01
|
|
9
|
0.01
|
1
|
0.01
|
|
10
|
0
|
1
|
0.01
|
Question 1: What is the average number of customers in the system (in line plus being served)?
Question 2: What is the average time a customer spends in the system?
Question 3: What is the average number of customers in line “behind” the customer receiving service (queue)?
Question 4: What is the average time customer spend waiting to receive service?
Question 5: What is the probability the customer service person is busy?
Question 6: What is the percent of time (or the probability) that there are zero customers at the serving window?
Question 7: What is the probability that there are more than two customers in the system?
Question 8: What is the probability that there are exactly two customers in the system?
Question 9: What is the probability that there are two or less customers in the system?
Question 10: What is the probability that there are less than two customers in the system?