Question

Question

The management of a busy petrol station is concerned that customers are being lost because of long waiting times sometimes required at their petrol pump. Over a two weeks period a careful study has been taken of the arrival of cars and the length of time taken to serve customers at the petrol station. The tables below show the arrival rates and the service time distribution:

Inter arrival time (minutes)	Percentage of customers		Service time (minutes)	Percentage of customers
0 - <2	60		0 - <4	20
2 - <4	25		4 - <6	30
4 - <6	10		6 - <8	20
6 - <8	5		8 - <10	15
			10 - <12	15

• Assuming that the petrol station has only one pump and only one member of staff attend the customers, simulate the arrival of the first 10 customers and calculate the following:

(a)     Average inter-arrival time.

2. Average service time.
3. Average waiting time.
4. Average queue length.

Use the random numbers given below for the simulation.

89,34,07,65,37,11,29,80,28,34,08,14,75,92,01,48,21,83,63,91.

				Service
Cust. No.	Random Number	Inter-Arrival Time	Clock time	Random Number	Service Time	Service Starts	Service Ends	Waiting Time	Queue Length
0	0	0	0	0	0	0	0	0	0
1
2
3
4
5
6
7
8
9
10

Answer 1

Inter arrival time (minutes)	Avg Inter-arrival time	Percentage of customers	Cummulative Probability	Random No. Assignment
0 - <2	1	60	60	01 to 60
2 - <4	3	25	85	61 to 85
4 - <6	5	10	95	86 to 95
6 - <8	7	5	100	96 to 100

Service time (minutes)	Average Service time	Percentage of customers	Cummulative Probability	Random No. Assignment
0 - <4	2	20	20	01 to 20
4 - <6	5	30	50	21 to 50
6 - <8	7	20	70	51 to 70
8 - <10	9	15	85	71 to 85
10 - <12	11	15	100	86 to 100

	Customer No	Random Number	Inter-Arrival Time	Clock time	Service Random Number	Service Time	Service Starts	Service Ends	Waiting Time	Queue Length
	1	89	0	0	8	2	0	2	0	0
	2	34	1	1	14	2	2	4	1	1
	3	7	1	2	75	9	4	13	2	1
	4	65	3	5	92	11	13	24	8	1
	5	37	1	6	1	2	24	26	18	2
	6	11	1	7	48	5	26	31	19	3
	7	29	1	8	21	5	31	36	23	4
	8	80	3	11	83	9	36	45	25	5
	9	28	1	12	63	7	45	52	33	6
	10	34	1	13	91	11	52	63	39	7
Total			13			63			168	30

a) Average Inter-arrival Time = [(Inter arrival times) ] / [ Number of arrivals-1]
= 13/9 = 1.4444 minutes

b) Average Service time = [(Service times) ] / [Number of Customers]
= 63/10 = 6.30 minutes

c) Average Waiting time = [(Waiting times in queue) ] / [Number of Customers]
= 168/10 = 16.8 minutes

d) Average queue length = [(Queue length) ] / [Number of Customers]
= 30/10 = 3

As is clearly evident from the simulation, the waiting time shows an increasing trend and length of queue keeps on increasing. The customers won't join the queue, if the queue is too long and would not use the services at the pump, leading to potential revenue losses.
There is a need of setting up a new pump and employing more staff. This will reduce queue length and would boost the revenues and increase customer satisfaction because of reduced waiting times.