An urban post is set up to have customers join a single line

An urban post is set up to have customers join a single line which is served by multiple workers. 30 customers arrive per hour (c_a = 0.92) during lunch and join a single line which is served by 5 clerks. It takes about 7 minutes to serve each customer (c_p = 0.37).

How long does the average customer wait in line (in minutes)?

a) 0.95

b) 4.82

c) 1.93

d) 0.70

Solution

To be calculated:

Average customer waiting time

Given values:

Number of customer arriving per hour = 30

This means 1 customer is arriving in every 2 minutes. Therefore,

Customer Inter-arrival time, a = 2 minutes, Ca = 0.92

Time taken to serve each customer = 7 minutes

Average activity time, P = 7 minutes, Cp = 0.37

Number of servers, m = 5 clerks

Solution:

Utilisation is calculated as;

U = Flow rate / Capacity = [(1/a) / m x (1/p)]

U = [p / (a x m)]

U = [7 / (2 x 5)] = 7/10 = 0.7

Time in Queue is calculated as;

TIQ = (Activity time/m) x [(Utilisation ^ 2(m+1) - 1) / (1 - Utilisation)] x (Ca^2 + Cp^2) / 2

TIQ = (7/5) x [(0.7 ^ 2.464) / 0.3)] x (0.846 + 0.137) / 2

TIQ = 0.952

Average customer waiting time in line = 0.95 minutes

Answer - Option (A)