Queues in a Bank


As a rule people dislike queueing in banks, and so in order to improve the level of service a bank manager is interested to find out:
  1. The average number of people waiting in the bank (i.e. the queue length)
  2. How much of the cashiers' time is spent idle (as a percentage of their time)
Depending on how many tellers she employs during lunchtime.  She is prepared to employ up to 5 cashiers, but not less than 1.

Model Assumptions and Details


Symbol Meaning, and value if in assumptions
c Number of cashiers (servers)
Lq Average length of the queue
W Percentage of cashiers idle (waiting) time
lamda Mean arrival rate (25 per hour)
mu 1/(mean time to server customers) = 30 per hour
p Average amount of work for each server, per hour


From the above we can easily calculate the following values, which are used to work out Lq and W:
Calculating r
Calculating p
p0 and Lq, W and the idle time can be calculated using the following equations:
p0 derivation
Lq derivation
W derivation
Direvation for the idle time
The results of these equations for 1 to 5 cashiers are summarised in the following table:
Cashiers p r^n / n! p0 Lq W Idle
1 0.833 0.833 0.167 4.167 0.2 0.167
2 0.417 0.347 0.412 0.175 0.04 0.583
3 0.278 0.096 0.432 0.022 0.034 0.722
4 0.208 0.02  0.434 0.003  0.033 0.792
5 0.167 0.003 0.435 0.000 0.033 0.833
From the table it can be seen that either 1 or 2 cashiers should be employed, as this will keep a short queue, but without the cashiers being idle too much of the time.

Henry Morgan, 14 June 1999