Question

Four servers are currently busy, each serving a different customer. You are the next in line to be served, as soon as one server becomes available. The servers have different levels of experience, resulting in slightly different expected service time per customer. Server 1 has mean service time 50 seconds, Server 2 has mean service time 55 seconds, Server 3 has mean service time 60 seconds, and Server 4 has mean service time 65 seconds. All servers have their service time exponentially distributed. A) Determine the probability that you will be served by Server i, for i=1,2,3,4. B) Determine the expected time that you have to wait before being served. C) Determine your own expected service time (in seconds), when you are still waiting in queue (so you do not know yet which server will be assigned to you). D) Determine the total time that you expect to spend waiting and then being served.

Answer 1

A) Each server has the equally likely chances to serve your request, so Probability that you will be served by Server i, for i = 1,2,3,4 is 25%.

B)

Servers	(1)	(2)	(3) = (1) * (2)
	Mean Serving Time (In Seconds)	Probability to be Served	Expected Time
Server 1	50	0.25	12.5
Server 2	55	0.25	13.8
Server 3	60	0.25	15.0
Server 4	65	0.25	16.3
Expected Time you have to Wait (In Seconds)			14.4

Expected Time you have to wait is the Average of column (3) i.e. 14.4 seconds

(C) My Expected Wait time is same as calculated in (B) i.e. 14.4 seconds.

But, my expected servicing time is the Average of column (1) of the above table i.e. 57.5 seconds.

(D) Total expected time inclusive of waiting and servicing time is the sum of 'Expected Time you have to wait' and 'Expected Servicing Time'.

Total expected time inclusive of waiting and servicing time = 'Expected Time you have to wait' + 'Expected Servicing Time'

Total expected time inclusive of waiting and servicing time = 14.4 seconds + 57.5 seconds

Total expected time inclusive of waiting and servicing time = 71.9 seconds