You need cash and FAST! You jump out of your car at a location known to have one

Question

You need cash and FAST! You jump out of your car at a location known to have one ATM, but a history of a line. Usually you can expect to wait 285 seconds from the time you arrive until leaving with the $$. Service time averages 35 seconds. SHOW ALL WORK!

Remembering

a.) What is the utilization of the server? _____%  

b.) What is the expected time between people arriving to this system? _____ seconds

c.) How long is the line expected to be upon arrival? _____ people

d.) What is the service rate of the server? _____ people/second

Explanation / Answer

Let arrival rate = “a” number of people per minute

Service rate = “s” number of people per minute

Basis M/M/1 queuing theory,

Average waiting time in the system = 1 /( s – a ) minutes

Given :

Waiting time in the system = 285 seconds = 4.75 minutes = 19/4 minutes

Service rate ( @ 35 seconds per person) =S = 60/35

Therefore .

1/( s -a ) = 19/4

Or, S -a = 4/19 ……….. ( I )

S = 60/35 …………. ( II)

Taking into account above two equations ,

60/35 – a = 4/ 19

Or, a = 60/35 – 4/19

Or, a = 1.714 - 0.210

Or a = 1.504

Therefore , expected time between people arriving to this system = 60/1.504 = 39.89 seconds

= Expected number of people waiting in queue + expected number of people getting served

= arrival rate xwaiting time in the system

= 1.504 x 4.75

= 7.144 people

Therefore , service rate of the server = 1/35 people per second = 0.02857 people per second

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