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 the equations:

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 the average arrival rate = a Persons / second

Let the service rate = s Persons/ second

Therefore ,

Waiting time in the system

= average waiting time in the queue + average waiting time in ATM

= a / S x ( s- a ) + 1/ S

= ( a + s – a ) / s x ( s – a )

= 1 / ( S - a)

Given ,

1/ ( s -a ) = 285

Or, S -a = 1/285

At the same time ,

Service time = 35 seconds

Therefore,

Service rate = S = 1/35 persons / second = 0.02857 persons/ second

Taking above two equations into account :

S – a = 1/285

Or, 1/35 – a = 1/285

Or a = 1/35 – 1/285

Or, a = 0.02857 – 0.00351

Or, a = 0.02506 persons / second

= 1/a seconds

= 1/0.02506

= 39.90 seconds

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