A queeing system has 3 servers with expected service times of 30 mintues. 20 min

Question

A queeing system has 3 servers with expected service times of 30 mintues. 20 minutes and 15 mintutes. The service time of each server have an exponetial distribution. Each server has been busy with a current customer for 10 mintues. Determine the expected remaining time until the next service completing the following steps.

a) Which properity(ies) of the exponetial distribution is(are) relevent to this analysis?

b) using the properties you identifired in (a), determine the expected remaining time until tge next service completion.

Explanation / Answer

Since the distributions are all exponential,

a) expected service time of each 3 servers is relevant to this analysis

we can ignore the fact that the servers have all been busy for the last 10 minutes (memoryless property).

b) The distribution of time to the next completion will also be exponential, with rate = sum of individual server rates, or µnext = 1/30 + 1 /20 + 1 /15 customers/minute.

This gives us the expected rate µnext 0.15 customers/minute

leading to our   expection that the time to next completion will be

1/ µnext 6.666 minutes.

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