Queuing Models?

Question

Guests enter the hotel and join a single que whch is served by five servers. ? how long it takes to custmers to check in ?
Given lamda=90 per hour, Mu = 20 per hour and Servers=5.
ASAP. thank you . [e-mail]

Answer 1

Depends if you are behind a coach loan of Americans :-)

OK, lets be a bit more serious ..

This looks like an exercise in probability .

The symbol μ (Greek: mu) is used to denote the arithmetic mean of an entire population .. in our case the 5 Servers .. thus together the 5 Servers would process 5x20 =100 per hour.

Not sure what "lamda" means = perhaps the sum of the squares of each Servers processing rate ? or is it the rate of Clients arrival (90 per hour ?) ..

In any event, for sure we need to know the rate and distribution of Client arrival at the hotel (so we can calculate how the queue will behave = see later)

We also need to know the 'distribution' of Server processing rates (mu 20 will result in all 5 Servers processing 20 or one server doing 10, 3 x 20 and the final at 30 .. worst case we might have 4 Servers each processing one per hour an the final one processing 96 !).

I thus assume that it is possible to use "lamda" to calculate the range of processing times (20 per hour = 3 miniutes each, but +/- what ? (i.e. whats the variation ?))

A simple straight 3mins +/- X will not lead to an answer for an individual Client however .. (in the 10, 20,20,20 30 = 6 mins, 3 mins, 2 mins) example above, the 'answer' would be 3 mins +3 / -1 ..plainly there is a straight 20% chance of of being processed in either 6 or 2 mins but 60% chance of 3 mins)


Things are complicated by how the queueing system is run ..

If a queue was formed behind each Server, you are stuck in the queue you have chosen (so the 20% each chance applies, UNLESS you choose to switch queues :-) )

However here we have a single queue system is used, thus (3x) more clients will be served by a fast Server (2 min) compared to a slow (6 min) one ... and thus your chances of being checked in by a fast Server is actually (3 times) higher than for the slower one ... PROVIDED the queue keeps all 5 Servers continuously busy ...

If, however, Clients arrive randomly at a rate that keeps some of the Servers busy (but never causes a queue to build up), you still more likely to find a 'fast' server free than a slower one BUT if more than one Server is free when you arrive, you will not be able to distinguish between a 'fast', 'medium' or 'slow' servers - so you could choose the slow one (leaving the fast / medium one(s) idle)

.. until finally, when Clients arrive singularly (and find all Servers free) the chances of choosing a fast, medium or slow Server are exactly the same as the 'queue behind each' system ...

Sounds to me like an ideal problem for a Monte Carlo simulation ..

Good luck !