I'm having a bit of an issue with a question likely to be on an exam I'm taking soon. Details are-
A messaging switching centre has three servers for passing messages to the network. If the servers are busy, the message passes into a queue for service on a first in first out basis. Message arrivals are Poissonian and server sesrvice times have a negative exponential distribution. During the busiest period, 400 messages arrive per minute. The average time taken to service a message is 5ms. Calculate the following:
- The fraction of messages that experience a delay
- The average delay for messages that experience a delay (in ms)
- The fraction of messages that are delayed for longer than 8.33ms during the busy period.
My process for trying to answer the first question is to calculate the probabilities of x or more calls arriving in a second, based on the 400/60 = 6.66 messages/second arrival rate-
$P(0) = \mu^x e^{-\mu} / x! = {6.66^0*e^{-6.66}}/0! = 0.0013$
$P(1) = 0.085$
$P(2) = 0.0283$
$P(3) = 0.0628$
So ultimately-
$P(x >= 4) = 1 - P(0)-P(1)-P(2)-P(3) = 0.8991$
So the chance of 4 or more calls in a second is 89.91%, but I'm just mind-blanking on how to then calculate the chance of calls being placed in the queue based on the service time.
The answers provided to us are-
- 0.444
- 2.22
- 0.01
Usually I'm pretty good at working backwards, but can't figure this one out. It's the method of this that I'm stuck with. Any help would be greatly appreciated.