# M/M/1: FIFO/N Model

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This is another model that is used in the queuing theory to get answers related to this system. In this system, both arrival and departure rates are taken to be as per the Poisson theory, with µ andλ. Though in case of this system, there is a single server available, the capacity associated with this system happens to be infinite.

Here are some of the sums that are there, solving which will help students get a better idea of this whole process.

Sum 1:  It can be seen that a single-man barber shop, customers keep arriving at the rate of 4 people per hour in the form of Poisson theory. Also, the time was taken to cut the hair was exponentially distributed with an average time-limit of 12 minutes. If there is no restriction on formation of the queue system, how to calculate the following?

• What are the fluctuations that are available in length of that queue?
• What’s the probability that there are 5 customers available in that queue?
• What percentage of time can itbe said that barber is sitting idle?
• What is the expected time in term of minutes that any customer would have to spend in the queue that is formed?

Sum 2:  A public booth is present, where arrival rate is calculated in regards to Poisson distribution, where average interval of time is taken to be at 10 minutes.If it is taken that service provided by telephone booth is distributed in an exponential manner, then time limit is taken to be at 2.5 minutes.

How to calculate?

• That fresh arrival who needs to wait for a call?
• What are the chances that any particular consumer finishes off that call in a span of 10 minutes and leaves?
• What can be taken as the average number of customers that are present in that booth?
• What is that probability that queue size extends a count of 5?

Once a student solves these sums, conceptual clarity will be better.

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