# M/M/S : FIFO/∞ Model

This is one model where total capacity of a system is specifically restricted and there is no chance of any extra individual joining this system in any manner. In this case, it’s taken that capacity of a system is fixed at N, where chances of (N+1) customer joining are nil.

The difference that is found is in the form of a differential equation, wherein differential equations of previous models would only be valid if there is if N is greater than n (N>n).

Hence a situation of N=n needs to be created.

Here are two sums, solving which will give students a deeper insight into this whole process.

Sum 1:  It is taken that a set of trucks are coming to the market area at the rate of 30 trucks on a regular basis. Also,it is taken that their arrival rate follows the process of exponential distribution. It is also assumed that unloading of trucks is done by following the exponential principle, with the average being at 42 minutes.

If, taken that a single market area can take up to 10 trucks at a single time, what is the ‘P’ when this market area is empty and what’s average length of this queue that is formed?

Also, if it is taken that time of unloading increases to 48 minutes, how would it affect the above mentioned questions?

Sum 2:  If a data is given that in a car testing center, cars come in the format of the Poisson center, and average rate of cars is 15 on an hourly basis.Also, it is said that this center can accommodate only 15 cars. How should then timing of service of the per car be calculated, when the mean rate is taken to be 10 cars per hour and the format of exponential distribution is to be followed.

How to get following data?

• What is the expected time for waiting until a car is leaves a testing center?
• What is the probability that a car which has arrived does not have to wait in that service center before getting a place?
• What are the chances that a car that’s arriving will find a specific place reserved for itself in the center?
• What’s taken as the effective rate of arrival of the cars at the center?

Solution to these doubts can help students get a deeper insight into the topic.

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