Queuing Model [M/M/S:∞,FIFO] Multi-Server and Infinite Capacity with Solved Problems

[M/M/S:∞, FIFO] is one of the Queuing Model with Multi-Server and Infinite Queuing Capacity. The service rate of each server in the system is the same i.e μ and the arrival rate in the system is λ.

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Important Formulae of [M/M/S:∞, FIFO]: (Operating Characteristics)

Read also:

Little's Formula

1. Average arrival rate at state 'n'  λ_n = λ
    Average service rate at state 'n'  μ_n= Sμ if n>=S
                                                             ,  nμ if n<S
where S is the number of server and n is the number of customer in the system.

2. Traffic Intensity:

It is also called the Utilization Factor. It gives the proportion of the time server is busy. It is denoted by rho(ρ).

3. Steady State Probability (Limiting probability): 

It is also the probability of Queue being in a state 'n'. It is denoted by 'Pn'.

4. The probability that the Server is idle :

5. The fraction of Time all Servers are busy = ρ
   The fraction of Time all servers are idle = 1 - ρ

6. The average number of Customer Waiting in Queue

7. The Average number of customer in System

8. The average waiting time of a customer in System/service:

9. The average waiting time of a customer in Queue:

10. The probability that arriving customer has to wait in Queue for service (Probability of Congestion):

11. The probability that the arriving customer do not have to wait in Queue for Service is:

  1 - P(N>=S)

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Solved Problems of (M/M/S: infinity, FIFO):

Solved Problem 1: A supermarket has two girls ringing up sales in the counters. Let service time for each customer be exponential with mean 4 minutes and people arrive in the queue in Poisson fashion at the rate of 10 an hour.

i. What is the probability that all servers are jobless?

ii. What is the probability that an arriving customer has to wait?

= Solution:

Here Given, 

Number of Servers (S) =  2 [as two girls are giving the service]

arrival rate (λ) = 10 per hour = 10/60 per min = 1/6 per minute

Service rate (μ) = 1/4 per minute

So, Traffic intensity (ρ) = λ/Sμ

                                    = 1/3

Also, Sρ = λ/μ = 2/3

Model Identification:

This is [M/M/S: infinity, FIFO] queuing model. Since there are more than one server and capacity of the system is infinite.

i. the probability that all servers are jobless = Po

Using the formula for Po from above, We get

Po = 1/12

ii.  the probability that an arriving customer has to wait = P(N>=S)

Using the formula from above, we get

P(N>=S) = 1/6

In this way, other operating characteristics of  [M/M/S: infinity, FIFO] queuing model can be find using the above-mentioned formulae.

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Solved Problem 2. A telephone exchange has two long-distance operators. The Telephone company finds that during the peak load, long distance calls arrive in a Poisson fashion at an average rate of 15 per hour. The length of service in these calls is approximately exponentially distributed with a mean length of 5 minutes. What is the probability that a subscriber will have to wait for his long-distance call during the peak hour of the day?
= Solution:
Here Given,

Number of Servers (S) =  2 

arrival rate (λ) = 15 per hour = 15/60 per min = 1/4 per minute

Service rate (μ) = 1/5 per minute

So, Traffic intensity (ρ) = λ/Sμ

                                    = 5/8

Model Identification:

This is [M/M/S: infinity, FIFO] queuing model. Since there are more than one server and capacity of the system is infinite.

Now, Po can be found using the formula above. We get,

Po = 3/13

Now, the probability that caller has to wait for service is: P(N>=S)
Using the formula we get,

 P(N>=S) = 25/52

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So these are a complete theory and Queuing Theory Solved problems related to the Queuing Model [M/M/S:∞, FIFO].

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• Introduction and Features of Queuing System and Queuing Theory
• Queuing Model [M/M/1:∞,FIFO] Single Server and Infinite Capacity with Solved Problems
• Queuing Model [M/M/S:∞,FIFO] Multi-Server and Infinite Capacity with Solved Problems
• Queuing Model [M/M/1:K,FIFO] Single Server and finite queuing Capacity with Solved Problems
• Erlang Queuing Model [M/Ek/1: infinity, FIFO] single server with infinite queuing capacity solved problems