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

[M/M/S: K, FIFO] is one of the Queuing Model with Multi-Server and finite 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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Queuing Model Multi-Server and finite Queuing Capacity

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

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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 and K is the system capacity.

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. Effective arrival rate:

It is denoted by λ_e .
λ_e = λ(1 - P_k)

4. Steady State Probability (Limiting probability):

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

5. The probability that the Server is idle :

6. The average number of Customer Waiting in Queue

7. The Average number of customer in System

8. The amount of Time Customer Spend in the System

9. The Amount of Time Customer spend waiting in Queue

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

Solved Problem 1: A dispensary has 2 doctors and 4 chairs in the waiting room. Patient arriving leave if all 4 chairs are occupied. The arrival rate of patients is 8 per hour and the mean service time is 10 minutes. find the probability that the arriving patient will not have to wait, effective arrival rate, the average number of customer in the System.

= Solution:

Here Given,

arrival rate (λ) = 8 per hour

Service rate (μ) = 1/10 per minute = 60/10 per hour = 6 per hour
number of servers (S) = 2
System capacity (K) = 2+4 = 6

So, Traffic intensity (ρ) = λ/Sμ

= 8/6*2 = 2/3
1. The probability that arriving patient will not have to wait(idle server) = Po
We know,
using the formula mention above, We get
Po = 0.2158
2. Effective arrival rate(λe) = λ(1-P_k) [K>S]
= 7.69
3. The average number of customer in the queue = Lq
using above formula we get,
Lq = 0.6205

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

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