Erlang Queuing Model [M/Ek/1: infinity, FIFO] single server with infinite queuing capacity solved problems

Erlang Queuing Model [M/Ek/1: infinity, FIFO]  is one of the queuing models with a single server and infinite queuing capacity. There are K identical stages (phases) in a series of services each with mean service time (1/Kμ) where μ is the average service rate.

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Important Formulae of [M/Ek/1: infinity, FIFO]: (Operating Characteristics)

1. The average number of Customer Waiting in Queue

2. The Average number of customer in System

Ls = Lq + λ/μ

3. The average waiting time of a customer in System:

Ws = Wq + 1/μ  or,

= Ls/λ

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

Wq = Lq/λ

5. Model value of service time, (most probable time spent in getting the service)

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Solved Problems of (M/Ek/1: infinity, FIFO):

Solved Problem 1: In a factory, the customers have to pass through three counters. The customers buy coupons at the first counter, select and collect the snacks at the second counter and collect tea at third counter. The server at each counter takes on an average 1.5 minutes although the distribution of service time is approximately Poisson at an average rate of 6 per hour. Find the average time a customer spends waiting in cafeteria. Also, find the most probable time spent in getting the service. 

= Solution:

Here Given,

arrival rate (λ) = 6 per hour 

Service time per phase = 1.5 minutes
so total service time = 3*1.5 = 4.5 minutes

Service rate (μ) = 1/4.5 per minute = 60/4.5 = 13.34 per hour
The number of phases (K) = 3 phases
i. The average time the customer spends waiting in cafeteria = Wq
Using the above formula we get
Wq = Lq/λ
               = 9/220 hour 
ii. Most probable time spent in getting serivce = (K-1)/Kμ
                                                                          = 1/20 hours

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So these are a complete theory and Queuing Theory Solved problems related to the Queuing Model [M/Ek/1: infinity, FIFO].

Related Posts:

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