# Queuing Theory

Topics: Queueing theory, Queueing model, Poisson distribution Pages: 5 (1225 words) Published: April 30, 2013
Introduction
Being in a queue (waiting line) is an inevitable fact of our daily life, such as waiting for checkout at a supermarket, or waiting to make a bank deposit. Queuing theory, started with research by Agner Krarup Erlang, is used to examine the impact of management decisions on these waiting lines (Anderson et.al, 2009). A basic Queuing Model structure consists of three main characteristics, namely behaviour of arrivals, queue discipline, and service mechanism (Hillier and Lieberman, 2001). In this assignment, New England Foundry’s queuing problem will be solved in Excel, and then, time and cost savings will be identified. First of all, current and new situation will be analysed in order to demonstrate the queuing model by using Kendall’s Notation (for the current queuing problem, queuing model is M/M/s). After that, arrival rate, queue size, and service rate will be defined, and added-in Excel file (Queuing models.xlsx). The results will be discussed at the end.

Description
New England Foundry (NEF) produces four different types of woodstoves for home use and additional products that are used with these four stoves. Due to the increase in energy prices, George Mathison president of the company wants to change the layout to increase the production of their bestselling type of Warmglo III. NEF has several operations in order to produce woodenstoves which are illustrated as a flow diagram in Figure 1. Current State Analysis

Current layout offers one counter with two personnel, namely Pete and Bob who are able to serve a total of 10 people per hour (5 per hour each), with an average of 4 people from maintenance and 3 people from molding arriving at the counter per hour randomly (4+3=7). An illustration of current layout can be seen in Figure 2.

Figure 1 Flow Diagram

Figure 2 Current State
This layout caused an additional walking time for a personnel to reach the counter from; Maintenance department: 3 minutes
Moulding department: 1minute
Therefore, it takes a total travel time of 6 mins for the maintenance personnel, and 2 minutes total for molding personnel to walk to and from the related departments. Characteristics of the current situation;

| One Counter|
Arrival rate (λ)| 7|
Service rate (µ)| 5|
Number of servers (s)| 2|
Queuing Model| M/M/s|

Assumptions:
Since there is no information about the size of arrivals, we can suppose that there is no limit on people whom both Bob and Pete can serve. The arrival rate for both the layouts is 4 per hour from maintenance and 3 per hour from the molding department, establishing a Poisson distribution. The arrival behaviour of the people involves no balking and reneging. This arrival probability distribution can be denoted by M (Markowian). Queue discipline is FIFO to serve people in a single line.

Since the queue size is not restricted, we can define the length of the queue as it is infinite. There is just one counter but the number of servers (s) is 2 i.e. Pete and Bob, through a single phase serving system. Both Bob and Pete can serve 10 people per hour establishing an exponential probability distribution. The service time probability distribution is denoted by M (Markowian) As result, the queuing model of current layout is M/M/s.

(See the queuing problem calculations of the current situation in Annex 1).

New State Analysis
After several week observations, George decided to separate the maintenance shop from the pattern shop. This modification provides a number of advantages by bringing changes in the queuing system (see Figure 3); * Walking time for a personnel from maintenance dropped from 3 to 1 minute. Therefore, the total walking time is 2 minutes from and to maintenance department. * The number of people per hour whom Pete and Bob serve increased (Pete: 7 people/hour, Bob: 6 people/hour)

New layout system can be seen in Figure 3;

Figure 3 New Layout

Pete (molding) and Bob (maintenance) will be solved...

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