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# What is Queuing Theory?

Module by: Lekulana Kolobe. E-mail the author

Summary: This module introduces queuing theory, its applications in telephony, and solutions to queuing problems. There is also an exercise to engage the reader.

Queueing theory (also commonly spelled queuing theory) is the mathematical study of waiting lines (or queues). There are several related processes, arriving at the back of the queue, waiting in the queue (essentially a storage process), and being served by the server at the front of the queue. It is applicable in transport and telecommunication and is occasionally linked to ride theory.

Application of queueing theory to telephony

The Public Switched Telephone Networks (PSTNs) are designed to accommodate the offered traffic intensity with only a small loss. The performance of loss systems is quantified by their Grade of Service (GoS), driven by the assumption that if insufficient capacity is available, the call is refused and lost. Alternatively, overflow systems make use of alternative routes to divert calls via different paths -- even these systems have a finite or maximum traffic carrying capacity. However, the use of queueing in PSTNs allows the systems to queue their customer's requests until free resources become available. This means that if traffic intensity levels exceed available capacity, customer’s calls are here no longer lost; they instead wait until they can be served. This method is used in queueing customers for the next available operator. A queueing discipline determines the manner in which the exchange handles calls from customers. It defines the way they will be served, the order in which they are served, and the way in which resources are divided between the customers. Here are details of three queueing disciplines:

First in First Out (FIFO) - customers are serviced according to their order of arrival

Last in First Out (LIFO) - the last customer to arrive on the queue is the one who is actually serviced first.

Processor Sharing (PS) - customers are serviced equally, i.e. they experience the same amout of delay.

Incoming traffic to queuing theory systems is modelled via a Poisson distribution,with the assumptons of Pure-Chance Traffic – Call arrivals and departures are random and independent events. Statistical Equilibrium – Probabilities within the system do not change. Full Availability – All incoming traffic can be routed to any other customer within the network. Congestion is cleared as soon as servers are free.

Solutions of queuing problems:

The most easily used methods to solve queing problems are analytical methods which make the following assumptions:

Arrival times are random and time between arrivals are distributed exponentially,

Service times are also distributed exponentially,

The queue is of FCFS type, and

There are no significant interdependencies

Average customer waiting time (Wq) is defined as:

Wq = (Sav)^2 / (Aav - Sav); For Aav>Sav (1.1)

Where Aav is the average time between arrivals; Sav is average service time.

Mean time required for customer to wait and be serviced is :

Wm = Wq + Sav (1.2)

## Exercise 1

Suppose the average service time per call at Telkom Call Centre is 5 minutes. On average calls arrive at 20 minutes interval. 1. What is the average time that a customer will hold before being serviced? 2. What is the mean time spend by a customer in total?

### Solution

1. From equation 1.1; Wq = (Sav)2 / (Aav - Sav), Wq = 52 / (20 - 5), Wq = 25 / 15 = 1.67 Minutes, 2. From equation 1.2; Wm = Wq + Sav, Wm = 1.67 minutes + 5 minutes = 6.67 Minutes

REFERENCES:

Ashley. Queuing Cost. Accessed on 13-02-2006 at http://www.bsbpa.umkc.edu/classes/ashley/Chaptr14/sld005.htm

Queuing Theory - A Straightforward Introduction. Accessed on 10-02-2006 at http://www.new-destiny.co.uk/andrew/past_work/queuing_theory/Andy

Introduction to Queuing Theory. Accessed on 9-02-2006 at http://staff.um.edu.nit/jskl1/simweb/intro.htm

Highly edited for Wikipedia 2006 Last Accessed on 13 March 2006

Krick E. Methods Engineering: Design and Measurement of Work Methods. John Wiley Inc. London. 1962. Problem 1.

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