Teletraffic queuing theory

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 [1]. Overflow systems make use of alternative routes to divert calls via different paths; however even these systems have a finite or maximum traffic carrying capacity [1].

However, the use of queuing in PSTNs allows the systems to queue their customers requests until congestion has abated. 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 [2].

A queuing discipline determines the manner in which the exchange handles calls from customers [2]. It defines the manner in which they are served, the order in which they are served, and the way in which resources are divided between customers [2, 3]. There are three main queuing disciplines:

• First In First Out – This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first [3].
• Last In First Out – This principle also serves customers one at a time, however it serves the customer with the shortest waiting time first [3].
• Processor Sharing – Customers are served equally. Network capacity is shared between customers and they all effectively experience the same delay [3].

Queuing is handled by control processes within exchanges, which can be modelled using state equations [2, 3]. Queuing systems use a particular form of state equations known as Markov chains which model the system in each state [2]. Traffic in these systems is modelled via a Poisson distribution and is subject to Erlang’s queuing theory assumptions viz. [1]:

• Pure-Chance Traffic – Call arrivals and departures are random and independent events [1].
• Statistical Equilibrium – Probabilities within the system do not change [1].
• Full Availability – All incoming traffic can be routed to any other customer within the network [1].
• Congestion is cleared as soon as servers are free [1].

A notation for describing Queues and their characteristics was developed by Kendall and can be viewed at [4]. Classic queuing theory involves complex calculations to determine call waiting time, service time, server utilisation and many other metrics which are used to measure queuing performance [2, 3].

Classic queuing theory suffers from a number of disadvantages. The first and most obvious is that it is too mathematically restrictive for real-life modelling [5]. The second reason is that heavily simplified uncertainty assumptions are made [5]. These disadvantages require that a different approach be used in a number of queuing applications. The technique described in [5] examines the use of simulation as an alternative to mathematical analysis of queues. It was found that the advantages of simulation help overcome the disadvantages of classic queuing theory and that a combination of the two provides the best analytical method to achieve the greatest accuracy.

References

[1] Flood, J.E. Telecommunications Switching, Traffic and Networks, Chapter 4: Telecommunications Traffic, New York: Prentice-Hall, 1998.

[2] Bose S.J., Chapter 1 - An Introduction to Queuing Systems, Kluwer/Plenum Publishers, 2002.

[3] Penttinen A., Chapter 8 – Queuing Systems, Lecture Notes: S-38.145 - Introduction to Teletraffic Theory, Helsinki University of Technology, Fall 1999.

[4] Penttinen A., Kendall’s Notation for Queuing Models, Lecture Notes: S-38.145 - Introduction to Teletraffic Theory, Helsinki University of Technology, Fall 1999.

[5] van Dijk N.M., To pool or not to pool? “the benefits of combining queuing and simulation”, Proceedings of the Winter Simulation Conference, 2002.