1. Modeling and Simulation Queuing theory
Princess Nora University
Modeling and Simulation Queuing theory
Arwa Ibrahim Ahmed

2. Queuing theory :
Queuing theory is the mathematical study of waiting lines, or queues. In queuing theory a model is constructed so that queue lengths and waiting times can be predicted. Queuing theory started with research by Agner Krarup Erlang when he created models to describe the Copenhagen telephone exchange. The ideas have since seen applications including telecommunications, traffic engineering, computing and the design of factories, shops, offices and hospitals.

3. Queuing theory :
The word queue comes, via French, from the Latin cauda, meaning tail. The spelling "queuing" over "queuing" is typically encountered in the academic research field.

4. Queuing theory
Queuing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide service. It is applicable in a wide variety of situations that may be encountered in business, commerce, industry, healthcare, public service and engineering. Applications are frequently encountered in customer service situations as well as transport and telecommunication. Queuing theory is directly applicable to intelligent transportation systems, call centers, networks, telecommunications, server queuing, and traffic flow.

5. Queuing theory
Notation for describing the characteristics of a queuing model was first suggested by David G. Kendall in Kendall's notation introduced an A/B/C queuing notation that can be found in all standard modern works on queuing theory, for example. The A/B/C notation designates a queuing system having A as inter-arrival time distribution, B as service time distribution, and C as number of servers. For example, "GI/G/S" would indicate a General (may be anything) arrival process, a General Distribution service process and a single server.

6. would indicate a type of distribution , e.g. :
Also:
M/M/1
would indicate a type of distribution , e.g. :
Mar cove distribution for arrival process, a Mar cove distribution for service process and a single server.

7. Queuing theory
Queuing is handled by control processes within exchanges, which can be modeled using state equations.Queueing systems use a particular form of state equations known as a Markov chain that models the system in each state. Incoming traffic to these systems is modeled via a Poisson distribution and is subject to Erlang’s queuing theory assumptions :

8. Queuing theory
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.

9. Queuing theory
Classic queuing theory involves complex calculations to determine waiting time, service time, server utilization and other metrics that are used to measure queuing performance. Networks of queues are systems which contain an arbitrary, but finite, number m of queues. Customers, sometimes of different classes, travel through the network and are served at the nodes. The state of a network can be described by a vector , where ki is the number of customers at queue i. In open networks, customers can join and leave the system, whereas in closed networks the total number of customers within the system remains fixed.