Nur Aini Masruroh Queuing Theory

Outlines IntroductionBirth-death processSingle server modelMulti server model

Introduction  Involves the mathematical study of queues or waiting line.  The formulation of queues occur whenever the demand for a service exceeds the capacity to provide that service.  Decisions regarding the amount of capacity to provide must be made frequently in industry and elsewhere.  Queuing theory provides a means for decision makers to study and analyze characteristics of the service facility for making better decisions.

Basic structure of queuing model  Customers requiring service are generated over time by an input source.  These customers enter the queuing system and join a queue.  At certain times, a member of the queue is selected for service by some rule know as the service disciple.  The required service is then performed for the customer by the service mechanism, after which the customer leaves the queuing system

The basic queuing process Input source Queue Service mechanism Customers Served Customers Queuing system

Characteristics of queuing models  Input or arrival (interarrival) distribution  Output or departure (service) distribution  Service channels  Service discipline  Maximum number of customers allowed in the system  Calling source

Kendall and Lee’s Notation Kendall and Lee introduced a useful notation representing the 6 basic characteristics of a queuing model. Notation: a/b/c/d/e/f where a = arrival (or interarrival) distribution b = departure (or service time) distribution c = number of parallel service channels in the system d = service disciple e = maximum number allowed in the system (service + waiting) f = calling source

Conventional Symbols for a, b M = Poisson arrival or departure distribution (or equivalently exponential distribution or service times distribution) D = Deterministic interarrival or service times Ek = Erlangian or gamma interarrival or service time distribution with parameter k GI = General independent distribution of arrivals (or interarrival times) G = General distribution of departures (or service times)

Conventional Symbols for d  FCFS = First come, first served  LCFS = Last come, first served  SIRO = Service in random order  GD = General service disciple

Transient and Steady States Transient state  The system is in this state when its operating characteristics vary with time.  Occurs at the early stages of the system’s operation where its behavior is dependent on the initial conditions. Steady state  The system is in this state when the behavior of the system becomes independent of time.  Most attention in queuing theory analysis has been directed to the steady state results.

Queuing Model Symbols n = Number of customers in the system s = Number of servers pn(t) = Transient state probabilities of exactly n customers in the system at time t pn = Steady state probabilities of exactly n customers in the system λ = Mean arrival rate (number of customers arriving per unit time) μ = Mean service rate per busy server (number of customers served per unit time)

Queuing Model Symbols (Cont’d) ρ = λ/μ = Traffic intensity W = Expected waiting time per customer in the system Wq = Expected waiting time per customer in the queue L = Expected number of customers in the system Lq = Expected number of customers in the queue

Relationship Between L and W If λ n is a constant λ for all n, it can be shown that L = λW L q = λ W q If λ n are not constant then λ can be replaced in the above equations by λ bar,the average arrival rate over the long run. If μn is a constant μ for all n, then W = W q + 1/μ

Relationship Between L and W (cont’d) These relationships are important because:  They enable all four of the fundamental quantities L, W, Lq and Wq to be determined as long as one of them is found analytically.  The expected queue lengths are much easier to find than that of expected waiting times when solving a queuing model from basic principles.

Birth and Death Process Most elementary queuing models assume that the inputs and outputs of the queuing system occur according to the birth and death process.  Birth :Refers to the arrival of a new customer into the queuing system.  Death: Refers to the departure of a served customer. Except for a few special cases, analysis of the birth and death process is very difficult when the system is in transient condition. However, it is relatively easy to derive the probability distribution of p n after the system has reached a steady state condition.

Rate Diagram for the Birth and Death Process  Rate In = Rate Out Principle  For any state of the system n, the mean rate at which the entering incidents occurs must equal the mean rate at which the leaving incidents.

Balance equation The equations for the rate diagram can be formulated as follows: State 0: μ 1 p 1 = λ 0 p 0 State 1: λ 0 p 0 + μ 2 p 2 = (λ 1 + μ 1 )p 1 State 2: λ 1 p 1 + μ 3 p 3 = (λ 2 + μ 2 )p 2 …. State n: λ n-1 p n-1 + μ n+1 p n+1 = (λ n + μ n )p n ….

Balance equation (cont’d) cncn

 Expected number of customers in the system  Expected number of customers in the queue:  Furthermore where is the average arrival rate over the long run It given by

Single server queuing models  M/M/1/FCFS/∞/∞ Model when the mean arrival rate λn and mean service μn are all constant we have

Single server queuing models (cont’d) Consequently

Single server queuing models (cont’d)

Multi server queuing models  M/M/s/FCFS/∞/∞ Model When the mean arrival rate λ n and mean service μ n, are all constant, we have the following rate diagram

Multi server queuing models (cont’d)

Some General Comments  Only very simple models allow analytic determination of quantities of interests.  That is, closed form solution can be obtained for simple queuing models only.  Transient versus steady state behavior  For some real world queuing systems, the transient behavior may be of interests to the decision makers.  For the more complex queuing systems, the quantities  of interests may be obtained through simulation.