queuing theory/ waiting line theory

QUEUING THEORY

Introduction • Queuing theory, also known as waiting line theory. • The theory owes its development to A K Erlang. • The theory is applicable to situations where

Customers arrive at a service station, Wait for their turn, are serviced and then leave the system.

GENERAL STRUCTURE OF QUEUING SYSTEM

• Waiting lines develop because the service to a customer may not be rendered immediately as the customer reaches service facility.

ELEMENTS OF QUEUING MODEL• 1.Arrival Process• According to source :finite & infinite. • According to numbers : individual or group. • According to time : certain & uncertain. • 2.Service System • According to Structure of service system. • According to Speed of service system.

Service structure • i) Single server facility • ii) Multiple, parallel facility with single queue• iii) Multiple, parallel facility with multiple queues• iv) Service facility in a series having multiple

servers

• 3.Queue structure • i) FCFS• ii)LCFS • iii)SIRO • iv) Priority

OPERATING CHARACTERISTICS• Length of the queue: avg.no. of customers in the queue

waiting for the service (Lq) • • Length of the system: avg no. Of customers in the system,

those waiting to be& those being serviced.(Ls)

• Waiting time in the queue: avg time a customer has to wait in the queue to get service. (Wq)

• Waiting time in the system: avg time a customer spends in the system, from entry to completion of the service.(Ws)

Assumptions :

• Arrivals and service both are assumed to follow distributions. • Arrivals follow Poisson Distribution. • Service follow Exponential Distribution. • Steady state condition. It means these operating

parameters would reach stable values.

Models

Deterministic Probabilistic

Arrival rate• The timings of arrival is described by specifying the

average rate of arrivals per unit of time.

Service Rate• Mean no.of customers that can be served per unit

time. Denoted by (u). • Ideally service rate should be more than the

arrival rate

Traffic intensity • This ratio is also called the average utilisation or

clearing ratio. • If the ratio is >1 the system would ultimately fail. • If the ratio is <1 the system works. Ratio is the

proportion of time server is busy.

Poisson - Exponential Single Server Model with lnfinite Population

M/M/1

M/M/1 formulae

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