Demo on Queuing Concepts

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Presentation transcript:

Demo on Queuing Concepts Refer to handout on web page. Basic queuing system: Customers arrive to a bank, they will wait if the teller is busy, then are served and leave. Scenario 1: Constant interarrival time and service time Scenario 2: Variable interarrival time and service time Objective: To understand concept of average waiting time, average number in line, utilization, and the effect of variability. IE 429, Parisay, January 2010

Scenario 1: Constant interarrival time (2 min) and service time (1 min) Scenario 2: Variable interarrival time and service time

Queuing Theory Arrival process Service process Queue Discipline Chap. 20, page 1051 Queuing Theory Arrival process Service process Queue Discipline Method to join queue IE 417, Chap 20, Jan 99

Queuing Theory Basic queuing system: Customers arrive to a bank, they will wait if the teller is busy, then are served and leave. Assume: Interarrival times ~ exponential Service times ~ exponential E(service times) < E(interarrival times) IE 429, Parisay, January 2010

Each Distribution for Random Variable Has: Definition Parameters Density or Mass function Cumulative function Range of valid values Mean and Variance IE 417, Chap 20, Jan 99

Triangular Distribution

Normal Distribution EGR 549

Continuous Uniform Distribution f(x) a b F(x) a b EGR 549

Exponential Distribution. = mean interval between consequent events = rate = mean number of counts in the unit interval > 0 X = distance between events >0 f(x) F(x) EGR 549

Notations used for QUEUING SYSTEM in steady state (AVERAGES) = Arrival rate approaching the system e = Arrival rate (effective) entering the system = Maximum (possible) service rate e = Practical (effective) service rate L = Number of customers present in the system Lq = Number of customers waiting in the line Ls = Number of customers in service W = Time a customer spends in the system Wq = Time a customer spends in the line Ws = Time a customer spends in service IE 417, Chap 20, May 99

Notations used for QUEUING SYSTEM in steady state j = State of the system, number of people in the system = Traffic intensity = / = P(j) = Probability that j units are in the system = P(0) = Probability that there are no units (idle) in the system Pw = P(j>S) = Probability that an arriving unit has to wait for service C = System capacity (limit) = Probability that a system is full (lost customer) = Probability that a particular server is idle EGR549, Chap 20, May 2012