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Chap. 20, page 1051 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. Each Distribution for Random Variable Has: Definition Parameters Density or Mass function Cumulative function Range of valid values Mean and Variance.

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Chap. 20, page 1051 Queuing Theory Arrival process Service process Queue Discipline

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

  2. 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

  3. Exponential Dist. Poisson Dist. IE 417, Chap 20, Jan 99

  4. Relation betweenExponential distribution ↔ Poisson distribution Xi : Continuous random variable, time between arrivals, has Exponential distribution with mean = 1/4 X1=1/4 X2=1/2 X3=1/4 X4=1/8 X5=1/8 X6=1/2 X7=1/4 X8=1/4 X9=1/8 X10=1/8 X11=3/8 X12=1/8 0 1:00 2:00 3:00 Y1=3 Y2=4 Y3=5 Yi : Discrete random variable, number of arrivals per unit of time, has Poisson distribution with mean = 4. (rate=4) Y ~ Poisson (4) IME 301

  5. Kendell-Lee Notation for Queuing System 1 / 2 / 3 / 4 / 5 / 6 Arrival / Service / Parallel / Queue / Max / Population Process Process Servers Discip- Cus- Size line tomer M, D, Er, G, GI IE 417, Chap 20, Jan 99

  6. Queuing System j = State of the system, number of people in the system Pij(t) = Probability that j people are in the system at time t given that i people are in the system at time 0 Steady state probability of j people in the system IE 417, Chap 20, Jan 99

  7. Laws of Birth-Death Process 1- : birth rate (arrival) in state j 2- : death rate (service ends) in state j 3- death and births are independent of each other, no more than 1 event in M/M/1 is considered a birth-death process Will not cover mathematical details of Section 20.3 IE 417, Chap 20, Jan 99

  8. 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

  9. Notations used for QUEUING SYSTEM in steady state = 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 IE 417, Chap 20, Mayl 99

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