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Basic Queueing Theory (I)

Basic Queueing Theory (I). Cheng-Fu Chou. Outline. Little result M/M/1 Its variant Method of stages. Queueing System. Kendall’s notations A/B/C/K C: number of servers K: the size of the system capacity; the buffer space including the servers A(t): the inter-arrival time dist.

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Basic Queueing Theory (I)

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  1. Basic Queueing Theory (I) Cheng-Fu Chou

  2. Outline • Little result • M/M/1 • Its variant • Method of stages

  3. Queueing System • Kendall’s notations • A/B/C/K • C: number of servers • K: the size of the system capacity; the buffer space including the servers • A(t): the inter-arrival time dist. • B(t): the service time dist. • M: exponential dist. • G: general dist. • D: deterministic dist.

  4. Time Diagram for queues • Cn: the n-th customer to enter the systsem • N(t): number of customers in the system at time t • U(t): unfinished work in the system at time t • tn: arrival time for Cn • tn: inter-arrival time between Cn-1 and Cn, i.e., A(t) = P[tn t] • xn: service time for Cn, B(t) = P[xn t] • wn: waiting time for Cn • sn: system time for Cn= wn+xn • Draw the diagram

  5. Little Result • a(t): no. of arrivals in (0,t) • d(t): no. of departures in (0,t) • lt: the average arrival rate during the interval (0,t) • r(t): the total time all customers have spent in the system during (0,t) • Tt: the average system time during (0,t) • proof

  6. M/M/1 • The average inter-arrival time is t = 1/l and t is exponentially distributed. • The average service time is x = 1/m and x is exponentially distributed. • Find out • pk : the prob. of finding k customers in the system • N : the avg. number of customers in the system • T : the avg. time spent in the system

  7. M/M/1 • Poisson arrival

  8. Discouraged Arrival • A system where arrivals tend to get discouraged when more and more people are present in the system • arrival rate: lk = a/(k+1) , where k = 0,1,2,… • service rate: mk = m, where k = 1,2,3,…

  9. Discouraged Arrival

  10. M/M/ • Infinite number of servers • there is always a new server available for each arriving customer. • arrival rate : l • service rate of each server: m

  11. M/M/ • We know • Arrival rate lk = l , k = 0, 1, 2, … • Departure rate mk = km , k = 1, 2, 3, …

  12. M/M/m • The m-server case • The system provides a maximum of m servers

  13. M/M/m • Arrival rate lk =l and service rate mk = min(km, mm)

  14. M/M/1/K • Finite storage: a system in which there is a maximum number of customers that may be stored ( K customers)

  15. M/M/1/K

  16. M/M/m/m • m-server loss system

  17. M/M/m/m (m-server loss system) • m-server loss systems

  18. M/M/1//m • Finite customer population and single server • A single server • There are total m customers

  19. M/M/1//m (finite customer population)

  20. PASTA • Poisson Arrival See Time Average

  21. Method of stages • Erlangian distribution

  22. Er: r-stage Erlangian Dist. • r-stage Erlangian dist.

  23. M/Er/1

  24. E2/M/1

  25. Bulk arrival systems • Bulk arrival system • gi = P[bulk size is i] • e.g. random-size families arriving at the doctor’s office for individual specific service

  26. Bulk Service System • Bulk service system • The server will accept r customers for bulk service if they are available • If not, the server accept less than r customers if any are available • HW : M/B2/1

  27. M/B2/1

  28. Response time in M/M/1 • The distribution of number of customers in systems : • How about the distribution of the system time ? • Idea: if an arrival who finds n other customers in system, then how much time does he need to spend to finish service?

  29. Response time (cont.) • rn: the proportion of arrivals who find n other customers in system on arrival • pn: the proportion of time there are n customers in system • Due to PASTA, {rn} = {pn}, given that there are n customers in the systems

  30. Response Time • Unconditioning on n

  31. Waiting time Dist. For M/M/c • For M/M/c queueing system, given a customer is queued, please find out his/her waiting time dist. is • (D| D>0) ~ exp(cm – l) • hint

  32. W = P(D>0)/(cm-l) • And

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