Cheng-Fu Chou, CMLab, CSIE, NTU Basic Queueing Theory (I) Cheng-Fu Chou

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 2 Outline  Little result  M/M/1  Its variant  Method of stages

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 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.

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 4 Time Diagram for queues  C n : 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   n : arrival time for C n  t n : inter-arrival time between C n-1 and C n, i.e., A(t) = P[t n  t]  x n : service time for C n, B(t) = P[x n  t]  w n : waiting time for C n  s n : system time for C n= w n +x n –Draw the diagram

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 5 Little Result   (t) : no. of arrivals in (0,t)   (t): no. of departures in (0,t)  t : 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

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 6 M/M/1  The average inter-arrival time is t = 1/ and t is exponentially distributed.  The average service time is x = 1/  and x is exponentially distributed.  Find out –p k : 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

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 7 M/M/1  Poisson arrival

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 8 Discouraged Arrival  A system where arrivals tend to get discouraged when more and more people are present in the system –arrival rate: k =  /(k+1), where k = 0,1,2,… –service rate:  k = , where k = 1,2,3,…

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 9 Discouraged Arrival

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 10 M/M/   Infinite number of servers –there is always a new server available for each arriving customer. –arrival rate : –service rate of each server: 

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 11 M/M/   We know –Arrival rate k =, k = 0, 1, 2, … –Departure rate  k = k , k = 1, 2, 3, …

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 12 M/M/m  The m-server case –The system provides a maximum of m servers

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 13 M/M/m  Arrival rate k = and service rate  k = min(k , m  )

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 14 M/M/1/K  Finite storage: a system in which there is a maximum number of customers that may be stored ( K customers)

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 15 M/M/1/K

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 16 M/M/m/m  m-server loss system

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 17 M/M/m/m (m-server loss system)  m-server loss systems

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 18 M/M/1//m  Finite customer population and single server –A single server –There are total m customers

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 19 M/M/1//m (finite customer population)

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 20 PASTA  Poisson Arrival See Time Average

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 21 Method of stages  Erlangian distribution

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 22 Er: r-stage Erlangian Dist.  r-stage Erlangian dist.

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 23 M/Er/1

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 24 E 2 /M/1

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 25 Bulk arrival systems  Bulk arrival system –g i = P[bulk size is i] –e.g. random-size families arriving at the doctor’s office for individual specific service

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 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/B 2 /1

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 27 M/B 2 /1

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 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?

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 29 Response time (cont.)  r n : the proportion of arrivals who find n other customers in system on arrival  p n : the proportion of time there are n customers in system  Due to PASTA, {r n } = {p n }, given that there are n customers in the systems

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 30 Response Time  Unconditioning on n

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 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(c  – ) –hint

Cheng-Fu Chou, CMLAB, CSIE, NTU P. 32  W = P(D>0)/(c  - )  And