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4. Models with General Arrival or Service Pattern

4. Models with General Arrival or Service Pattern. 4.1 M / G /1 4.2 M / G / c 4.3 G / M /1 , G / M / c. 4.1 M / G /1. General Service Time. b ( t ) : the probability density function of service time S. B ( t ) : the cumulative distribution function of service time S.

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4. Models with General Arrival or Service Pattern

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  1. 4. Models with General Arrival or Service Pattern 4.1 M/G/1 4.2 M/G/c 4.3 G/M/1, G/M/c

  2. 4.1 M/G/1 • General Service Time b(t) : the probability density function of service time S B(t) : the cumulative distribution function of service time S • Poisson Arrival Process X(t) : the number in the system

  3. 4.1 M/G/1 The service times of customers are assumed to be independent. Notably,

  4. 4.1.1 The Pollaczek-Khintchine Formula

  5. 4.1.1 The Pollaczek-Khintchine Formula Since Poisson arrival sees time average, we have L = L(D).

  6. 4.1.1 The Pollaczek-Khintchine Formula • Example Consider a M/M/1 queue with arrival rate of  = 10/hr and mean service time 5 min.The training course will decreases the standard deviation of service time from 5min to 4 min but increases the mean service time from 5 min to 5.5 min. Should they have further training? Performance is much more sensitive to mean than to its variation.

  7. 4.1.2 Departure-Point S-S {n} • Transition probabilities of M/G/1 Queue

  8. 4.1.2 Departure-Point S-S {n} • Example Machines break down according to a Poisson process with rate of  = 5/hr. The repairing time is 9 min with probability 2/3 and 12 min with probability 1/3.Find the probability that more than three machines will be down at any time?

  9. 4.1.2 Departure-Point S-S { n } • Example (Cont’d)

  10. 4.1.3 Proof that n = pn • Proof X(t): system size at time t An(t): number of unit upward jumps from state n occuring in (0, t) Dn(t): number of unit downward jumps to state n occuring in (0, t)

  11. Generalization of Little’s Formula 4.1.4 Waiting Times • Generalization of Little’s Formula Laplace-Stieltjes Transform where W*(s) is the LST of W(t). where L(k) is the factorial k-th moment of the system size and Wk is the regular k-th moment of W(t).

  12. 4.1.4 Waiting Times • Waiting Time Distribution of M/G/1 Here, B*(s) is the LST of the service time B(t).

  13. 4.1.4 Waiting Times • Waiting Time Distribution of M/G/1 where R*(s) is the LST of the residual-service-time R(t) with CDF

  14. 4.1.4 Waiting Times • Waiting Time Distribution of M/G/1 • Example In the previous example, if it loses $5000 per hour that a machine is down, and that an additional penalty must be incurred because of the possibility of an excessive number of machines being down. It is decided to cost this variability at $10000(std deviation of delay).

  15. 4.1.5 Busy Period Analysis

  16. Mean of Busy Period 4.1.5 Busy Period Analysis

  17. 4.1.6 M/G/1/K Queueing Models • Transition Matrix

  18. 4.1.6 M/G/1/K Queueing Models • Departure Point and Arrival Point Probabilities

  19. 4.1.7 Some Additional Results • M/G/1 with Balking b: probability that an arrival decides to actuaaly join the system • M/G/1 with Non-preemptive Priorities M/M/1priority model can be extended to nonexponential service time system. But

  20. 4.1.7 Some Additional Results • Output Process of M/G/1 M/M/1 has a Poisson output process, but M/G/1 no. C(t) : CDF of interdeparture times of M/G/1 queue

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