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Symmetric Queues Served in Cyclic Order

Symmetric Queues Served in Cyclic Order. S. W. Fuhrmann AT&T Bell Laboratories Operations Research Letters, Oct. 1985. Present by Mike Hsiao, 2005/5/3. Outline. Introduction A stochastic decomposition result Mean relationship Mean waiting times. Outline. Introduction

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Symmetric Queues Served in Cyclic Order

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  1. Symmetric Queues Served in Cyclic Order S. W. Fuhrmann AT&T Bell Laboratories Operations Research Letters, Oct. 1985 Present by Mike Hsiao, 2005/5/3

  2. Outline • Introduction • A stochastic decomposition result • Mean relationship • Mean waiting times

  3. Outline • Introduction • A stochastic decomposition result • Mean relationship • Mean waiting times

  4. Symmetric Queues Served in Cyclic Order • Symmetric Queues • In this paper, we consider n queues handled by a single server in cyclic order. • The system is symmetric in the sense that defining characteristics for each queue are the same. • λ / n : arrival rate of each queue (Poisson) • H(.) : the service time distribution function (General) • h, h(2) : the first and second moment of H(.)

  5. Symmetric Queues Served in Cyclic Order (cont’d) • Changeover Time • also called “walk time” • begins when the server departs form a queue,and ends when it arrivals to the next queue. • C(.) : the changeover time distribution function • c, c(2) : the first and second moment of C(.)

  6. Symmetric Queues Served in Cyclic Order (cont’d) changeover time cycle time

  7. 2 3 1 2 1 1 3 2 exhaustive service stop here gated service stop here 2 1 3 3 2 3 2 1 3 2 3 2 4 3 4 1 2 3 1 2 2 1 2 1 2 1 3 2 3 3 1 2 2 1 1 2 1 3 2 2 1 3 2 1 3 2 1 1 1 nth queue 3 2 1 (n+1)th queue 1 Symmetric Queues Served in Cyclic Order (cont’d) • Cyclic Order • Gated service • Exhaustive service • Limited service • until either k customersor empty Server …

  8. The Result • For a M/G/1 queue • arrival rate λ • service time distribution function H(.) • System time = queueing time + service time

  9. The Result (cont’d) • The mean system time of three services with n queues

  10. The Result (cont’d) • And more

  11. Outline • Introduction • A stochastic decomposition result • Mean relationship • Mean waiting times

  12. Server vacation model for a single queue, server takes a vacation (leaves the queue) until he returns to that queue after cycling through the other (n-1) queues. Changeover vacation for the whole system, the server takes a vacation every time changeover occurs Point of view on vacation

  13. Notations • ψ(.) = the p.g.f for the stationary number of customers in the whole system at arbitrary time • ζ(.) = the p.g.f for the stationary number of customers in the whole system when the server begins an arbitrary vacation (changeover)

  14. Notations (cont’d) • α(.) = the p.g.f for the total number of customers that arrival during an arbitrary vacation (changeover) • π(.) = the p.g.f for the stationary number of customers in a standard M/G/1 system at arbitrary time

  15. Cooper[10] says • Moreover, by definition

  16. Find ψ’(1) • ψ’(1) is the mean value of stationary number of customers in the whole system at arbitrary time • So, we need to find ζ’(1).

  17. Outline • Introduction • A stochastic decomposition result • Mean relationship • Mean waiting times

  18. Mean relationship • Let T be cycle time, • S be the time the server stays at any particular queue per visit.

  19. Mean relationship (cont’d) • m1 = the mean number of customers present at (any) one of the queue just after the server departs from that queue in equilibrium. • m2 = the mean number of customers present at (any) one of the queue just after an arbitrary customer departs from that queue in equilibrium.

  20. Relationship between m1 & ξ’(1) • The mean number of customers present at queue (n-1) when server departs from queue (n-1) is equal to m1. • The mean number of customers present at queue (n-1) when server departs from queue n is equal to

  21. Relationship between m2 & ψ’(1)

  22. Outline • Introduction • A stochastic decomposition result • Mean relationship • Mean waiting times

  23. Results • Exhaustive Service • m1 = 0, (no any customer left in the queue) • Gated Service • , (arrival rate * time period) • And more, • E[WG] = E[WE] + E[S]

  24. Results • Limited Service with k = 1 • p0 = prob.{ no customer in this queue, when server arrives}

  25. Results • Limited Service with limit k

  26. Relationship under limited service I • Yj donate the # of customers present (at this queue) after the jth service attempt. (j=1 to k) • Y0 donate the # of customers present when the server arrives to the queue. • if Yj-1 > 0, then Yj will actually happen. • if Yj-1 = 0, then Yj=0. • E[Y0]> E[Y1]> …> E[Yk] • E[Yk] = m1.

  27. Relationship under limited service II • N donate the # of customers served per visit by server. • Zj donate the # of customers present (at this queue) after the jth customer departs.

  28. Relationship under limited service II

  29. Thanks for your attention.

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