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ENGM 732 Queuing Applications. Motivation Idea: We want to minimize the total cost of a queuing system Let SC = cost of service WC = cost of waiting TC.

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Presentation on theme: "ENGM 732 Queuing Applications. Motivation Idea: We want to minimize the total cost of a queuing system Let SC = cost of service WC = cost of waiting TC."— Presentation transcript:

1 ENGM 732 Queuing Applications

2 Motivation Idea: We want to minimize the total cost of a queuing system Let SC = cost of service WC = cost of waiting TC = total cost of system

3 Motivation Idea: We want to minimize the total cost of a queuing system Let SC = cost of service WC = cost of waiting TC = total cost of system min E[TC] = E[SC] + E[WC]

4 Motivation E[TC] = E[SC] + E[WC] E[TC] E[SC] E[WC] Service Level Cost

5 Example Suppose we have 10 CNC machines, 8 of which are required to meet the production quota. If more than 2 machines are down, the estimated lost profit is $400 per day per additional machine down. Each server costs $280 per day. Time to failure is exponential ( =0.05). Service time on a failed machine is also exponential (  =0.5). Should the firm have 1 or 2 repairmen ?

6 Example (rate diagrams) 031210 8/20 8/20 8/20 7/20 1/20 1/2 1/2 1/2 1/2 1/2 031210 8/20 8/20 8/20 7/20 1/20 1/2 1 1 1 1 M/M/1 Queue M/M/2 Queue

7 Example (rate diagrams) 031210 8/20 8/20 8/20 7/20 1/20 1/2 1/2 1/2 1/2 1/2 M/M/1 Queue PCP nn  0 CC n nn nn n n n        120 11 1 1......

8 Example (rate diagrams) 031210 8/20 8/20 8/20 7/20 1/20 1/2 1/2 1/2 1/2 1/2

9 Example (rate diagrams) 031210 8/20 8/20 8/20 7/20 1/20 1/2 1/2 1/2 1/2 1/2

10 Example (rate diagrams) 031210 8/20 8/20 8/20 7/20 1/20 1/2 1/2 1/2 1/2 1/2

11 Example (rate diagrams) 031210 8/20 8/20 8/20 7/20 1/20 1/2 1/2 1/2 1/2 1/2

12 Example (rate diagrams) 031210 8/20 8/20 8/20 7/20 1/20 1/2 1 1 1 1

13 Waiting Costs ( g(N) form ) The current rate at which costs are being incurred is determined primarily by the current state N. gN n nn (),,, (),,,...,    R S T 0012 40023410

14 Example (rate diagrams) 031210 8/20 8/20 8/20 7/20 1/20 1/2 1/2 1/2 1/2 1/2

15 Example (rate diagrams) 031210 8/20 8/20 8/20 7/20 1/20 1/2 1/2 1/2 1/2 1/2

16 Example (rate diagrams) 031210 8/20 8/20 8/20 7/20 1/20 1/2 1/2 1/2 1/2 1/2 031210 8/20 8/20 8/20 7/20 1/20 1/2 1 1 1 1

17 Waiting Costs ( g(N) form ) The current rate at which costs are being incurred is determined primarily by the current state N.

18 Waiting Costs For g(n) linear; g(n) = C w nP n EWCEgN gnP n n [][()] ()      0

19 Waiting Costs For g(n) linear; g(n) = C w nP n EWCEgN gnP n n [][()] ()      0 E gnPCnP C CL n n wn n wn n w []()            00 0

20 Example 2 A University is considering two different computer systems for purchase. An average of 20 major jobs are submitted per day (exp with rate =20). Service time is exponential with service rate dependent upon the type of computer used. Service rates and lease costs are shown below. ComputerService RateLease Cost MBI computer (  = 30) $5,000 / day CRAB computer (  = 25)$3,750 / day

21 Example 2 Scientists estimate a delay in research costs at $500 / day. In addition, due to a break in continuity, an additional component is given for fractional days. h(w) = 500w + 400w 2 where w = wait time for a customer

22 Waiting Costs ( h(w) model ) Ehwforcustomerwait hwfwdw w [()] ()()    z expectedcost 0

23 Waiting Costs ( h(w) model ) Since  customers arrive per day Ehwforcustomerwait hwfwdw w [()] ()()    z expectedcost 0 EWCEhw hwfwdw w [][()] ()()    z 0

24 Waiting Costs ( h(w) model ) Recall, for an M/M/1 queue, the distribution of the wait time is given by fwe w w ()() ()     EWChwfwdw wwe w w []()() ()() ()      z z   0 2 0 20500400

25 Example 2 (rate diagram) 031210 20 20 20 20 20 25 25 25 25 25 031210 20 20 20 20 20 30 30 30 30 30 MBI Comp. CRAB Comp.

26 MBI Computer (  – = 10) EWCwwedw w []()  z  2050040010 2

27 MBI Computer (  – = 10) EWCwwedw wedwwe w ww []() ()()   z zz   2050040010 20500102040010 2 2

28 MBI Computer (  – = 10) z EWCwwedw wedwwe we we w ww ww []() ()() ()()    zz zz    2050040010 20500102040010 20500102040010 2 2 21 31

29 MBI Computer (  – = 10) z

30 CRAB Computer (  – = 5) z EWCwwedw w []()   205004005 25

31 CRAB Computer (  – = 5) z EWCwwedw we we w ww []() ()()   zz   205004005 205005204005 25 215315

32 CRAB Computer (  – = 5) z EWCwwedw we we w ww []() ()(), (), ()    zz   205004005 205005204005 50000 2 5 40000 3 5 25 215315 23 

33 CRAB Computer (  – = 5) z EWCwwedw we we w ww []() ()(), (), (), $,      zz   205004005 205005204005 50000 2 5 40000 3 5 2 640 2 25 215315 23 

34 Expected Total Cost EWC MBI CRAB [],,  1160 2640 ETC MBI CRAB [],,,,,,     11605000 26403750 6160 6390

35 Decision Models Unknown s Let C s = cost per server per unit time Obj: Find s s.t. min E[TC] = sC s + E[WC]

36 Example (Repair Model) min E[TC] = sC s + E[WC] ssCsE[WC]E[TC] 1280 280 561 2560 48 608 3840 0 840

37 Decision Models Unknown  & s Let f(  ) = cost per server per unit time A = set of feasible  Obj: Find , s s.t. min E[TC] = sf(  ) + E[WC]

38 Example For MBI  = 30 CRAB  = 25 f(),,,,       500030 375025 ETCfEWC[]()[],,,,,,        5000116030 3750264025

39 Example For MBI  = 30 CRAB  = 25 f(),,,,       500030 375025 ETCfEWC[]()[] ,,,,      616030 639025

40 Decision Models Unknown & s Choose both the number of servers and the number of service facilities Ex: What proportion of a population should be assigned to each service facility # restrooms in office building # storage facilities

41 Decision Models Unknown & s Let C s = marginal cost of server / unit time C f = fixed cost of service / facility – unit time p = mean arrival rate for population n = no. service facilities = p /

42 Decision Models Unknown & s Cost / facility = fixed + marginal cost of service + expected waiting cost + travel time cost = C f + C s +E[WC] + C t E[T]

43 Decision Models Unknown & s Cost / facility = C f + C s +E[WC] + C t E[T] Min E[TC] = n{ C f + C s +E[WC] + C t E[T] }

44 Example Alternatives one tool crib at location 2 two cribs at locations 1 & 3 three cribs at locations 1, 2, & 3 12 3

45 Example Each mechanic is assigned to nearest crib. Walking rate = 3 mph 12 3 ET alt [].,.,.,  0041 02782 0023

46 Example Fixed cost / crib = $16 / hr (C f ) Marginal cost / crib= $20 / hr (C s ) Travel cost = $48 / hr (C t ) p = 120 / hr.  = 120 / hr (1 crib) 12 3

47 Example 12 3 ETCnsEWCCET nsE n ET t []{[][]} {[]()[]}   1620 1620 120 48

48 Example 12 3 ETCnsEWCCET nsE n ET t []{[][]} {[]()[]}   1620 1620 120 48 EWCCL w []  ETCnsL n ET[]{()[]}  162048 120 48 But,

49 Example 12 3 ETCnsL n ET[]{()[]}  162048 120 48 Consider 1 facility, 2 servers ( M/M/2 ) P 0 = 0.333 L q = 0.333 L = L q + /  = 1.333

50 Example 12 3 P 0 = 0.333 L q = 0.333 L = L q + /  = 1.333 ETCLET[]{()()[]} (.)()(.).    1162024812048 164048133312048004 35040

51 Example 12 3

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