1. 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 min E[TC] = E[SC] + E[WC]

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

4. 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 (l=0.05). Service time on a failed machine is also exponential (m=0.5). Should the firm have 1 or 2 repairmen ?

5. Example (rate diagrams)
M/M/1 Queue 3 1 2 10
8/ / / / /20
1/ / / / /2
M/M/2 Queue 3 1 2 10
8/ / / / /20 1/

6. Example (rate diagrams)
M/M/1 Queue 3 1 2 10
8/ / / / /20
1/ / / / /2 C n = - l m 1 2 . P C n =

7. Example (rate diagrams)3 1 2 10
8/20 8/ /20 7/ /20
1/ / /2 1/ /2

8. Waiting Costs ( g(N) form )
The current rate at which costs are being incurred is determined primarily by the current state N. g N n ( ) , . = - 1 2
400 3 4 10

9. å Waiting Costs å E WC g N n P [ ] ( ) = For g(n) linear; g(n) = CwnPnå
For g(n) linear; g(n) = CwnPn E WC g n P C nP L w [ ] ( ) = å

10. 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 l=20). Service time is exponential with service rate dependent upon the type of computer used. Service rates and lease costs are shown below.
Computer Service Rate Lease Cost
MBI computer (m = 30) $5,000 / day
CRAB computer (m = 25) $3,750 / day

11. 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 + 400w2 where w = wait time for a customer

12. Waiting Costs ( h(w) model )
for
customer
wait f dw [ ( )] ) =
expected
cost ʃ
Since l customers arrive per day E WC h w f dw [ ] ( ) = l ʃ

13. Waiting Costs ( h(w) model )
Recall, for an M/M/1 queue, the distribution of the wait time is given by f w e ( ) = - m l E WC h w f dw e [ ] ( ) )( = + - l m 2 20
500
400 ʃ ʃ

14. Example 2 (rate diagram)
MBI Comp. 3 1 2 10
CRAB Comp. 3 1 2 10

15. MBI Computer (m – l = 10) ʃ ʃ ʃ ʃ ʃ E WC w e dw we [ ] ( ) , $ = + 20- 20
500
400 10
100
000 2 80 3 1
160 G ʃ ʃ ʃ ʃ

16. CRAB Computer (m – l = 5) ʃ ʃ ʃ E WC w e dw [ ] ( ) , $ = + 20 500 400- 20
500
400 5 50
000 2 40 3
640 1 G ʃ ʃ

17. Expected Total Cost E WC MBI CRAB [ ] , = 160 2 640 1 E TC MBI CRAB [+ 1
160 5
000 2
640 3
750 6
390

18. Decision Models
Unknown s Let Cs = cost per server per unit time Obj: Find s s.t. min E[TC] = sCs + E[WC]

19. Example (Repair Model)
min E[TC] = sCs + E[WC] s sCs E[WC] E[TC]

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

21. Example For MBI m = 30 CRAB m = 25 f ( ) , m = 5 000 30 3 750 25 E TCWC [ ] ( ) m = + , 6
160 30
390 25

22. Decision Models
Unknown l & 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

23. Decision Models
Unknown l & s Let Cs = marginal cost of server / unit time Cf = fixed cost of service / facility – unit time lp = mean arrival rate for population n = no. service facilities = lp/l

24. Decision Models
Unknown l & s Cost / facility = fixed + marginal cost of service + expected waiting cost + travel time cost = Cf + Cs +E[WC] + lCtE[T]

25. Decision Models
Unknown l & s Cost / facility = Cf + Cs +E[WC] + lCtE[T] Min E[TC] = n{ Cf + Cs +E[WC] + lCtE[T] }

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

27. Example 1 2 3
Each mechanic is assigned to nearest crib. Walking rate = 3 mph E T
alt [ ] . , = 04 1
0278 2 02 3

28. Example 1 2 3
Fixed cost / crib = $16 / hr (Cf) Marginal cost / crib = $20 / hr (Cs) Travel cost = $48 / hr (Ct) lp = 120 / hr. m = 120 / hr (1 crib)

29. Example But, E WC C L [ ] = E TC n s L T [ ] { ( ) } = + 16 20 48 1203 E TC n s WC C T t [ ] { } ( ) = + 16 20
120 48 l
But, E WC C L w [ ] = E TC n s L T [ ] { ( ) } = + 16 20 48
120

30. Example E TC n s L T [ ] { ( ) } = + 16 20 48 1203 E TC n s L T [ ] { ( ) } = + 16 20 48
120
Consider 1 facility, 2 servers ( M/M/2 )
P0 = 0.333
Lq = 0.333
L = Lq + l/m = 1.333

31. Example E TC L T [ ] { ( ) } . )( = + 1 16 20 2 48 120 40 333 04 350
Lq = 0.333
L = Lq + l/m = 1.333 E TC L T [ ] { ( ) } . )( = + 1 16 20 2 48
120 40
333 04
350

32. Example 1 2 3