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 = total cost of system min E[TC] = E[SC] + E[WC]

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

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 ?

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

Example (rate diagrams) M/M/1 Queue 3 1 2 10 8/20 8/20 8/20 7/20 1/20 1/2 1/2 1/2 1/2 1/2 C n = - l m 1 2 . P C n =

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

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

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

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

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

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

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

Example 2 (rate diagram) MBI Comp. 3 1 2 10 20 20 20 20 20 30 30 30 30 30 CRAB Comp. 3 1 2 10 20 20 20 20 20 25 25 25 25 25

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

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

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

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

Example (Repair Model) min E[TC] = sCs + E[WC] s sCs E[WC] E[TC] 1 280 280 561 2 560 48 608 3 840 0 840

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]

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

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

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

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

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

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

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

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)

Example But, E WC C L [ ] = E TC n s L T [ ] { ( ) } = + 16 20 48 120 3 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

Example E TC n s L T [ ] { ( ) } = + 16 20 48 120 3 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

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

Example 1 2 3