Queuing Systems Analysis and Conclusions

Chapter 15 Queuing Systems 151 a Efficiency 10029 71 b For average waiting time 3 minutes at least 5 cashiers are needed For efficiency 70 the associated idle percentage is 10 The corresponding number of cashiers is at most 2 Conclusion The two conditions cannot be satisfied simultaneously At least one of the two conditions must be relaxed Set 153a Av interarrival time time units arrival rate λ in customersunit time Let λ 20 arrivalsλh I av interarrival time i λ 6070 6 arrivalsλh I 10 minutes 1λ hour ii λ 603 20 arrivalsλh I 6 3 minutes 120 λh iii λ 2 106030 20 arrivalsλh I 3070 3 minutes 1λ hour iv λ 15 2 arrivalsλh I 5 λ b Let S av service time i μ 6012 5 servicesλh S 12 minutes 2 λh ii μ 6075 8 servicesλh S 75 min 125 λh iii μ 53060 10 servicesλh S 330 6 min 1 λh iv μ 13 333 servicesλh S 3 λ a λ 2 failuresλh μnet 2247 336 failuresμ Prob Pat least one failure in 2 hours Ptime until failure 2 Pt 2 1 e22336 33 Pt 3hrs 1 Pt 3 e233 55 d Pt 1 1 e21 18 CA 18 per hour CB 25 per hour Length of queue A 4 jobs Length of queue B 7x4 28 jobs Cost of A 18 4x10 58 per hour Cost of B 25 25x10 53 per hour Decision Select Model B λ 40 arrivalsλh Pt 1 e40t Pt 3 e40160 5134 Pt 15 e401560 4866 Jims exp payoff arriving customers 25134 24866 10536 cent Jims exp payoff8 hours 00536840 1715 cent Conclusion Jim will pay Ann an average of 17 cents every 8 hours 152 a λ 6 arrivalsλh b Pt 1560 e61560 223 c Pt 6060 1 e66060 865 a Pt 260 1 6886 b Pt 260 Pt 3 Pt 1560 Pt 260 e401560 e40260 4034 Pt 2 e35260 1738 Situation Customer Server a Plane Runway b Passenger Taxi c Machinist Clerk at tool crib d Student Clerk e Caus Judge f Shipper Cashier g Car Parking Space a λ green 1 stepmin λ 17 shpmin λ combined 114 014286 shpmin P25 24286 x 5 e24286 x 5 0219 The two bases could be 2G 2R or 1G and 1R b Pt 2 1 e243x2 03849 153 Ent λt n eλt λtnn λt e2λt Ent n2 λt n eλtn1 λt e2λt n n1 λtn1n1 λt eλt λt2 λt λt2 λt Queuing Situation Customers p0t λ P0t 1 p1t λ P1t λ P0t 2 or p0t λ P1t λ eλt e eλt A eλt C i1 Because P00 1 A 1 P0t e2λt 1 Arrival of orders Orders 2 Processing single machine Push orders 3 Processing Prod line Read jobs 4 Processing Prod line Read jobs 5 Processing compiled jobs Completed orders 6 Tool crib Tools 7 Machine Breakdown machines Set 154b Situation Calling Source Customers arrival a Individual b Individual c Individual d Individual e Individual f Individual g Individual h Individual Set 155a Situation Interarrival time Service time a Probabilistic Time to clear runway b Probabilistic Ride time c Probabilistic Time to receive tool d Deterministic Time to process letter e Probabilistic Time to process engine f Probabilistic Tire time g Probabilistic Checkout time h Probabilistic Parking time λ 3652479000 973 failuresyr Pt 13 1 e9731 622 Set 155a Situation Queue Capacity Queue Discipline a FIFO b FIFO c FIFO d Random e FIFO f FIFO g FIFO h None Lack of memory property applies a The waiting time for the green bus is exponential with mean 10 minutes ft λ eλt t 0 b The waiting time for the red bus is exponential with mean 7 minutes ft λ eλt t 0 Set 155a a None b None c None d None e Jockey or bulk f None g Jockey h None Set 156a Set 156b L₅ proserved 19 cars of idle time new p₀new 100 100 025 25 The device can be justified based on the number of visiting customers L₅ in the existing system but not on the basis of idle time in the network Pj in queue j 2 Pn in system n 2 j2 pj Thus expected number n2 n pn p1 n2 pn p0 1 1 1 p 1 1 p wt M λ eM λt for t 0 wλ λ μ λ and μ λ Thus Exp waiting time in queue for those who must wait ρ1 ρ λ 1 μ λ ρ 1 ρ a λ₀ 3654 a P₀ 000002 d Wₑ Wₛ Lμ λₑₓₑ 510038 491 custhr Wₛ Lₛλₑₓₑ 1286491 02675 hours Wₑ 02675 13 1424 hours TORA input 8 5 2 0 TORA input 16 5 4 0 TORA input 20 12 3 0 Pa customer is waiting Pat least c1 in system nc1 pn nc pn fc P0 pc c 1ρc Pc P0 11ρc 1 P0 1 ρP c1ρ Wqτ j0 Wqτ j qj where qj c c1 µcj j 0 Hence for τ 0 Wqτ i1 µcµcτj1eµcτ c j1 P0 µc eµcτ j0 µcτjj P0 µc eµcτ c Pτ y y Wqτ dτ cµP0 c ecyλτ dt P0 cµ ccλβ ecλβτ a C LS LQ 4 424 154 13 cases b p0 004468 c N 10 d λ 6010 6μ e μ 6030 2λ N18 f p0 05 g N 10 spaces including the pumps f p0 p0 p1 p2 pn 1 pn29 1 097533 02467 a p0 0 b pn10 1 pn29 1 c pn40 pn29 07771 013787 063923 d LS 36 Net annual equity 100 x 36 103 09115 839780 R1 λeff λ22Ls 52212004 4998 R4 λeff 52221 99 b No of idle repair persons 4 LsLq 4 2111 201 c P0 10779 d R3 P2 or 3 available P0 P1 34492 R Ls Lq λeff Ws Wq λeff 1μ Since λeff μR P0 32768 λ 6045 1333 h μ 6020 3 h R4 K4 Set 157a Set 159a Set 159a C1 20 C2 45 λ 175 callshr N 10hr Rate of breakdownmachine λ 576 36125hr C1 12 C2 5 λ 147 1428 breakdownshr C1 12