06IP/IM74 OPERATIONS RESEARCH. UNIT - 5: Queuing Theory

Transcription

1 0I/IM74 OERATIONS RESEARCH UNIT - 5: Queuig Theory The Basic structure of ueuig model Itductio Queues are a part of everyday life. e all wait i ueues to buy a movie ticket, to make bak deposit, pay for gceries, mail a package, obtai a food i a cafeteria, to have ride i a amusemet park ad have become adjustmet to wait but still get aoyed by uusually log waits. The Queuig models are very helpful for determiig how to operate a ueuig system i the most effective way if too much service capacity to operate the system ivolves excessive costs. The models eable fidig a apppriate balace betwee the cost of service ad the amout of waitig. Iput Iput Source Queue Disciplie Service Mecahaiso Served Uits

2 Iformatio reuired to solve the ueuig pblem: Characteristics of the ueuig system-: (a) Iput source (a) Iput source (b) Queue disciplie (c) Service mechaism Oe characteristic of the iput source is the size. The size is the total umber of uits that might reuire service fm time to time. It may be assumed to be fiite or ifiite. The customer assumptio is that they geerate accordig to oisso Distributio D at a certai average rate

3 Therefore, the euivalet assumptio is that they geerate accordig to expoetial distributio betwee cosecutive arrivals. To solve the pblems use & assume customer populatio as (b) Queue Disciplie A ueue is characterized by maximum permissible umber of uits that it cotais. Queues are called fiite or ifiite, accordig to whether umber is fiite or ifiite. The service disciplie refers to the order i which umber of ueues are selected for service. Ex: It may be FIFO, radom or priority; FIFO is usually assumed uless stated otherwise. (c) Service mechaism This cosists of oe or more service facilities each of which cotais oe or more parallel service chael. If there is more tha oe service facility, the arrival uit may receive the service fm a seuece of service chaels. At a give facility, the arrival eters at the service facility ad is completely served by that server. The time elapsed fm the commecemet of the service to its completio for a uit at the service facility is kow as service time usually, service time follows as expoetial distributio. Classificatio of ueuig models usig kedal & ee otatios Geerally, ay ueuig models may be completely specified i the followig symbolic form a / b/ c : d / e a Type of distributio of iter arrival time b Type of distributio of iter service time c Number of servers d Capacity of the system e Queue disciplie M Arrival time follows oisso distributio ad service time follows a expoetial distributio. 3

4 Model I : here M M / M / : / FCFS Arrival time follows a oisso distributio M Service time follows a expoetial distributio Sigle service model FCFS Capacity of the system is ifiite Queue disciplie is first come first served Model II : here N Model III : here SIRO M / M / : N / FCFS Capacity of the system is fiite M / M / : / SIRO Service i radom order Model IV : M / O / : / FCFS here D Model V : here G Service time follows a costat distributio or is determiistic M / G / : / FCFS Service time follows a geeral distributio or arbitrary distributio Model VI : M / E k / : / FCFS here E k Model VII : here K Model VIII : Service time follows Erlag distributio with K phases. M / M / K : / FCFS Multiple Server model M / M / K : N / FCFS Model I: M / M / : / FCFS Formulas:. Utilizatio factor traffic itesity / Utilizatio parameter / Busy period ρ here Mea arrival rate ; μ mea service rate Note : μ > i sigle server model oly 4

5 . bability that exactly ze uits are i the system o 3. bability that exactly uits i the system o 4. bability that or more uits i the system or more more the meas should be 5. Expected umber of uits i the ueue / ueue legth ( ). Expected waitig time i the ueue 7. Expected umber of uits i the system 5

6 8. Expected waitig time i the system 9. Expected umber of uits i ueue that fm time to time (OR) o - empty ueue size D 0. bability that a arrival will have to wait i the ueue for service bability - o. bability that a arrival will have to wait i the ueue more tha w ( where w > o), the waitig time i the ueue bability e ( ) w. bability that a arrival will have to wait more tha v(v > o) waitig time i the system is e ( ) v 3. bability that a arrival will ot have to wait i the ueue for service o

7 Model - blems. Arrivals at a telephoe both are cosidered to be oisso at a average time of 8 mi betwee our arrival ad the ext. The legth of the phoe call is distributed expoetially, with a mea of 4 mi. Determie (a) Expected fractio of the day that the phoe will be i use. (b) Expected umber of uits i the ueue Expected waitig time i the ueue. (c) Expected umber of uits i the system. (e) Expected waitig time i the system (f) Expected umber of uits i ueue that fm time to time. (g) hat is the pbability that a arrival will have to wait i ueue for service? (h) hat is the pbability that exactly 3 uits are i system (i) hat is the pbability that a arrival will ot have to wait i ueue for service? (j) hat is the pbability that there are 3 or more uits i the system? (k) hat is the pbability that a arrival will have to wait more tha mi i ueue for service? (l) hat is the pbability that more tha 5 uits i system (m) hat is the pbability that a arrival will have to wait more tha 8 mi i system? () Telephoe compay will istall a secod booth whe coviced that a arrival would have to wait for attest mi i ueue for phoe. By how much the flow of arrival is icreased i order to justify a secod booth. Solutio: The mea arrival rate /8 x / hour. The mea service x 0 5 / hour. a) Fractio of the day that the phoe will be i use 7.5 ρ (b) The expected umber pf uits i the ueue ( ) 5( 5 7.5) ( uits) perso 7

8 (c) Expected waitig time i the ueue hrs (d) Expected umber of uits i the system:- / perso (e) Expected waitig time i the system (f) Expected umber of uits i the ueue that form fm time to time:- D persos 8

9 (g) bability that a arrival will have to wait i the system:- o o (h) The bability that exactly ze waits i the system:- o (i) The pbability that exactly 3 uits i the system:- o 3 ( 0.5) (j) bability that a arrival will ot have to wait for service:- o 0.5 9

10 (k) bability that 3 or more uits i the system:- or more or more (l) bability that a arrival will have to wait more tha mis i ueue for service ω mi 0.5 e 0.3 e ( ) hrs 0 ( ) 0 (m) bability that more tha 5 uits i the system 0.5 ω 0.05 () bability that a arrival will directly eter for service o 0.5 0

11 (O) bability that arrival will have to wait more tha 8mis i the system. V 8 / 0 hrs (p) e e ( ) ( 7.5 5) per v hrs ( ) ( ) hour. 0. hr To justify a secod booth should be icreased fm 7.5 to 9 per hour ) I a self service store with oe cashier, 8 customers arrive o a average of every 5 mis. ad the cashier ca serve 0 i 5 mis. If both arrival ad service time are expoetially distributed, the determie a) Average umber of customer waitig i the ueue for average. (3.) b) Expected waitig time i the ueue (0.033) c) hat is the pbability of havig more tha customers I the system (0.09)

12 Solutio: Mea arrival rate. x 0 9 / hour Mea service rate x 0 0 / hour. (a) Average umber of customers waitig i ueue for service ( ) 0( 0 9) 3. customers 9 (b) Expected waitig time i the ueue (c) bability of havig more tha customers i the system or more where ) Cosider a box office ticket widow beig maed by a sigle server. Customer arrives to purchase ticket accordig to oisso iput pcess with a mea rate of 30/hr. the time reuired to serve a customer has a ED with a mea of 90 secods determie:

13 (a) Mea ueue legth. (.5) (b) Mea waitig time i the system. (0.) (c) The pbability of the customer waitig i the ueue for more tha 0mi. (0.4) (d) The fractio of the time for which the server is busy. (0.75) Solutio: The mea arrival rate The mea service rate (a) Mea ueue legth 30 / hr / hr customers (b) Mea waitig time i the system hr 40 ( ) 40( 40 30) 3

14 (c) bability of the customer waitig i ueue for more tha 0mi. 0 0 hour ( e ) w ( e ) (d) Fractio of time the serve is busy ρ hr 4) A T.V repairma repair the sets i the order i which they arrive ad expects that the time reuired to repair a set has a ED with mea 30mis. The sets arrive i a oisso fashio at a average rate of 0/8 hrs a day. (a) hat is the expected idle time / day for the repairma? (0.375x8) b) How may TV sets will be there awaitig for the repair? (.04) 4

15 Solutio Mea arrival rate 0 8 hours Mea service rate 30 x 0 hours (a) Expected idle time / day of the repair Busy eriod idle time o hour idle time / day hrs / day (b) Number of T.V sets awaitig for the repair:-.5 ( ) (.5).04 5) I a bak there is oly o widow. A solitary employee performs all the service reuired ad the widow remais cotiuously ope fm 7am to pm. It has discovered that a average umber of cliets is 54 durig the day ad the average service time is 5mis / perso. Fid a) Average umber of cliets i the system (3) b) Average waitig time (0.5) c) The pbability that a cliet has to sped more tha 0mis i a system. (0.0) 5

16 Solutio The mea arrival rate The mea service rate 54 9 cliets / hour 0 5 cliets / hour (a) Average umber of customer i the system ( ) 9 ( 9) 3 cliets (b) Average waitig time:- 9 9 ( ) ( 9) 0.5 (c) bability that a customer has to sped more tha 0mi i a system. ϑ 0 0 e hr ( ) v ( 9 ) e 0.0

17 ) A departmetal Secretary receive a average of 8 job / hr. may are short jobs, while other are uiet log. Assume however, that the time to perform a job has a ED mea of mis determie a) The average elapsed time fm the time the secretary receives a job, util it is completed. (0.5) b) Average umber of jobs i a system (4) c) The pbability that the time i the system is greater tha ½hr. (0.3) d) bability of more tha 5 jobs i the system. (0.8) Solutio Mea arrival rate 8 jobs / hrs Mea service rate x 0 0 jobs / hrs. (a) Average elapsed time fm the time the secretary receives a job o till it is completed 0 ( ) 8 ( 0 8)

18 (b) Average umber of jobs i the system:- 4 0 ( ) 8 ( 0 8) jobs 8 0 (c) bability that the customer speds time i the system is greater tha ½ hr. v 0.5 hr e e ( ) v ( 8 0 ) (d) bability of more tha 5 jobs i the system: ) At public telephoe booth i a post office arrivals are cosidered to be oisso fashio with a average iter arrival time of mis. The legth of the phoe call is ED with a mea of 4mis. Determie: 8

19 (a) The pbability that the fresh arrival will ot have to wait for the phoe. (0.) (b) hat is the pbability that the a arrival will have to wait for more tha 0mis before the phoe is free (0.09) (c) hat is the average legth of the ueue that forms fm time to time (.5) Solutio: Mea arrival rate Mea service rate 4 x 0 5 / hr x 0 5 / hr (a) bability that fresh arrivals will ot have to wait for the phoe: o (b) bability that a arrival will have to wait more tha 0 mi before the phoe is free : 0 0 e 5 e hr ( ) ω ( 5 5) 9

20 (c) Average legth of the ueue that form fm time to time: D ) There is cogestio o the platform of a railway statio. The trais arrive at a rate of 30/days. The service time for ay trai is ED with a average of 3mis. Calculate: (a) Mea ueue size (.5) (b) bability that there are more tha 0 trais i the system. (0.04) Solutio Mea arrival rate Mea service rate (a) Mea ueue size: / days.5 / hr 4 0. / hr 3 ( ).(..5).95 per hr.5 (b) bability that tha 0 trais i the system ) The arrival rate for a waitig lie system obeys a.d with a mea of 0.5 uits/hr. it is reuired that the pbability of oe or more uits i the system does ot exceed 0.5. what is the miimum service rate that must be pvided if the service duratio will be distributed expoetially? (/hr) 0

21 Solutio 0.5 uits / hr or more / hr 0) I a muicipality hospital patiets arrival are cosidered to be oisso with a arrival iterval time of 0mis. The doctors (examiatio ad dispesig) time may be assumed to be ED with a average of mis fid : a) hat is the chace that a ew patiet directly sees the doctor? (0.4) b) For what pportio of the time the doctor is busy? (0.) c) hat is the average umber of patiets i the system? (.5) d) hat is the average waitig time of the system? (.5) e) Suppose the muicipality wats to recruit aother doctor, whe a average waitig time of a arrival is 30mis i the ueue. Fid out hose large should be to justify a d doctor? ( 8.33) Solutio 0 / hr 0 0 0/ hr

22 (a) bability that a ew patiet straight away sees the doctor: (b) portio of time the doctor is busy:- ρ 0 0.hr (c) Average umber of patiets i the system (d) Average waitig i the system:- ( ) 0( 0 ) ( ) 0 0 ( )

23 (e) The value of ( ) 05hr ( ) / hr has to be icreased fm to 8.33 justify a secod doctor. ) At a oe ma barber shop customers arrive accordig to.d with a mea arrival rate of 5/hr. The hair cuttig time is ED with a hair cut takig 0 mi o a average assumig that the customers are always willig to wait fid: a) Average umber of customer i the shop [5] b) Average waitig time of a customer [0.833] c) The percet of time a arrival Ca walkright with out havig to wait [.%] d) The pbability of a customer waitig more tha 5mis [0.7] Solutio 5 / hr Mea service rate 0 0 / hr 3

24 (a) Average umber of customer s i the shop. ( ) 5 5 ( 5) 5 customers (b) Average waitig time of a customer. ( ) 5 ( 5) 0.833hr (c) ercet of time arrival ca walk right without havig to wait. p o % 4

25 d) bability of a customer waitig more tha 5mis. 5 0 / ( e ) ( 5 e ) w ) At a stamp veder widow of a post office 0 customers arrive o a average every 0 mi. the veder clerk ca serve 5 customers i mi. Determie a)average umber of customer i the System [4] b) Average waitig time of a customer [0.0] c) bability of a customer waitig more tha 3mis before beig served[0.785] d) Idle time of the veder clerk i a shift of 8hrs [.] Solutio / hr 0 5.5/ mi.5 0 (a) Average umber of customers i the system: ( 0) ( ) 0 50 (b) Average waitig time of a customer:-. 4 customers hr 0 50 / hr 5

26 (e) bability of a customer waitig more tha 3mis before beig received ( ) w e ( 0 50 e ) (d) Idle time of the vedor clerk i a shift of 8hours. o hr 3) Arrivals of machiist at a tool crib are cosidered to be.d at a average rate of /hr. the legth of time the machiist must remai at the tool crib is ED with a average time beig 0,05 hrs. a) what is the pblem that the machiist arrivig at the tool crib will have to wait? [0.3] b) hat is the average umber of machiist at the tool crib? [0.48] c) The compay will istall a d tool crib whe coviced that a machiist would expect to spet at least mis waitig ad beig served at the tool crib. By how much the flow of machiist to the tool crib icrease to justify the d tool crib. [x 0] Solutio / hr / hr

27 (a) bability that the machiist arrivig at the tool crib will have to wait. o (b) Average umber of machiist at the tool crib. ( C ) ( ) ( 0 ) 0 / hr 0.hr ( ) ( 0 ) 0 4) Jobs arrive at a ispectio statio accordig to oisso pcess at a mea rate of /hr ad are ispect oe at a time o a FIFO basis. The uality cotl egieer both ispects ad makes mior adjustmets. The total service time for the job appears to be ED with a mea of 5mis. Jobs that arrive but caot be ispected immediately by the egieer must be stored util the egieer is free to take them. Each job reuires s mts space determie a) The waitig lie legth [4.] b) The waitig time [.08] c) % of idle time of the egieer [.%] d) The floor space to be pvided i the uality cotl om. [5] 7

28 Solutio / hr / hr (a) ( ).4(.4 ) (b) c) Idle time of the egieer:- o %.08 (d) Floor space to be pvided i the uality cotl om m ) The arrival of aircraft at a iteratioal teds to follow a oisso fashio, i spite of schedule flight time, due to high operatig variability i the schedule time. It ca be assumed that the aircraft arrives at a average rate of /hr. The ladig service is pvided thugh a sigle ruway by a cotl tower accordig to ED with a average service time of mis/flight: 8

29 (a) Fid the pb. that will more tha 0mis all together to wait for ladig ad to lad a aircraft. [0.53] (b) hat is pb. that the ruway will be free for a icomig flight? [0.4] Solutio (a) V / hr e e ( ) ( 0 ) / hr v / b) bability that the ruway will be free for a icomig flight. o o ) At what rate must the clerk of a super market work i order to esure a pb. Of 0.9 that the Customer will ot have to wait loger tha mis i the system. It is assumed that the arrivals follows a oisso fashio at the rate of 5/hr. The legth of service by the clerk has a ED. (a) Also fid the average umber of customers ueuig for service. [0.738] (b) The b. of havig more tha 0 customers i the system [.9x0-3 ] 9

30 Solutio: v / hr 0. e 0.9 e 0. e 0. ( ) ( 5 ) 0. ( 5 ) ( 5 ).5 0. (a) Average umber of customer s ueuig for service ( ) (.5 5) (b) bability of havig more tha 0 customers i the system:

31 7) A mechaic is to hired to repair a machie which breaks dow at a average rate of 3/hr. breakdows are distributed i time i a maer that may be regarded as oisso. The opductive time o ay machiee is cosidered to cost the Compay. Rs. 5/hr. The T Compay has the choice to mechaics A & B.. The mechaic A repairs the machies at a average rate of 4/hr ad he will demad d Rs. 3/hr. The mechaic B costs Rs. 5/hr ad ca repair the machies expoetially at a average rate of /hr. Decide which mechaic should be hired. Solutio Cosider the mechaic A 3 / hr, 4 / hr The umber of break dow machie i the system 3 4 ( ) 3 ( 4 3) m / c' s 3 4 Mechaic A 4 / hr cost Rs 3 / hr Mechaic B / hr cost Rs 5 / hr The o-pductive time the compay / hr 3 x 5 5 Rs The amout paid to the mechaic A per hour 3 Rs The total expected cost per hour 53 8 Rs 3

32 Cosider the mechaic B 3 / hr, / hr The umber of break dow machie i the system ( ) 3 ( 3) m / c' s 3 The o-pductive time cost of the compay / hr x 5 5 Rs Amout paid to the mechaic / hr 5 Total expected cost per hour 55 0 Rs Selected mechaic B as the total expected cost per hour of mechaic B is less tha the total expected cost per hour of mechaic A. 3

33 33 QUEUEING THEORY ROBEMS : MODE Multiple server Model M / M / K : / FCFS Formulae Multiple server Model () ( ) k k K K K o!! 0 () o.! (3) ( ) ( ) o K k k! (4) (5)

34 () (7) ρ k ) A Commercial bak has 3 cash payig assistats customers are foud to arrive i a oisso fashio at a average rate of /hr for busiess trasactio. The service time is foud to have a E.D with a mea of 8 mis. The customers are pcessed o FCFS basis. Calculate a) Average umber of customers i the system b) Average time a customer speds i the system c) Average ueue legth d) How may hours a week ca a cash payig assistat sped with the customers. Solutio: K 3 / hr o / hr K 0! K! K k k o 0! ! 3.33! !

35 (a) Average umber customers i the system:- K ( k )!( k ) 3.33! ( ) ( b) ( c) 3 o 0.45 Average time a customer speds i the system 0.53 Average ueue legth

36 (d) Assumig 5 days a week ad 8 hrs a day the umber of hrs i a week the cash payig assistat speds with the customers ρ 5 8 k hrs () A telephoe exchage has two log distat operators. The telephoe compay fids that durig the fashio at a average rate of 5/hr. The legth mea of 5 mis. (a) what is the pbability that a subscriber will have to wait for his log distat dials o the peak hour of the day. (b) what is the average waitig time for the customers. Solutio: K 5/ hr 0 / hr 5 o 0 5 0! 5! 5! 5 o

37 (a) bability that a subscriber will have to wait for his log distat call is 5! (b) average waitig time for the customer ( k )!( k ) 5! o! ( 4 5) k hr. o o 3) A isurace compay has 3 clerks i its brach office. eople arrive with claims agaist the compay are foud to arrive i a oisso fashio at a average of 0 per 8 hours a day. The amout of time that a clerk speds with the cliet is foud to have ED with a mea time of 40 mis. The cliets are pcessed i the order of their appearace. (a) How may hours a week ca a clerk expect to sped with the cliets. (b) How much time a average does a cliet sped i the brach office. 37

38 Solutio: K 3 0.5/ hr 8 0.5/ hr 40 o o K 0.5 0! ! 0 K!.5!.5 K k k.5!.5.5 3! (a) The umber of hours per week a clerk expects to sped with the cliet k ρ x 5 x 8 assumig 5 days a week ad 8 hrs / day hrs 38

39 ( b) Average time a clerk speds i the brach office ( k )! ( k ).5.5.5! k 3 o ( 3.5.5) 0.8 hrs ) A bak has tellers workig o savig accouts. The st tellers hadles withdrawal's oly ad the d teller hadles deposits oly, it has bee foud that service time distributio for depositors ad withdrawal's. Both are E.D with a mea service time of 3mi /customer. Depositor are foud to arrive i a oisso fashio with a arrival rate of /hr ad withdrawal's also drive i a oisso with a mea rate of 4/hr. hat would be the effect o the average waitig time for the depositors ad withdrawal's if each teller would hadle both withdrawal's ad deposits. hat would be the effect if the time would oly be accomplished by icreasig the service time to 3.5mis. 39

40 Solutio: aitig time i the ueue for the depositors D ( ) 0( (0 ) waitig time i the ueue for thee withdrawal's w ( ) 4 0(0 4) 0..0hrs Cosider it as a multi server model. 0.hrs K D 0 / hr 30 / hr 40

41 o o o! ! 30 0! aitig time i the ueue w k o ( K )!( K )! ( 0 30) 0.04hrs is the waitig time i the ueue If the service time is icreased fm 3 to 3.5mis o ! hrs 30! ! ( ) ! _ 30 By icreasig the service time fm 3 to 3.5 mi the waitig time i the ueue for the depositors is decreased fm 0. to 0.9hrs. But i the case of withdrawal's the waitig time i the ueue icreased fm 0.hrs to 0.9hrs

42 5) A tax cosultig firm has 3 couters i its offices to receive the people who have pblems cocerig their icome ad the sales tax. O a average 48 persos arrive i 8hrs a day. Each tax advisor speds 5 mi a o average for a arrival of the arrival time follows a oisso distributio ad the service time follows a E.D. (a) Fid the average umber of customer i the system. (b) Average waitig time of the customer i the system. (c) Average umber of customers waitig the ueue for service. (d) Average waitig time of the customers i the ueue. (e) How may hours each week a tax advisor speds performig his job. (f) bability that a customer has to wait before he gets service. (g) Expected umber of idle tax advisors at ay specified time Solutio: K 3 48 hrs / hrs ! 4! 4 0.0! 4 3 3!

43 43 a) Average umber of customers i the system: ( ) ( ) ( ) customers K K K ! ! ) ( 3 0 b) Average waitig time of the customer i the system. ( ) ( ) ( ) ( ) hour K K K ! ! 3 0 (c) Average umber of customer i the ueue ( ) ( ) ( ) ! 0 K K

44 44 (d) Average time of the customers i the ueue (e) Assumig 5 days a week ad 8 hours per day the umber of hours the tax advisors spets with customers durig the week. hours K ρ (f) bability that a customer has to wait before he gets service ! 0 0 K !

45 g) Expected umber of idle tax advisors at ay specified time % Referece Books:. Taha H A, Operatio Research - A Itductio, retice Hall of Idia, 7 th editio, 003. Ravidra, hillips ad Solberg, Operatios Research : riciples ad ractice, Joh iely & Sos, d Editio 3. D.S.Hira, Operatio Research, S.Chad & Compay td., New Delhi,