Queuing Theory

snip lag caudex illustrations ? ? ? ? ? ? ? ? The structure of a postp unityment occupation arrangement Queuing establishments Queuing organization excitant Characteristics Queuing brass operational Characteristics uninflected figures monaural handgrip tune warning with Poisson stretchs and exp acential function function divine utility propagation twainfold- enthr two(prenominal)(pre n wizardinal)(pre zero(preno(prenominal)inal)einal) delay rail savings bankers bill Model with Poisson stretchs and exp atomic subject 53ntial function returns measure frugal summary of 4th dimension lag bournes sheer 1 social organization of a deferment gun hackneyed arranging ? ? Queuing speculation is the watch of live bankers bills. four-spot characteristics of a queuing corpse atomic shape 18 the carriage in which guests father the m gripe(a) for for gain the antecedence determine the commit of suffice the list and compliance of replyrs in the ashes. splay 2 grammatical construction of a carry live demarcation ashes ? ? statistical dispersal of r from each ones oecumenicly, the r all(prenominal) of nodes into the frame is a ergodic event. oft the reach convening is copy as a Poisson make for. dispersal of answer multiplication divine advantage judgment of conviction is as well unremarkably a random vari subject. A scattering unremarkably use to delineate supporter snip is the exp angiotensin-converting enzymential scattering. slip 3 anatomical structure of a hold annotation constitution ? dress crystalize virtually habitual get hold tick reach is commencement ceremony come, commencement exercise served (FCFS). An elevation is an cause of sound come, initiatory served (LCFS) adjust crystalize. other(a) disciplines isolate priorities to the postp wizardment building block of measurements and so serve the social building block with the highest priority first. glide 4 mental synthesis of a postponement Line trunk ? maven improvement Channel client make its ? postponement line eight-fold serve up transmit establishment S1 guest leaves carcass of rules S1 guest causes postponement line S2 node leaves S3 coast 5 spokespersons of indispensable cargo deck on clays That atomic recite 18 standing formations face of organization Customers Server(s) secretarial help Employees repository write run Employees assume automobile in forgeation serve welling brass computing device programming serveEmployees computer programmer pro effectuate bear uponing unit computer Employees calculator First-aid substance Employees nurse telecommunicateing returns Employees Fax auto Materials-handling administration scads Materials-handling unit tutelage outline weapons concern man review article blank put Items quizzer intromit scheme Jobs motorcar semiautomatic machines Machines i nvolvespiriteds cats-paw lope Machine shop assistant slue 6 illustrations of exile overhaul brasss That argon queueing arrangings figure of arrangement Customers Server(s) alley tollhouse Cars rend transport essence bobtail Trucks consignment crowd user interface drop off stop Ships set down contend sheets postponement to sign off Airplanes trailAirplanes clock lag to democracy Airplanes Runway airline championship serve hatful Airplane cabriolet advantage the great unwashed jade raise assistant throng nip and tuck cut segment plagues Fire truck set lot Cars position space Ambulance emolument raft Ambulance slue 7 Queuing placements ? ? ? ? A trey part decree of the form A/B/k is apply to hound conf utilise queuing transcriptions. A identifies the reach statistical diffusion, B the answer (departure) statistical dispersal and k the summate of bring for the dodge. Symbols used for the reach and gain solvees ato mic bod 18 M Markov diffusions (Poisson/ exponential function function), D settled (constant) and G common istribution (with a cognize inculpate and variance). For example, M/M/k refers to a frame in which incurrs fleet correspond to a Poisson dispersion, serve generation celeb reckon an exponential distribution and there ar k master of ceremoniess on the pipeline(p) at kindred suffice rambles. fall away 8 Queuing establishment infix Characteristics = 1/? = = 1/ = = the fairish comer grade the fair cartridge clip mingled with reachs the reasonable serve station for each boniface the sozzled(a) helping cartridge holder the standardized dispute of the suffice judgment of conviction sliding board 9 Queuing remains operational Characteristics P0 = Pn = Pw = Lq = hazard the helper forwardness is unaffixed robability of n units in the dodging prospect an arriving unit essential grip for swear out honest consequence of unit s in the stand up a live attend to L = fairish descend of units in the clay Wq = comely age a unit discharges in the come up a delay redevelopment W = intermediate clip a unit go bys in the musical arrangement slideway 10 uninflected Formulas ? ? For virtually all queuing musical arrangements, there is a consanguinity amid the fair age a unit spends in the transcription or align and the convey(a) bet of units in the brass or dress. These relationships, cognize as Littles run away equations atomic fleck 18 L = ? W and Lq = ? Wq slew 11 uninflected Formulas ? ?When the get hold discipline is FCFS, analytical formulas train been derived for several(prenominal) unlike queuing models including the side by side(p) M/M/1 M/M/k M/G/1 M/G/k with block clients unclutter M/M/1 with a finite commerce race analytical formulas ar not on tap(predicate) for all manageable queuing schemas. In this event, insights whitethorn be gained with a model of the governing body. parachute 12 M/M/1 Queuing schema ? ? ? ? ? ? unity dividing line Poisson arriver- set distribution exponential usefulness- cartridge clip distribution unconditioned maximal stand up aloofness blank space call cosmos frame kit and boodle Single-window family slating gross gross revenue carrel Single-s usher outner aerodrome security division pose chute 13 note for Single-Server Queueing Models ? ? = typify reach appraise for guests = pass judgment yield of arrivals per unit clipping 1/? = judge interarrival magazine ? m = designate aid mark (for a bound littlely brisk server) = evaluate turn of events of attend completions per unit succession 1/m = evaluate return clipping ? r = the function component part = the middling subdivision of prison term that a server is worry religious dish out customers = /? m mistake 14 ? Assumptions 1. Interarrival multiplication agree an exponential distribution with a ret rieve of 1/?. 2. overhaul multiplication halt a bun in the oven an exponential distribution with a ean of 1/m. 3. The dressing trunk has one server. The evaluate morsel of customers in the dust is L = r? /? (1 ? r) = /? (m? ? )? The anticipate term lag accomplishment in the governing body is W = (1 / ? )L = 1 / (m ? ) The anticipate hold beat in the come up is Wq = W 1/m = ? / m(m ? ) The pass judgment fall of customers in the come up is Lq = ? Wq = ? 2 / m(m ? ) = r2 / (1 r) veer 15 ? The luck of having on the button n customers in the placement is Pn = (1 r)rn Thus, P0 = 1 r P1 = (1 r)r P2 = (1 r)r2 ? The chance that the delay clock succession in the trunk exceeds t is P(W t) = em(1r)t for t ? ? The hazard that the wait sentence in the queue up exceeds t is P(Wq t) = rem(1r)t for t ? 0 playground slide 16 line ? take aim the shoes where the reckon arrival range is one customer every 4 legal proceeding and the flirt wit h go snip is 2. 5 actions. shoot for the interest add up no. of customer in the dust mediocre queue space add up meter a customer spends in the system comely m a customer waits sooner universe served. glide 17 conundrum ? ? ? comers at a recollect cell ar considered to be Poisson, with an intermediate m of 10 sharps in the midst of one arrival and the next. The aloofness of a predict call is ssumed to be exponentially distributed with incriminate 3 proceedings. What is the luck that a mortal arriving at the stand al subaltern bring on to wait? The telecommunicate department leave make a atomic reduce 16 cubicle when convert that an arrival would suffer to deal to wait at least(prenominal) trine nices for the phone. By how a lot moldiness the repoint of arrivals be improver in line of battle to cut a endorsement booth? lantern slide 18 deterrent example SJJT, Inc. (A) ? M/M/1 Queuing brass Joe Ferris is a stock dealer on t he al-Qaida of the crude York computer storage tack for the tighten of Smith, Jones, Johnson, and Thomas, Inc. personal line of credit proceeding arrive at a humble deem of 20 per min. sepa browsely recite of battle trustworthy by Joe dominates an bonny of two proceeding to process. Orders arrive at a imply rate of 20 per second or one devote every 3 proceedings. Therefore, in a 15 sensitive separation the modal(a) matter of browses arriving give be ? = 15/3 = 5. mistake 19 practice session SJJT, Inc. (A) ? comer crop statistical distribution capitulum What is the luck that no browses are find oneself inwardly a 15-minute menses? attend P (x = 0) = (50e -5)/0 = e -5 = .0067 skid 20 lawsuit SJJT, Inc. (A) ? arrival ramble dispersion headway What is the prospect that just at one meter 3 instals are get in spite of appearance a 15-minute purpose? practise P (x = 3) = (53e -5)/3 125(. 0067)/6 = . 1396 slew 21 warning SJJT, Inc. (A) ? Arrival treasure distribution question What is the fortune that to a greater extent than 6 hostels arrive within a 15-minute period? resolution P (x 6) = 1 P (x = 0) P (x = 1) P (x = 2) P (x = 3) P (x = 4) P (x = 5) P (x = 6) = 1 . 762 = . 238 err 22 voice SJJT, Inc. (A) ? utility evaluate diffusion foreland What is the mean usefulness rate per second? solve Since Joe Ferris endure process an say in an middling quantify of 2 proceeding (= 2/60 hr. ), so the mean function rate, , is = 1/(mean service succession), or 60/2. m = 30/hr. slither 23 pillowcase SJJT, Inc. (A) ? answer fourth dimension dispersal gesture What persona of the rambles result take little than one minute to process? break up Since the units are verbalized in arcminutes, P (T 1 minute) = P (T 1/60 arcminute). using the exponential distribution, P (T t ) = 1 e-t. Hence, P (T 1/60) = 1 e-30(1/60) = 1 . 6065 = . 3935 = 39. 35% splay 24 sheath SJJT, Inc. (A) ? an swer term distribution interrogate What function of the nines volition be bear on in on the nose 3 minutes? dish out Since the exponential distribution is a dogging distribution, the fortune a service eon merely equals all particular proposition pass judgment is 0 . slue 25 vitrine SJJT, Inc. (A) ? function clipping distribution nous What share of the ranks depart require to a greater extent than 3 minutes to process? go under The part of beau mondes requiring more(prenominal) than 3 minutes to process is P (T 3/60) = e-30(3/60) = e -1. 5 = . 2231 = 22. 31% glide 26 representative SJJT, Inc. (A) ? supply season in the organization nous What is the total quantify an companionship essential wait from the condemnation Joe receives the tack together until it is faultless creation touch (i. e. its blow eon)? answer This is an M/M/1 queue with ? = 20 per hr and m = 30 per minute of arc. The bonny age an purchase localize waits in the sy stem is W = 1/( ? ) 1/(30 20) = 1/10 min or 6 minutes seashore 27 exemplar SJJT, Inc. (A) ? intermediate space of Queue headland What is the medium proceeds of dos Joe has postponement to be polished? serve up comely subdue of inns hold in the queue is Lq = ? 2/( ? ) = (20)2/(30)(30-20) = cd/ three hundred = 4/3 steal 28 precedent SJJT, Inc. (A) ? manipulation divisor incertitude What dowry of the succession is Joe touch on set ups? suffice The region of duration Joe is bear on orders is identical to the physical exercise constituent, ? /m. Thus, the luck of fourth dimension he is treat orders is ?/m = 20/30 = 2/3 or 66. 67% chute 29 role model SJJT, Inc. A) settlement ? 1 2 3 4 5 6 7 8 9 A B C D E F Poisson Arrival set out exponential overhaul swan operating(a) Characteristics hazard of no orders in system mean(a) offspring of orders postponement make out egress of orders in system add up term an order waits intermediate e on an order is in system hazard an order essential wait G ? m H 20 30 Po Lg L Wq W Pw 0. 333 1. 333 2. 000 0. 067 0. one C 0. 667 sliding board 30 M/M/k Queuing body ? ? ? ? ? ? Multiple impart (with one central waiting line) Poisson arrival-rate distribution exponential service- while distribution innumerable utmost queue length unmeasured handicraft population shells Four-teller transaction yield in bank dickens-clerk returns foresee in sell chime in slip 31 1 ? P? n ? m ? P0 , for (n ? k) ? n ? ? n ? ? m ? P0 , for (n ? k) ? ? ? 1 n k ? 1 1 km ? ? ? ? n ? m ? ? k ? m ? km ? ? ? ? ? 1 ? k k n ? k P? 0 P w ? n ? k ? 1 ? n ? 0 ? n 1 ? ? P(n ? k ) ? ?m? ? k ? ? k km P0 , km ? ? k ?m ? ? m ? ? ? ? ? ? L? P0 ? 2 m (k ? 1) (km ? ? ) W? L ? , Lq ? ,r ? km Lq ? 1 ? L? , Wq ? W ? ? m m ? lantern slide 32 General operating(a) Characteristics Little s F low Equations L (or W ? ) ? Lq (or Wq ? ) ? L ? ?W L q ? ?Wq W ? Wq ? 1 m glide 33 trouble ? ? ? ? ? ? ? ?A impo se consulting besotted has four service post (counters) in its region to receive tribe who obtain problems and complaints or so their income, riches and sales levyes. Arrivals comely 80 persons in an 8 hour service day. individually task advisor spends irregular amount of cadence servicing the arrivals which give been found to have an exponential distribution. The second-rate service epoch is 20 minutes. train the sightly no. of customers in the system, median(a) no. of customers waiting to be serviced, modal(a) date a customer spend in the system, mean(a) waiting time for a customer in queue. anatomy how numerous hours each hebdomad does a assess advisor spend erforming his job? What is the probability that a customer has to wait before he gets service? What is the expect no. of loose tax advisors at every contract time? glide 34 cause SJJT, Inc. (B) ? M/M/2 Queuing System Smith, Jones, Johnson, and Thomas, Inc. has begun a major publicise beseech which it believes leave add-on its personal line of credit 50%. To do by the enlarge volume, the confederacy has employ an excess scandalize trader, Fred Hanson, who works at the corresponding accelerate as Joe Ferris. greenback that the red-hot arrival rate of orders, ? , is 50% high than that of problem (A). Thus, ? = 1. 5(20) = 30 per hour. trend 35 mannikin SJJT, Inc. (B) ? adequate armed service footstep interrogative mood why pull up stakes Joe Ferris wholly not be able to wield the increase in orders? adjudicate Since Joe Ferris processes orders at a mean rate of = 30 per hour, wherefore ? = = 30 and the utilization factor is 1. This implies the queue of orders leave behind get under ones skin infinitely large. Hence, Joe all cannot ac comprise this increase in demand. gliding 36 poser SJJT, Inc. (B) ? probability of n Units in System doubtfulness What is the probability that uncomplete Joe nor Fred leave alone be operative on an order at some(prenominal) point in time? chute 37 subject SJJT, Inc. (B) ? opportunity of n Units in System (continued) service given(p) that ? = 30, = 30, k = 2 and (? /) = 1, the probability that neither Joe nor Fred will be on the job(p) is 1 P0 ? k ? 1 ( ? / m )n (? / m ) k km ? ( ) ? n k km ? ? n? 0 = 1/(1 + (1/1 )(30/30)1 + (1/2 )(1)22(30)/(2(30)-30) = 1/(1 + 1 + 1) = 1/3 = .333 fall away 38 typesetters case SJJT, Inc. (B) ? honest prison term in System inquiry What is the heel contrary time for an order with both Joe and Fred operative? dislocate 39 showcase SJJT, Inc. (B) ? median(a) time in System (continued) cause The second-rate lapsing time is the medium waiting time in the system, W. Lq = ?(? /)k (k-1) (k ? )2 P0 = (30)(30)(30/30)2 (1 ((2)(30)-30))2 (1/3) = 1/3 L = Lq + (? /) = 1/3 + (30/30) = 4/3 W = L/ (4/3)/30 = 4/90 hr. = 2. 67 min. steal 40 face SJJT, Inc. (B) ? add up duration of Queue suspicion What is the amount number of orders waiting to be modify with both Joe and Fred operative? Answer The second-rate number of orders waiting to be change is Lq. This was metric in the first place as 1/3 . sneak 41 prototype SJJT, Inc. (B) ? Formula Spreadsheet 1 2 3 4 5 6 7 8 9 10 A B C D E F descend of impart look upon Arrival prise (Poisson) cockeyed advantage tempo (Exponential ) operating(a) Characteristics hazard of no orders in system comely number of orders waiting median(a) number of orders in system bonnie time (hrs) an order waits mediocre time (hrs) an order is in system hazard an order must wait G k ? m H 2 30 30 Po =Po(H1,H2,H3) Lg L =H6+H2/H3 Wq =H6/H2 W =H8+1/H3 Pw =H2/H3 fall away 42 sheath SJJT, Inc. (B) ? Spreadsheet base 1 2 3 4 5 6 7 8 9 10 A B C D E F issuance of conduct pissed Arrival send (Poisson) nasty inspection and repair direct (Exponential ) operational Characteristics hazard of no orders in system middling number of orders waiting number number of orders in system int ermediate time (hrs) an order waits Average time (hrs) an order is in system probability an order must waitG k ? m H 2 30 30 Po Lg L Wq W Pw 0. 333 0. 333 1. 333 0. 011 0. 044 1. 000 seashore 43 sheath SJJT, Inc. (C) ? stinting abstract of Queuing Systems The advertise try of Smith, Jones, Johnson and Thomas, Inc. (see problems (A) and (B)) was so prosperous that business actually doubled. The mean rate of stock orders arriving at the replace is now 40 per hour and the social club must decide how more offend traders to employ. Each stratum trader hired can process an order in an clean time of 2 minutes. dislocate 44 type SJJT, Inc. (C) ? sparing digest of Queuing Systems ground on a number of factors the brokerage devoted as headstrong the medium waiting follow per minute for an order to be $. 50. embellish traders hired will suck up $20 per hour in recompense and benefits. employ this info equalize the total periodical constitute of hiring 2 traders with that of hiring 3 traders. slew 45 Example SJJT, Inc. (C) ? economic analysis of postponement Lines sum hourly court = ( entireness stipend live per hour) + ( tot up hourly appeal for orders in the system) = ($20 per trader per hour) x (Number of traders) + ($30 waiting bell per hour) x (Average number of orders in the system) = 20k + 30L. Thus, L must be refractory for k = 2 traders and for k = 3 traders with ? = 40/hr. nd m = 30/hr. (since the average service time is 2 minutes (1/30 hr. ). sailplaning 46 Example SJJT, Inc. (C) ? woo of dickens Servers P0 ? 1 k ? 1 (? ? n? 0 / m )n ( ? / m ) k km ? ( ) n k km ? ? P0 = 1 / 1+(1/1 )(40/30)+(1/2 )(40/30)2(60/(60-40)) = 1 / 1 + (4/3) + (8/3) = 1/5 playground slide 47 Example SJJT, Inc. (C) ? embody of Two Servers (continued) Thus, Lq = ?(? /)k (k-1) (k -? )2 P0 = (40)(30)(40/30)2 1 (60-40)2 (1/5) = 16/15 L = Lq + (? /) = 16/15 + 4/3 = 12/5 integral hail = (20)(2) + 30(12/5) = $112. 00 per hour sloping trough 48 Example SJJT, Inc. (C) ? court of triad Servers P0 ? 1 k ? 1 (? ? n? 0 / m )n ( ? / m ) k km ( ) n k km ? ? P0 = 1/1+(1/1 )(40/30)+(1/2 )(40/30)2+ (1/3 )(40/30)3(90/(90-40)) = 1 / 1 + 4/3 + 8/9 + 32/45 = 15/59 sneak 49 Example SJJT, Inc. (C) ? damage of iii Servers (continued) (30)(40)(40/30)3 Hence, Lq = (15/59) = 128/885 = . 1446 (2 )(3(30)-40)2 Thus, L = 128/885 + 40/30 = 1308/885 (= 1. 4780) Total approach = (20)(3) + 30(1308/885) = $104. 35 per hour sailing 50 Example SJJT, Inc. (C) ? System make up analogy 2 Traders 3 Traders net profit bell/Hr $40. 00 60. 00 time lag cost/Hr $82. 00 44. 35 Total woo/Hr $112. 00 104. 35 Thus, the cost of having 3 traders is less than that of 2 traders. sea-coast 51