Transporaion Sysem Engineering Chaper 66 Queuing Analysis 66.1 Inroducion One of he major issues in he analysis of any raffic sysem is he analysis of delay. Delay is a more suble concep. I may be defined as he difference beween he acual ravel ime on a given segmen and some ideal ravel ime of ha segmen. This raises he uesion as o wha is he ideal ravel ime. In pracice, he ideal ravel ime chosen will depend on he siuaion; in general, however, here are wo paricular ravel imes ha seem bes suied as benchmarks for comparison wih he acual performance of he sysem. These are he ravel ime under free flow condiions and ravel ime a capaciy. Mos recen research has found ha for highway sysems, here is comparaively lile difference beween hese wo speeds. Tha being he case, he analysis ofdelay normally focuses on delay ha resuls when demand exceeds is capaciy; such delay is known as ueuing delay,andmaybesudiedbymeansofueuingheory. This heory involves he analysis of wha is known as a ueuing sysem, which is composed of a server; a sream of cusomers, who demand service; and a ueue, or line of cusomers waiing o be served. 66.2 Queuing Sysem Figure 66.1 shows a schemaic diagram illusraing he concep of a ueuing sysem. Various componens are discussed below. 66.2.1 Inpu parameers Mean arrival rae Mean service rae The number of servers Queue discipline Arrival rae Inpu source (Cusomers) Queue Queue discipline Service faciliy Service rae Served cosumers (leaving) Figure 66.1: Componens of a basic ueuing sysem These are explained in he following secions. Mean Arrival rae () I is rae a which cusomers arrive a a service faciliy. I is expressed in flow (cusomers/hr or vehicles/hour in ransporaion scenario) or ime headway (seconds/cusomer or seconds/vehicle in ransporaion scenario). If iner arrival ime ha is ime headway (h) isknown, he arrivalraecanbefound oufromhe euaion: = 3600 (66.1) h Mean arrival rae can be specified as a deerminisic disribuion or probabilisic disribuion and someimes demand or inpu are subsiued for arrival. Mean service rae () I is he rae a which cusomers (vehicles in ransporaion scenario) depar from a ransporaion faciliy. I is expressed in flow(cusomers/hr or vehicles/hour in ransporaion scenario) or ime headway (seconds/cusomer or seconds/vehicle in ransporaion scenario). If iner service ime ha is ime headway (h) is known, he service rae can be found ou from he euaion: = 3600 h (66.2) Dr. Tom V. Mahew, IIT Bombay 66.1 March 8, 2017
Transporaion Sysem Engineering Number of servers The number of servers ha are being uilized should be specified and in he manner hey work ha is hey work as parallel servers or series servers has o be specified. Queue discipline Queue discipline is a parameer ha explains how he cusomers arrive a a service faciliy. The various ypes of ueue disciplines are 1. Firs in firs ou (FIFO) 2. Firs in las ou (FILO) 3. Served in random order (SIRO) 4. Prioriy scheduling 5. Processor (or Time) Sharing 1. Firs in firs ou (FIFO): If he cusomers are servedinheorderofheirarrival,henhisisknown as he firs-come, firs-served (FCFS) service discipline. Prepaid axi ueue a airpors where a axi is engaged on a firs-come, firs-served basis is an example of his discipline. 2. Firs in las ou (FILO): Someimes, he cusomers are serviced in he reverse order of heir enry so ha he ones who join he las areservedfirs. For example, assume ha leers o be yped, or order forms o be processed accumulae in a pile, each new addiion being pu on he op ofhem. The ypis or he clerk migh process hese leers or orders by aking each new ask from he op of he pile. Thus, a jus arriving ask would be he nex o be serviced provided ha no fresh ask arrives before i is picked up. Similarly, he people who join an elevaor firs are he las ones o leave i. 3. Served in random order (SIRO): Under his rule cusomers are seleced for service a random, irrespecive of heir arrivals in he service sysem. In his every cusomer in he ueue is eually likely o be seleced. The ime of arrival of he cusomers is, herefore, of no relevance in such a case. 4. Prioriy Service: Under his rule cusomers are grouped in prioriy classes on he basis of some aribues such as service ime or urgency or according o some idenifiable characerisic, and FIFO rule is used wihin each class o provide service. Treamen of VIPs in preference o oher paiens in a hospial is an example of prioriy service. 5. Processor (or Time) Sharing: The server is swiched beween all he ueues for a predefined slice of ime (uanum ime) in a round-robin manner. Each ueue head is served for ha specific ime. I doesn maer if he service is complee for a cusomer or no. If no hen i ll be served in i s nex urn. This is used o avoid he server ime killed by cusomer for he exernal aciviies (e.g. Preparing for paymen or filling half-filled form ). 66.3 Sysem performance measures The following noaion assumes ha he sysem is in a seady-sae condiion (A a given ime ): 1. Uilizaion facor ρ = 2. Pn = probabiliy of exacly n cusomers in ueuing sysem (waiing + service). 3. L= expeced(avg) number of cusomers in ueuing sysem. [someimes denoed as Ls] 4. L=expeced (avg) ueue lengh (excludes cusomers being served) or no of Cusomers. 5. W = Expeced waiing ime in sysem (includes service ime) for each individual cusomer or ime a cusomer spends in he sysem. [someimes denoed as Ws] 6. W = waiing ime in ueue (excludes service ime) for each individual cusomer or Expeced ime a cusomer spends in a ueue 66.3.1 Relaionships beween L, W, L and W: Assume ha n is a consan for all n. I has been proved ha in a seady-sae ueuing process, ( may be considered as avg): 1. L = W 2. L = W 3. W = W + 1 66.3.2 Queuing Paerns A variey of ueuing paerns can be encounered and a classificaion of hese paerns is proposed in his secion. The classificaion scheme is based on how he arrival and March 8, 2017 66.2 Dr. Tom V. Mahew, IIT Bombay
Transporaion Sysem Engineering m l l m Sv Sv l m Figure 66.2: Consan arrival and service raes service raes vary over ime. In he following figures he op wo graphs are drawn aking ime as independen variable and volume of vehicles as dependan variable and he boom wo graphs are drawn aking ime as independen variable and cumulaive volume of vehicles as dependan variable. Figure 66.3: Consan arrival rae and varying service rae 66.3.3 Consan arrival and service raes In he lef hand par of he Fig.66.2 arrival rae is less han service rae so no ueuing is encounered and in he righ hand par of he figure he arrival rae is higher han service rae, he ueue has a never ending growh wih a ueue lengh eual o he produc of ime and he difference beween he arrival and service raes. 66.3.4 Consan arrival rae and varying service rae In he lef hand of Fig. 66.3 he arrival rae is consan overime while he serviceraesvaryoverime. Ishould be noed ha he service rae mus be less han he arrival rae for some periods of ime bu greaer han he arrival rae for oher periods of ime. One of he examples of he lef hand par of he figure is a signalized inersecion and ha of he righ hand side par of he figure is an inciden or an acciden on he roads which causes a reducion in he service rae. 66.3.5 Varying arrival rae and consan service rae In he lef par of Fig. 66.4 he arrival rae vary over ime bu service rae is consan. Boh he lef and righ pars are examples of raffic variaion over a day on a Figure 66.4: Varying arrival rae and consan service rae Dr. Tom V. Mahew, IIT Bombay 66.3 March 8, 2017
Transporaion Sysem Engineering 2. Average number of cusomers in he sysem = L = ρ 2 3. Expeced waiing ime in he sysem W = L = (1/) = 1 4. Expeced waiing ime in he ueue W = L 1 2 ( ) = ( ) = Figure 66.5: Varying arrival and service raes faciliy bu he lef hand side one is an approximaion o make formulaions and calculaions simpler and he righ hand side one considers all he ransiion periods during changes in arrival raes. 66.3.6 Varying arrival and service raes In he Fig.66.5he arrivalrae follows a suarewaveype and service rae follows invered suare wave ype. The diagrams on he righ side are an exension of he firs one wih ransiional periods during changes in he arrival and service raes. These are more complex o analyzed using analyical mehods so simulaion is ofen employed paricularly when sensiiviy parameer is o be invesigaed. 66.4 Queuing models There are various kinds of ueuing models. These ueuing models have a se of defined characerisics like some arrival and service disribuion, ueue discipline, ec. The ueuing models are represened by using a noaion which is discussed in he following secion of ueue noaion. 66.4.1 M/M/1 model In his model he arrival imes and service raes follow Markovian disribuion or exponenial disribuion which are probabilisic disribuions, so his is an example of sochasic process. In his model here is only one server. The imporan resuls of his model are: 1. Average number of cusomers in he sysem = L = ρ Numerical example Vehicles arrive a a oll booh a an average rae of 300 per hour. Average waiing ime a he oll booh is 10s per vehicle. If boh arrivals and deparures are exponenially disribued, wha is he average number of vehicles in he sysem, average ueue lengh, he average delay per vehicle, he average ime a vehicle is in he sysem? Soluion Mean arrival rae = 300 vehicles/hr. Mean service rae = 3600 10 vehicles/hr. Uilizaion facor = raffic inensiy = ρ = = 300 360 = 0.833. Percen of ime he ollboohwillbeidle=p(0)=p(x=0)=ρ 0 () = (0.833) 0 (1 0.833) = 0.139(60min)=8.34 min. Average number of vehicles in he sysem = E[X] = ρ =4.98. Averagenumber ofvehicles in he ueue =E[L ] = ρ2 = 4.01. Average a vehicle spend in he sysem =E[T] = = 0.016 hr = 0.96 min = 57.6 sec. Average ime a 1 vehicle spends in he ueue =E[T ] = = 0.83 min = 50 sec. 66.4.2 M/M/N model ( ) = 0.013hr The difference beween he earlier model and his model is he number of servers. This is a muli -server model wih N number of servers whereas he earlier one was single server model. The assumpions saed in M/M/1 model are also assumed here. Here is he average service rae for N idenical service couners in parallel. For x=0 P(0) = [ N 1 x=0 ( ρ x x! + ρ N ) ] 1 (N 1)!(N ρ) (66.3) The probabiliy of x number of cusomers in he sysem is given by P(x). For 1 x N P(x) = ρx P(0) (66.4) x! March 8, 2017 66.4 Dr. Tom V. Mahew, IIT Bombay
Transporaion Sysem Engineering / N / N / N / N Arrivals Dispaching Deparures Arrivals Deparures =arrival rae discipline =arrival rae / N Server N Figure 66.6: Muli-server model / N Server N Figure 66.7: Muliple single server For x > N ρ x P(x) = P(0) (66.5) N!Nx N The average number of cusomers in he sysem is ρ N+1 E[X] = ρ+[ (N 1)!(N ρ) 2]P(0) (66.6) The average ueue lengh ρ N+1 E[L ] = [ (N 1)!(N ρ) 2]P(0) (66.7) The expeced ime in he sysem E[T] = E[X] The expeced ime in he ueue E[T ] = E[L ] 66.4.3 Numerical example (66.8) (66.9) Consider he earlier problem as a muli-server problem wih wo servers in parallel. Soluion Average arrival rae = = 300 vehicles/hr. Average service rae = = 3600 10 vehicles/hr. Uilizaion facor = raffic inensiy = ρ = = 300 360 = 0.833. P(0) = [ N 1 x=0 x! + ρ N ) ] 1 (N 1)!(N ρ) ( ρ x = 0.92(60) = 55.2min Average number of vehicles in he sysem is = L = ρ E[X] = ρ + [ N+1 (N 1)!(N ρ) ]P(0) = 1.22. The average 2 number of cusomers in he ueue = L = E[L ] = ρ N+1 [(N 1)!(N ρ) ]P(0)= 0.387. Expeced ime in he sysem =W = E[X] 2 = 0.004 hr = 14 sec. The expeced ime in he ueue =W = L = 0.00129 hr = 4.64 sec. 66.4.4 Muliple single servers model In his model here are N numbers of idenical independen parallel servers which receive cusomers from a same source bu in differen parallel ueues (Compare o M/M/N model. I has only one ueue) each one receiving cusomers a a rae of N. Fig. 66.7 shows how a ypical muliple single servers model looks like. 66.4.5 Numerical example Consider he problem 1 as a muliple single server s model wih wo servers which work independenly wih each one receiving half he arrival rae ha is 150 vehicles/hr. Soluion Mean arrival rae = = 150 vehicles/hr. Mean service rae = = 3600 10 vehicles/hr. Uilizaion facor = raffic inensiy = ρ = = 150 360 = 0.416. The percen of ime he oll booh will be idle = P(0) = P(X=0) = (0.416) 0 (1 0.416) = 0.584(60min)=35.04 min. The average number of vehicles in he sysem = E[X] = ρ = 0.712. The average number of vehicles in he ueue =L = ρ2 = 0.296. The average a vehicle spend in he sysem =E[T] = W = 1 = 0.0047 hr = 0.285min = 17.14sec. The averageime a vehicle spends in he ueue =E[T ] = W = ( ) = 0.0022hr = 0.13 min = 8.05 sec Dr. Tom V. Mahew, IIT Bombay 66.5 March 8, 2017
Transporaion Sysem Engineering M/M/1 model M/M/2 model Muliple single server model Idle ime of oll 8.34 55.2 35.04 boohs(minues) Number of vehicles 4.98 1.22 0.712 in he sysem(unis) Number of vehicles 4.01 0.387 0.296 in he ueue(unis) Average waiing ime 57.6 14 17.14 in sysem(seconds) Average waiing ime 50 4.64 8.05 in ueue(seconds) Comparison of he hree models From he Table 1 by providing 2 servershe ueue lengh reduced from 4.01 o 0.387 and he average waiing ime of he vehicles came down from 50 sec o 4.64 sec, bu a he expense of having eiher one or boh of he oll boohs idle 92% of he ime as compared o 13.9% of he ime for he single-server siuaion. Thus here exiss a rade-off beween he cusomers convenience and he cos of running he sysem. 66.4.6 D/D/N model In his model he arrival and service raes are deerminisic ha is he arrival and service imes of each vehicle are known. Assumpions 1. Cusomers are assumed o be paien. 2. Sysem is assumed o have unlimied capaciy. 3. Users arrive from an unlimied source. 4. The ueue discipline is assumed o be firs in firs ou. 66.4.7 Numerical example Morning peak raffic upsream of a oll booh is given in he able 2. The oll plaza consiss of hree boohs, each of which can handle an average of one vehicle every 8 seconds. Deermine he maximum ueue, he longes delay o an individual vehicle. Time period 10 min volume 7.00-7.10 200 7.10-7.20 400 7.20-7.30 500 7.30-7.40 250 7.40-7.50 200 7.50-8.00 150 Soluion The arrival volume is given in he able. Service rae is given as 8 seconds per vehicle. This implies for 10 min, 75 vehicles can be served by each server. I is given here are 3 servers. Hence 225 vehicles can be served by 3 servers in 10 min. In he firs 10 min only 200vehiclesarrivewhichare servedso he serviceraefor res 50 min is 225 veh/10 min as here is a ueue for he res period. The soluion o he problem is showed in he able 3 following. The cumulaive arrivals and services are calculaed in columns 3 and 5. Queue lengh a he end of any 10 min inerval is go by simply subracing column 5 from column 3 and is recorded in column 6. Maximum of he column 6 is maximum ueue lengh for he sudy period which is 300 vehicles. The service rae has been found ou as 225 vehicles per hour. From proporioning we ge he ime reuired for each ueue lengh o be served and as 475 vehicles is he max ueue lengh, he max delay is corresponding o his ueue. Therefore max delay is 21.11 min. 66.5 Conclusions The ueuing models ofen assume infinie numbers of cusomers, infinie ueue capaciy, or no bounds on inerarrival or service imes, when i is uie apparen ha hese bounds mus exis in realiy. Ofen, alhough he bounds do exis, hey can be safely ignored because he differences beween he real-world and heory is no sa- March 8, 2017 66.6 Dr. Tom V. Mahew, IIT Bombay
Transporaion Sysem Engineering Time 10 min Cum. Service Cumulaive Queue Delay period flow (3) rae(4) service(5) =(3)-(4) (6) 7.00-7.10 200 200 200 200 0 0 7.10-7.20 400 600 225 425 175 7.78 7.20-7.30 500 1100 225 650 450 20.00 7.30-7.40 250 1350 225 875 475 21.11 7.40-7.50 200 1550 225 1100 450 20.00 7.50-8.00 150 1700 225 1325 375 16.67 isically significan, as he probabiliy ha such boundary siuaions migh occur is remoe compared o he expeced normal siuaion. Furhermore, several sudies show he robusness of ueuing models ouside heir assumpions. In oher cases he heoreical soluion may eiher prove inracable or insufficienly informaive o be useful. Alernaive means of analysis have hus been devised in order o provide some insigh ino problems ha do no fall under he scope of ueuing heory, alhough hey are ofen scenario-specific because hey generally consis of compuer simulaions or analysis of experimenal daa. Deparmen of Civil engineering, Indian Insiue of Technology Bombay, India. References 1. James H Banks. Inroducion o ransporaion engineering. Taa Mc-Graw Hill, 2004. 2. Adolf D. May. Fundamenals of Traffic Flow. Prenice - Hall, Inc. Englewood Cliff New Jersey 07632, second ediion, 1990. 3. C S Papacosas. Transporaion engineering and planning by Papacosas. C. S, 3rd ediion, Prenice- Hall of India in 2001. Prenice-Hall of India, 2001. Acknowledgmens I wish o hank several of my sudens and saff of NPTEL for heir conribuion in his lecure. 66.6 Acknowledgmens I wish o hank my sudens Mr. Pradham Kumar and Dillip Rou for heir assisance in developing he lecure noe, and my saff Mr. Rayan in ypeseing he maerials. I also wish o hank several of my sudens and saff of NPTEL for heir conribuion in his lecure. I also appreciae your consrucive feedback which may be sen o vm@civil.iib.ac.in. Prof. Tom V. Mahew, Dr. Tom V. Mahew, IIT Bombay 66.7 March 8, 2017