Examples Of Applications Of Queueing Theory In Canada

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infor.46.4.003INFOR 046 004Techset Composition Ltd, Salisbury, U.K.1/10/2009INFOR, Vol. 46, No. 4, November 2008, pp. 29–46ISSN 0315-5986 j EISSN 1916-0615Examples of Applications of Queueing Theory in CanadaMarvin MandelbaumDepartment of Computer Science and Engineering, York University, 4700 Keele Street, Toronto, Canada M3J 1P3,e-mail: mandel@cse.yorku.caMyron HlynkaDepartment of Mathematics & Statistics, University of Windsor, Windsor, Canada N9B 3P4, e-mail: hlynka@uwindsor.caOKeywords Queueing theory, applications, history.NLYAbstract—As part of the 50th anniversary of the Canadian Operational Research Society, wereviewed queueing applications by Canadian researchers and practitioners. We concentrated onfinding real applications, but also considered theoretical contributions to applied areas that havebeen developed by the authors based on real applications. There were a surprising number ofapplications, many not well documented. Thus, this paper features examples of queueing theoryapplications over a spectrum of areas, years and types. One conclusion is that some of thesuccessful queueing applications were achieved and ameliorated by using simple principles gainedfrom studying queues and not by complex mathematical models.1. INTRODUCTIONThis paper is a collection of examples of applications of queueing analysis in Canada or applications by Canadian researchersand practitioners, showing their contributions since the 1950’s.This paper is dedicated to the 50th anniversary of CORS(Canadian Operational Research Society)1.We concentrated on finding applications to real queueingsituations as much as possible, but we also considered theoretical contributions to applied areas that have been discovered byauthors based on real applications. Hence, we were interested,to a lesser extent, in theoretical work directly motivated byreal applications and done by someone closely involved, withthe intent of helping the client organizations. Given that theintent was to obtain concrete and practical results, the workmay not have resulted in a publication, but rather led toimplementation or an internal report. Moreover, these applications often do not involve academic (or even consulting)queueing theorists. Thus, a survey of this nature can never becomplete since many of the applications are silent; there maybe no articles in the open literature and often there is no reportor documentation, and memories of what happened long agoPROOFhave faded. But our lack of completeness is also explained bythe time constraints of this survey.Queueing applications are abundant in Canada. The diverseareas where queueing analysis has been applied include: shipping and canals, grocery store checkout line count estimation,vehicle traffic flow, airline operations, airport terminal planning, forest fire management, medical wait times for surgery,patient scheduling, hospital service management, hospitalemergency room management, ambulance management,banking, bus, truck, railroad and train operations, productionand manufacturing, border crossings and customs, mail services, telecommunications and computer design, and callcentre operations and staffing.There is no good published description of the role ofCanadians in queueing applications or theory. There are anumber of papers on the birth of operational research inCanada, of which queueing was a fundamental part, and fromthem there is a hint at the role of Canadians in queueingresearch and application (Sandiford, 1963). Also, there are histories of queueing theory that include a little on Canadianresearchers’ contribution to theory. See the queueing historylink on Hlynka’s Queueing Theory Page (2008).In order to prepare the current article and given the issues ofincompleteness, lack of publications and fading memories, andknowing that we could not be all inclusive, our methodologywas as follows. We first issued a call for information bysending an e-mail to all current CORS members through theReceived 00 March 2008; Revision 00 September 2008, Accepted00 November 20081See the website http://www.cors.ca/29

infor.46.4.00330INFOR 046 004Techset Composition Ltd, Salisbury, U.K.1/10/2009MARVIN MANDELBAUM AND MYRON HLYNKAhighly likely not to be used, and may be incapable of influencingpolicies. Such studies do increase understanding, in a way similarto the second category. An example is a traffic system in which weconsider the effects of a behaviour change by vehicles users. If thebehaviour change is not likely to happen and if the parameters ofthe model are theoretical rather than based on data, we have a category four example.The presentation of the review below is ordered by application area.NLY2. TRANSPORTATIONIn this section we discuss queueing applications in the area oftransportation. This includes transportation on water, land andin the air. Transportation research played an important role inthe early days of the application of operational research andthis is reflected in the queueing applications we have selected.O2.1 The Welland CanalOne of the more successful projects that involved queues inCanada was a study3 done on the Welland Canal for theSt. Lawrence Seaway Authority by KCS, a firm established bya remarkable entrepreneur, Joe Kates, whose firm was involvedin pivotal operational research work in the 1960’s and 1970’s.(See section 2.4.2 for more on Joe Kates.) The Seaway, madeup of a succession of locks and canals, was built so that bothlarge inland vessels and “Salties” (ocean going vessels) couldgo all the way up the St. Lawrence river and Lake Ontario tothe Upper Great Lakes, including Chicago, Thunder Bay,Duluth and other important ports. For general descriptions ofthe seaway, see The St. Lawrence Seaway ManagementCorporation (2008a,b). This allowed a new influx of shippingtraffic to go through the Welland Canal. Canada faced a majorproblem when this new traffic and a rise in grain exportsclogged the Welland Canal in the early 1960’s, not long afterthe Seaway, as we know it today, had been built. The numberof ships wanting to get through the locks approached thesystem’s ultimate capacity and when it crossed the limit due tofluctuations in demand, queues started to build up. It appearedthat the system was unable to get rid of the queues withoutdenying access to some of the ships. As well, the shipping companies were troubled by the waiting times and the great capitaland operating expense of the ships caused by the delays.Canada’s foreign trade was also seriously threatened by thedelays and actual limitations of export capacity.Given the importance to Canada’s economy of shipping onthe Great Lakes, government authorities wanted to expand thelocks by twinning (creating parallel locks) but the cost amountedto billions of dollars. Moreover, the twinning project involvedmuch work and would have taken a long time. In 1962–63,PROOFCORS list server and to a list of Canadian queueing researchersculled from Hlynka’s Queueing Theory Page2. These attemptsresulted in a very low response rate and minimal success. Usingour knowledge and experience of people who worked in thefield, we telephoned many practitioners directly, as well as contacted companies or organizations in areas where we thoughtthere might be applications. Many of the people we contactedgave us information about others (living or dead, regardless ofacademic background) whose work included applied queueing.The interview approach with individuals, using the snowball technique, resulted in information on many of the applications discussed in this paper. We relied on the integrity of researchersand practitioners in telling us about these applications, becauseof the lack of documentation or references. We also had tomake choices as to what was included and what we thought metthe spirit of our intent. For example, what is meant by a“Canadian” application is somewhat vague. However everythingreferred to in this article has some Canadian connection and weused our discretion on an individual basis as to what was included.Hence, this is a very personal and unscientific study of queueingapplications from our memory and the memory of those interviewed and is not meant to be exhaustive. Accordingly, we consider this paper as a collection of examples. We apologize inadvance for any exclusions, omissions, errors, misrepresentations, or misinterpretations.The many applications we consider in this paper can be categorized in different ways. A first category includes cases when knowledge of queueing methods was actually applied to a real problemand affected the choice of policies used. Traffic signal changesmade by people with an understanding of queueing or using software with a queueing theoretical basis would be an example.Simulation techniques could be used in such situations as well.A second category actually applies queueing techniques successfully to model a real system. This includes data collectionand simulation techniques. The modelling does not necessarilyresult in changes to the system but rather increases one’s understanding and has the potential to lead to a change in policy withimproved performance. A study of ambulance services thatresults in no change would be an example.A third category is the use of common sense and includessuch areas as hospital emergency room triage nurses. Thesenurses certainly apply their expertise to assign priorities topatients and to provide the best service in the shortest time.But their applications of queueing could be labeled as applyingsound intuition to a difficult situation.A fourth category includes real situations that allow for aqueueing model, but a model with an abstraction level thatmight be of limited value to the practitioner. An example is anytheoretical queueing paper with a very real underlying modelbut which has not used data to select parameters and which is2This website contains information on queueing theory collected byMyron Hlynka of the University of Windsor. OR, Vol. 46, No. 4, November 2008, pp. 29–46ISSN 0315-5986 j EISSN 1916-06153Most of this story comes from interviews and correspondencewith Frank Collins, Lee Sims and Andrew Elek, former KCSpersonnel.DOI 10.3138/infor.46.4.029Copyright # 2008 INFOR Journal

INFOR 046 004Techset Composition Ltd, Salisbury, U.K.1/10/2009EXAMPLES OF APPLICATIONS OF QUEUEING THEORY IN CANADA31the company KCS was called in and given carte blanche to figureout what to do to get more ships through the canal, avoiding thehuge expenditure. By this time much engineering of the twinning project had already been done and no one gave muchhope to the success of the KCS contract. Joe Kates sold the government on the idea that by slightly increasing the capacity of thelocks it would be possible to eliminate the queues. His reasoningwas based on the nonlinearity of queueing performance tochanges in capacity. A slight increase in capacity can make adrastic reduction in queue size and waiting times or delays.We can see Joe’s idea by looking at the waiting time for asimple M/M/1 queue:1:m l.PROOFIf the arrival rate approaches the service rate (i.e. the system isnear capacity), then the wait is very large. But if we increase thecapacity by increasing the service rate by a small percentage,the average wait falls substantially more in percentage terms.For example, if the service rate is 1.1 and the arrival rate isone, then the expected wait is 10; however, if we raise theservice rate to 1.2 (which is less than a 10% increase), theexpected waiting time falls to five, a 50% drop.Actually, the locks would operate more like an M/D/1 queue.As a simple hypothetical but telling illustration, Joe explained it inthe following way: If the canal could handle 15 down bound shipsa day and 16 wanted to pass, in 270 days (by December) therewould be 270 ships in the queue, with a waiting time of 17 days.If the capacity were to be increased by just 10 percent, therewould be little or no queue. This is a case in which the rate ofarrival exceeds the service capacity. Joe promised that he couldincrease capacity enough to lower wait times to acceptable levels.Around 1965, he put a team of consultants on the job. Some ofthose participating in the work, besides Joe Kates, were FrankCollins, Lee Sims, Arthur Mittermaier and Peter Sandor, allprofessionals4 with industrial engineering and operational researchbackgrounds, and Peter was also an economist. The study costabout 10 million dollars, including all the experiments, and actually saved the Seaway Authority several billion dollars. Joe wasa very enthusiastic thinker and came up with many ideas on howto carry out the study and to make the system better. Several operational improvements were suggested, concerning vessel tie-upsand manoeuvres, meant to make the service more efficient,some of which were implemented rather promptly.Some operational measures to increase the capacitysuggested by KCS were:OE½Ws ¼after a ship had been dropped via a down-bound lockage,the lock was immediately filled with water again, withoutwaiting for the immediate availability of an up-boundvessel (i.e. from a small up-bound queue). This speeded upservice since an empty lockage could actually be accomplished in a few minutes instead of the long time it wouldhave taken to move a ship up as part of a complete cycle.Previously, this was only done occasionally in obvious circumstances. Now, with a simple calculation, it could bedetermined how many empty lockages the system couldafford, given the asymmetry of the demand. New trafficcontrol strategies led to much more use of this procedure tomaximize overall traffic flow.Multiple small ships in one lockage. Doubling and tripling up(or even more) of small ships in one lockage was suggested.When small ships showed up requesting passage they weretold to wait until enough showed up to fill a lock. Sincethis was contrary to past practice, this policy at firstangered many users. Later a booking system was introducedso that ships could sign up days in advance for specificpassage times, which could then be coordinated.Flying lockage. The usual procedure was to let a ship into thelock, immediately tie it to the lock sides, close the lock andthen raise or lower the water in the lock. At Lock 8 (PortColborne), a long lock with a low lift, the flying lockage procedure was not to tie the ship up but to allow the ship to keepon going in the lock and by the time the boat reached theother end the water level had changed and the locks wereopened and the boat sailed out. This speeded up the serviceoperation of the lock enormously.NLYinfor.46.4.003.Empty (turn back) lockages. When the traffic was imbalanced( primarily at the beginning of the season) KCS suggested thatsome of the lockages should be “empty”. This meant that4Most of these people later joined Kates, Peat, Marwick whenKCS merged with Peat, Marwick to form KPM around 1967. Somelater went on to work at IBI, an off shoot of KPM, like Neal Irwin,Lee Sims and Arthur Mittermaier. Peter Sandor has since passed away.INFOR, Vol. 46, No. 4, November 2008, pp. 29–46ISSN 0315-5986 j EISSN 1916-0615Some of the ideas to advance the study were:.Air time-lapse photos: Time-lapse photographs were takenfrom a helicopter to understand the interconnection of theSeaway’s system of locks and canals.Consultants imbedded with the locals: Local staff resisted theideas of a newcomer, hired by the Board of Directors, whoknew nothing about shipping. The local Seaway staff oftenpleaded that a new idea was unsafe or not operationallyimplementable. KCS had an engineer consultant live nearthe canal (at Lock 7) and become close with the operationalsupervisor and his staff. By developing an informal friendlyrelationship with the consultant, the supervisor agreed to tryout new procedures.The local consultant spent months watching the operations atthe locks, especially Lock Seven, which had the tightestspace available for operations and was the major bottleneck.A simulation model of the operation of the lock system interms of a series of queues was built to determine strategiesto improve the throughput of the system. One heuristic,based on a necessary condition for optimal throughput, wasto be sure that the locks where always full when there wasbalanced traffic. When this occurred there were queues inevery reach, the waterway between locks of the system.DOI 10.3138/infor.46.4.029Copyright # 2008 INFOR Journal

Techset Composition Ltd, Salisbury, U.K.1/10/2009MARVIN MANDELBAUM AND MYRON HLYNKAAnd if the locks were always full then at each move of a locka boat went up and down the lock so that queues betweenlocks always remained the same and the idea was to determine the optimal number of ships in each of the queues.PROOFImprovements were made to traffic control procedures, hardware and software, controller training, etc. As part of this, anextensive traffic control centre was built with vastly improvedcommunications, telemonitoring of operations and a dynamicphysical model of the system showing the current status of alllocks and ships.As a result of all the work there were dramatic improvementsin a relatively short time. Frank Collins indicated that through amultiplicity of improvements, the Canal’s service time wasreduced and resulted in a 40% increase in the ship handlingcapacity. People ceased talking about twinning the canal, andJoe became a hero. KCS established its reputation withTransport Canada. This reputation was inherited by KPMGand has lasted to this day.The improvements described above were significant but notunlimited. What about the long term anticipated growth? Alarger “Seaway-class ship” was born: a ship that exactlymatched the dimensions of the Seaway locks, with not aninch to waste. These larger ships had great economies ofscale compared to the older smaller ships. Gradually, as olderships were retired, fewer ships were needed to carry the samecargo.In the long run, growth of grain exports was negligible, asfarmlands have not grown and productivity improvementswere limited. But more importantly: much of the exportmarket shifted from the Atlantic to the Pacific. So, today,Seaway capacity is no longer a topic of concern.requesting service sometimes involved considerable waiting,suggesting that it might be beneficial to increase the numberof servers. To substantiate this hypothesis, it was necessary tocollect data on inter-arrival times, waiting times, and servicetimes. In his book, Lee presents some of the real data whichallowed a prediction of how many extra servers were neededand the resulting improvement in service. As part of the analysis, an M/M/c model was used.Passenger Check-in Process: Alec Lee also began usingqueueing theory to determine airline check-in counter capacityand policies in an airline terminal. Chapter 10 discusses thispassenger check-in process. Passengers often arrived ingroups and data were collected on the group size. One of themain points that emerged here was that it is worthwhile tohave an operational research team in place that can proposeand study alternative policies with the aim of improvingquality of service. The studies involved data collection, simulation, and validation aspects. Theoretical models were usedto estimate the effect of policy change before it wasimplemented, often using simulation. Some of this workappeared in Lee and Longton (1959).Passenger Air Terminal Sizing: Chapter 13 addresses theissue of passenger terminal sizing where one of the majorissues is space. Statistics were collected on previous terminalsto help make decisions. Short term parking is a type of queueing problem. This led to collecting real data on loss and probability of rejection. Pricing can be used as a control on thearrival and service rates of parking. Loading times for taxiswere recorded. Loading and unloading times for buses wererecorded. Terminal design must address not only the currentstate of affairs but also

have faded. But our lack of completeness is also explained by the time constraints of this survey. Queueing applications are abundant in Canada. The diverse areas where queueing analysis has been applied include: ship-ping and canals, grocery store checkout line count estimation, vehic

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