Hypercube Queueing Models for Emergency … · Hypercube Queueing Models for Emergency Re- ......

Hypercube Queueing Models for Emergency Re-sponse Systems

Burak Boyacı

Nikolas Geroliminis

Urban Transport Systems Laboratory (LUTS) May 2014

STRC 14th Swiss Transport Research Conference

Monte Verità / Ascona, 14 – 16 May, 2014

Hypercube Queueing Models for Emergency Response Systems May 2014

Urban Transport Systems Laboratory (LUTS)

Hypercube Queueing Models for Emergency Response Sys-tems

Burak BoyacıManagement ScienceLancaster University Management SchoolB58A, LA1 4YX, Lancaster, United Kingdomphone: +44-1524-592732fax: [email protected]

Nikolas GeroliminisUrban Transport Systems LaboratoryEcole Polytechnique Fédérale de LausanneGC C2 389, Station 18, 1015 Lausanne,Switzerlandphone: +41-21-693 24 81fax: +41-21-693 50 [email protected]

May 2014

Abstract

Spatial queueing systems (SQS) can be defined as a type of queue that mobile servers areassigned to travel to the customer and provide on-scene service or the customers travel to servicefacilities to have service. It has a lot of application areas in literature from emergency responseto vehicle repair services, dial-a-ride to paratransit.

In this research, our aim is to find a rapid approach to calculate performance measures ofSQS. Our ultimate aim is to utilize this rapid approach as an instance solver inside someoptimization algorithms such as simulated annealing (SA) and variable neighborhood search(VNS) to find better location for systems such as ambulances, fire brigades. For this purpose, wehave developed two methods to calculate performance measures of an instance of SQS. To checkaccuracy and efficiency, the approach is compared with simulation results on some instances.Then the two methods are used with SA and VNS to improve server locations. Results showthat the approach is promising and can be applied as a tool inside some optimization algorithms.

Keywordsemergency response, spatial queues, hypercube queueing models, approximation algorithms

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1 Introduction

Emergency response systems are important for modern societies. They protect public health,provide assistance and ensure safety. Response areas of ambulances, design of police-beats orlocations of fire brigades are important decisions for these systems. Although the demand rate islow on average for emergency response systems, the service availability is important when theyare needed. In other words, in addition to adequate coverage, rapid and reliable response timesare also important for emergency response systems.

Location-allocation problems are the set problems in the operations research literature that areanalyzed extensively. Different than the general location-allocation problems, in emergencyresponse system location problems, both the demand and service time are stochastic whichresults in congestion and losses in every system. This stochasticity is addressed in manyresearches but the server based state of the system has been taken into consideration in a fewapproaches applicable to small-sized systems.

The model proposed by Larson (1974) models the problem as a spatial queueing system (SQS)which is also known as 2n hypercube queueing model (HQM). In this approach, each emergencyresponse unit is modeled as a server with two states available and busy. If n is the number ofservers in the system, there are 2n states in these models. In Larson (1974)’s model, time spenton the way to service is assumed to be negligible. He assumed, service rate is a function ofthe dispatched server but not the region receiving the service. This may be acceptable for firebrigades but not for ambulances.

In this research, we are proposing a new 3n HQM which enables to apply different service ratesfor different server-region pairs. In 3n HQM, each server has 3 states: available, busy insideits primary service area and outside its primary service area. However this new model can beintractable even more small sized problems (with more than 8 servers). For this purpose, wealso propose and aggregate model namely, 3n aggregate HQM (AHQM). In this last model,instead of keeping each servers’ condition separately at each state, it keeps number of servers indifferent states at each bin (i.e. set of servers). In our research for small cases we use 3n HQMmodel whereas for larger instances we implemented an algorithm that combines set of iterationscontaining partitioning, 3n HQM and AHQM, mix aggregate hypercube queueing algorithm(MHQA). Both 3n HQM and MHQA are used to calculate the performance of the locationsof emergency vehicles in Euclidean networks. We also implement two optimization methods,variable neighborhood search (VNS) and simulated annealing (SA) to improve the locations ofemergency vehicles on two networks.

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We continue the paper with literature survey. Afterwards, the existing 2n HQM and newlyproposed two models 3n HQM and AHQM are defined. In the next section, MHQA is described.The last two sections contain computational results and, conclusions with future researchdirections.

2 Previous Related Research

The early models dealing with the location of emergency response systems assume deterministicdemand. They ignored stochastic nature of the problem and dealt on coverage and medianmodels.

Median problems locate the facilities on discrete candidate locations that minimize averageresponse time or distance. Hakimi (1964) proposed p-median problem in which the main aimis to locate p facilities on a finite set of candidate locations in such a way that minimizes totaltransportation cost of n customers. Although it is a combinatorial optimization problem, thereare some exact algorithms (Galvão and Raggi, 1989, Avella and Sassano, 2001) and heuristicmethods (Daskin and Haghani, 1984, Schilling et al., 1993) as well. Mladenovic et al. (2007)wrote a survey which covers most of the literature on meta-heuristics about this subject.

Coverage models are used to locate limited number of facilities (i.e. emergency responsesystems) which maximize total coverage. Toregas et al. (1971) and Church and ReVelle (1974)approach coverage models from two different directions. In the probabilistic version of thisproblem, namely maximum availability location problem (MALP), the maximized value is theregions which are covered with α-reliability (Marianov and ReVelle, 1996). Daskin and Stern(1981) and Gendreau et al. (1997) altered the MCLP and proposed two models that maximizethe number of regions that are covered more than once.

Although the literature mainly covers static and deterministic location models, in recent modelsuncertainty is also taken into account. This uncertainty can be either related to planning futureperiods (dynamic models) or input model parameters (probabilistic models). Dynamic models

are suitable for models which, are considering the relocation of vehicles. The first article on thissubject is written by Ballou (1968) in which the main aim is to relocate a warehouse in sucha way that maximizes the profit in a finite horizon. Scott (1971) works with the extension ofthis problem with more than one facilities. Schilling (1980) extends MCLP with additional timeconstraint.

For urban problems, it is obvious that probabilistic models are the most suitable ones. For

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location and allocation of the emergency response systems, it is more convenient to model boththe demands and the duration of the time the facility serving these demands with probabilisticmodels. In these models, with some probability, it is always possible to have demand whichcannot be intervened by any facility, because of stochasticity in both demand and service times.Manne (1961), Daskin (1983), ReVelle and Hogan (1989) and, Marianov and ReVelle (1996)are some of the important articles written in this literature.

Larson (1974) proposed a hypercube queueing model (HQM) which is the first model that embedsthe queueing theory in facility location allocation problems. This model analyzes systems suchas emergency services (e.g. police, fire, ambulance, emergency repair), door-to-door pickupand delivery services (e.g. mail delivery, solid waste collection), neighborhood service centers(e.g. outpatient clinics, libraries, social work agencies) and transportation services (e.g. busand subway services, taxicab services, dial-a-ride systems) which has response district designand service-to-customer mode (Larson and Odoni, 1981). The solution of this model providesstate probabilities and associated system performance measures (e.g. workload, average servicerate, loss rate) for given server locations. “The HQM is not an optimization model; it is only adescriptive model that permits the analysis of scenarios” (Galvão and Morabito, 2008). HQMmodels the current state as a continuous-time Markov process but does not determine the optimalconfiguration. Police patrolling (Sacks and Grief, 1994) and ambulance location (Brandeau andLarson, 1986) are two applications modeled by HQM. Marianov and ReVelle (1996) extendedthe MALP and developed queueing maximum availability location problem.

The first model proposed by Larson (1974) assumes that the service time is independent ofthe locations of the calls for service and the dispatched unit. This argument was supported bythe idea that time spend on the road is negligible compared to service time. This can be a factfor fire brigades but not for the ambulances and on-demand vehicles. However even with thissimplification, as number of servers (n) increases, number of states (2n) grows exponentially. Asan extension, Atkinson et al. (2008) proposed a partial 3n HQM that assumes different servicerates for each server in the system with equal interdistrict or intradistrict responses. Iannoniand Morabito (2007) and Iannoni et al. (2008) embedded hypercube in a genetic algorithmframework to locate emergency vehicles along a highway. They extend the problem to enablemultiple dispatch (e.g. more than one server can intervene for the same incident). Geroliminiset al. (2009) integrate the location and distracting decisions in the same optimization and solvethe problem by using steepest descent for up to 10 servers. Geroliminis et al. (2011) extendedthe previous work to deal with larger instances with spatially homogeneous demand. They use agenetic algorithm that is using an entity called superdistrict which is similar to bin. However,they have not taken interactions between superdistricts into consideration which seems importantas it is shown later in this paper.

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3 Hypercube Queueing Models

In this section, we start by describing the Larson (1974)’s model. Then we describe the twomodels we propose: 3n HQM and 3n AHQM.

3.1 2n Hypercube Queueing Model

2n HQM proposed by Larson (1974). Each state name contains n binary variables where n standsfor the number of servers in the system. ith digit of the state name contains condition of serveri: available (0) or busy (1). For each region, which is named as atom in HQM literature, thereexists a priority list of servers. Atoms are served by the available server that has the highestpriority in their list. If there are no available server that can serve the atom, either the request islost or joins to a queue to be served later depending on the system structure. Both interarrivaltimes of incidents at each atom (λ j) and service times of servers (µi) are exponentially distributed.A transition diagram with three servers for a 2n HQM can be seen in Figure 1

010

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μ 3μ 3

Preference List

R1: S1 S2 S3

R2: S2 S3 S1

R3: S3 S1 S2

Figure 1: Larson (1974)’s 2n HQM for three servers with equal intra and interdistrict service rates(µi). State “011”, “111” and the transition connecting them is shown with differentcolors.

3.2 3n Hypercube Queueing Model

In our first proposed model, we define three states for each server. More precisely, dispatchingservers to primary service area (intradistrict) and secondary service area (interdistrict) are

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differentiated from each other. We use three states: available (0), busy with intradistrict (1) andbusy with interdistrict (2). A transition diagram of a 3n HQM with 3 servers can be seen inFigure 2. In this figure, µi and µ′i stands for the intradistrict and interdistrict service rates ofserver i respectively and λ j is the interarrival rate in atom j. Not that, in this system the serverhas always priority for its own intradistrict area. However, this does not prevent to have statessuch as “222”.

Preference List

R1: S1 S2 S3

R2: S2 S3 S1

R3: S3 S1 S2

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λ2+λ3

Figure 2: 3n HQM model for two servers with different intra (µi) and interdistrict (µ′i) servicerates. State “210”, states directly connected to it and transitions are colored differently.

To have an illustrative example, the transition equation for the state “210” is given below. Instate “210”, the rightmost digit (“0”) represents the first server and shows it is available; themiddle digit (“1”) shows the condition of the second server is busy with intradistrict responseand; the leftmost digit (“2”) indicates the third server is busy with interdistrict response. Pleasenote Pr shows the steady state probability of state r.

P210(λ1 + λ2 + λ3 + µ′3 + µ2

)= µ1P211 + µ′1P212 + λ2P200 + λ2P010 (1)

3.3 3n Aggregate Hypercube Queueing Model

As stated above, the size of 3n HQM makes it intractable even for medium sized problems. Forhis reason, we propose 3n AHQM which is less accurate but more efficient than 3n HQM. Inthis new model, we propose a new concept called bin instead of individual servers. Bins can be

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00

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Preference List

R1: S1 S2

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Figure 3: 3n AHQM for two bins containing two servers in each bin with different intra (µb)and interdistrict (µ′b) service rates, and primary demand areas (λ j). State “10/01” andstates connected to it are filled with different colors to show an example of transitionequations.

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regarded as group of servers. Instead of keeping each servers’ conditions separately in the states,the number of servers in different states at each bin is kept. Since the total number of serversis known for each bin, we represent the condition of each bin with two values. The number ofservers in intradistrict and interdistrict response. For µb and µ′b stands for the intradistrict andinterdistrict service rates of each server in bin b respectively and λ j is the interarrival rate inatom j, the transition diagram for a system with two bins with two servers each can be illustratedas in Figure 3.

In Figure 3 the transition equation for the state “10/01” (shown with red) can be written as inEquation 2. Please note, the first line of the state name “10” shows the condition of the first bin.The value on the left (“1”) shows the number of busy servers in intradistrict response and thevalue on the right (“0”) shows the number of busy servers in interdistrict response. Similarly,“0” and “1” shows the number of busy servers in intra and interdistrict responses in bin 2. Tosum up, in state “10/01” there are one busy server in intradistrict response in bin 1 and one busyserver in interdistrict response in bin 2.

Pr

1 (∃n : T (r, b, free) , 0

)∑j

λ j +∑

b

T (r, b, intra)µb +∑

b

T (r, b, inter)µ′b

=

∑q,b:

D(q,r,b,intra)=1

Pqµb +∑q,b:

D(q,r,b,inter)=1

Pqµ′b +

∑q,b:

D(r,q,b,intra)=1

Pb

∑j∈Rb

λ j +∑q,b:

D(r,q,b,inter)=1

Pb

∑j∈S (r,b)

λ j

(2)

For a 3n AHQM, if Cb equals to the number of servers in bin b, total number of states equals∏b

(Cb+2)(Cb+1)2 which is far less than 3n. For most of the cases with two bins, this value is even

less than 2n (i.e. for the cases with 8 or more servers). For instance, the system with 20 servershas over one million3 states in 2n and 3.5 billion states in 3n HQM whereas a 3n AHQM of twobins with 10 servers each has only 4356 states. In the next section, we describe the MHQA thatutilizes the two new 3n models, i.e. 3n HQM and 3n AHQM.

4 Mix Aggregate Hypercube Queueing Algorithm

The increase in the number of states makes 3n HQM not applicable to real life instances. Becauseof that, we propose 3n AHQM. However, 3n should be applied in a way that will keep the resultsin some accuracy level in an efficient way. In this section we will briefly describe this method.Further information will be given in a journal paper which is under review right now.

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Figure 4: An illustration of the partitioning approach.

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An illustration of the procedure can be seen in Figure 4. Figure 4a shows the whole problemarea with all servers (red dots). Dark color represents atoms of high demand and lighter colorthe ones of lower demand. Figure 4b shows the primary areas of responsibilities of each serverestimated with a Voronoi approach based on Euclidean distance. Figures 4b-d shows the resultsof the sequential partitioning method which is briefly described in this section.

In this method, the core subregions are merged to larger compound subregions, which aremodeled as a 3n AHQM. When compound regions are merged to form larger subregions, again3n AHQM is used. This process is repeated until the whole problem area is covered. Note that,at each merging step, only two subregions are used.

As stated above we developed MHQA to have accurate results in reasonable time. In order tohave that, we also need a partitioning algorithm that partitions the whole problem area as wewant. We try to develop a partitioning algorithm that satisfies the following properties:

1. The number of servers in each partition should be the parameter of the partitioningalgorithm. We need to set the number of servers in each subregion. There is a maximumsize that is efficiently solvable with both hypercube models and over partitioning (i.e.using more partitions than needed) decreases the accuracy of the final result.

2. Servers in the same partition should be adjacent to each other. This prevents disconnectedatoms and helps to have connected subregions which improves accuracy of the method.

3. Partitioning should be sequential in order to apply the approximation algorithm.4. Partitioning should be efficient. Our aim in developing an approximation algorithm is to

evaluate instances in an optimization framework. To do that, we need efficient algorithmsin all steps of the evaluation.

For this purpose an algorithm is developed that generates “cuts” composed of paths on theproblem area and creates subregions. This algorithm is applied on the Voronoi diagram of serverlocations. Afterwards, we solve one or more flow problems that divides the whole problemarea into two or more partitions. Readers who are interested in the mathematical model of thisapproach can refer to our journal paper which is under review right now.

5 Computational Results

After describing our methods briefly, in this section we first evaluate the accuracy of the modeldescribed in the previous two sections, by comparing them with a discrete event simulation.Then these methods are combined with two optimization methods variable neighborhood search

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22.2 17.7 20.5 20.9 22.0 20.5 19.7 18.8 15.9 20.9 10.1 9.4 11.4 10.7 9.5 9.1 11.3 8.4 8.3 10.4 10.7 8.9 9.7 8.6 11.8 11.4 8.0 9.7 9.3 9.9 8.2 8.2 11.9 11.5 8.8 11.1 10.3 9.5 10.5 11.6

20.0 22.7 16.7 22.3 17.8 17.0 16.6 19.7 23.7 16.6 11.2 8.3 10.7 10.3 8.0 11.6 10.3 9.8 10.4 11.7 11.8 9.5 11.4 10.2 8.2 8.5 11.2 9.1 11.4 11.8 10.7 9.1 11.9 11.4 9.7 9.2 9.2 11.1 8.2 11.8

22.1 18.0 20.3 22.0 15.9 23.3 22.3 17.5 17.8 21.0 9.1 10.1 8.9 11.0 10.5 11.6 8.6 8.3 11.9 10.4 10.7 10.1 11.4 10.9 10.0 9.0 10.6 8.2 9.8 10.5 10.1 11.7 11.1 9.5 10.5 11.9 8.9 11.6 8.0 10.1

22.0 17.0 16.6 23.2 16.6 23.8 23.3 21.8 20.4 21.0 8.7 8.1 8.0 9.5 8.8 10.3 8.1 11.5 8.6 10.8 8.0 11.7 11.8 10.7 11.2 10.9 11.2 11.1 11.8 11.5 9.4 9.1 10.5 11.0 10.5 11.9 10.7 9.2 8.8 11.2

19.2 22.8 17.5 18.2 21.5 22.1 21.6 22.1 17.2 23.0 8.8 9.1 8.9 11.1 11.7 9.3 10.1 10.8 11.9 10.1 11.2 8.1 9.4 11.4 8.7 10.3 9.0 10.8 9.6 8.5 8.5 9.5 9.1 10.3 8.0 11.1 10.9 10.4 9.0 8.4

20.8 17.6 17.5 17.0 23.2 18.0 20.9 22.5 23.4 19.8 11.7 11.0 10.6 9.4 11.4 10.1 10.5 9.0 11.1 8.1 11.4 8.0 8.1 10.2 10.6 10.6 8.8 8.4 9.2 9.1 8.3 8.0 8.5 8.9 9.7 9.0 9.5 11.1 11.4 8.4

22.6 22.6 19.0 22.4 20.5 23.3 19.2 19.1 20.3 22.2 11.0 10.9 10.1 10.7 11.6 10.0 8.0 10.0 8.8 9.6 11.2 11.3 11.8 11.4 10.5 9.3 9.9 9.4 9.0 11.0 8.4 9.6 9.4 10.1 11.0 9.0 11.5 9.8 11.8 10.7

22.4 20.7 19.2 21.3 18.1 19.4 23.2 16.6 20.8 17.7 8.8 11.7 8.7 8.2 9.4 9.7 8.6 9.5 11.3 10.4 8.5 11.4 10.8 11.1 11.9 10.2 8.4 11.1 11.3 10.4 11.5 8.2 11.5 10.7 8.8 8.9 10.2 10.9 9.0 9.6

18.4 22.0 19.2 23.1 16.4 21.7 21.3 18.2 18.3 19.0 11.8 9.0 8.1 11.4 9.4 11.1 8.1 8.9 8.4 8.7 11.6 11.0 11.9 9.9 11.6 9.1 11.7 11.1 11.3 9.6 8.9 9.4 11.0 9.1 11.8 9.8 10.3 10.6 8.1 10.8

17.2 19.6 21.1 19.7 18.1 17.1 21.6 18.1 19.1 16.5 11.0 10.3 8.5 9.2 11.1 9.6 10.8 10.0 11.5 11.8 10.9 11.1 10.0 10.9 11.1 10.3 10.2 10.4 11.2 8.5 9.7 8.2 9.6 10.5 11.6 10.1 10.1 11.8 11.5 9.8

9.6 10.6 11.4 9.1 8.9 10.7 10.2 9.0 9.4 11.8 9.0 11.3 10.1 8.5 11.5 8.7 8.2 8.8 10.0 9.6 9.9 11.9 8.7 10.1 11.8 9.6 9.2 9.2 8.7 11.9 18.3 19.5 21.9 17.6 17.4 16.5 19.1 18.5 21.2 19.0

11.7 10.8 9.1 8.8 11.8 9.6 8.7 9.2 8.8 10.4 10.8 8.0 11.4 11.5 9.3 11.4 9.9 8.9 8.2 9.8 11.0 8.3 8.0 8.0 11.3 10.5 10.2 11.3 9.2 9.4 21.6 16.3 17.5 16.8 21.0 22.6 20.5 18.2 23.4 18.8

9.1 8.7 8.6 9.9 10.7 11.2 11.4 11.7 11.6 10.7 8.3 8.1 8.3 9.1 11.7 9.8 10.2 9.3 11.3 11.0 11.8 11.7 10.2 8.3 10.4 11.0 9.3 8.6 11.0 10.4 21.9 22.6 21.8 16.5 16.2 21.8 17.5 17.1 22.3 21.8

8.7 10.1 9.0 10.6 11.1 11.6 9.2 8.0 10.7 10.0 9.2 10.6 10.5 10.0 11.6 10.0 8.1 11.2 9.5 11.7 11.7 11.3 8.8 10.2 11.0 11.7 9.1 10.0 9.0 8.3 21.5 23.3 20.7 19.1 19.5 23.5 18.6 23.6 19.5 19.8

8.5 8.0 10.1 11.8 9.8 11.0 9.6 8.7 9.9 9.4 8.7 10.7 8.4 8.7 11.0 8.6 8.2 10.3 9.2 8.4 11.7 10.2 8.2 10.1 11.3 8.0 9.7 8.6 8.8 9.8 20.9 23.8 19.9 22.8 18.1 22.6 18.4 19.8 18.5 23.2

11.7 11.1 8.0 8.0 8.1 9.0 11.2 11.1 10.1 8.9 9.1 8.5 8.3 10.4 9.7 10.0 11.9 8.8 11.8 10.0 11.9 8.8 10.1 10.1 8.6 9.0 10.7 11.9 11.4 8.7 16.2 22.7 18.4 20.9 21.5 23.4 20.6 22.0 18.9 17.3

9.5 9.0 10.5 8.4 11.3 9.4 8.4 10.6 11.0 9.6 11.7 11.9 10.9 11.5 8.6 10.6 11.0 11.0 11.8 11.8 9.5 10.9 10.7 8.1 10.2 9.3 9.8 11.5 11.7 11.4 16.0 16.0 21.2 17.2 23.5 23.4 20.4 20.2 21.3 23.0

11.6 9.6 8.9 8.0 9.1 10.0 11.8 8.0 8.8 11.6 8.9 9.8 9.4 11.0 10.6 11.9 10.1 11.0 11.8 9.5 11.1 9.7 8.2 11.2 8.1 8.7 10.8 11.8 10.3 9.0 21.7 16.2 19.7 17.2 22.5 21.7 21.5 20.1 22.4 18.9

8.7 9.0 10.6 8.0 11.6 9.4 9.8 10.8 10.1 8.7 9.0 9.1 8.5 9.6 8.0 9.0 10.7 10.9 9.8 9.7 11.6 11.6 9.6 10.1 9.0 11.3 8.1 10.5 11.3 9.3 20.4 17.0 17.0 21.9 20.2 20.1 16.6 18.2 18.4 18.2

8.6 9.7 9.1 9.2 9.8 11.8 8.1 11.3 9.5 8.8 9.6 11.6 11.4 11.4 9.8 11.0 11.8 8.1 8.9 8.7 11.5 10.3 8.1 10.4 11.3 10.5 8.3 9.1 10.6 11.6 21.7 18.9 19.8 19.3 17.7 18.4 23.9 17.9 19.2 18.1

Figure 5: The demand distribution of the network used in experiments.

(VNS) (Mladenovic and Hansen, 1997) and simulated annealing (SA) (Kirkpatrick et al., 1983)to find better facility locations. All of the algorithms in this work are developed under C# .NETenvironment. For partitioning algorithm IBM ILOG Cplex 12.4 is used through Concert userinterface. MATLAB 7.9 through MATLAB Automation Server interface is used for matrixoperations. All experiments are conducted on a PC with Intel Core2 Quad 3.00 Ghz processor.

We use an experimental network given in 5. Each square in this figure shows qkm2 area. Thevalue at each square shows 104 times the ratio of the arrival rate to total arrival rate. Euclideannorm is used in distance calculation but other norms can also be used with minor changes in thealgorithm. It is assumed that total service time equals to the sum of total travel time (going toscene and coming back from scene) and on scene service time.

5.1 Accuracy of 3n HQM

In this part, MHQA is compared with discrete event simulation for a case with 12 servers. Wetested following instances with three demand (5, 15, 45 instance/hour), average on-scene servicetime (5, 10, 20min) and accessibility distance (10, 15, 20kms) for each demand distributionwhich makes 27 scenarios in total and compare lost rates. We set the speed of servers to1km/min. On-scene service time and inter-arrival times are distributed exponentially. 500random instances with 12 facilities were generated for each setting. We used partitions of6 servers in approximation algorithm. Both simulation and our method are run over overexperimental network. The percentage of error in loss rates are reported in Figure 6. The ratioof the difference of lost rate between each simulation result and the MHQA result to the lost rate

10


found by simulation are reported as error.

0%

2%

4%

6%

8%

10%

12%

Pe

rce

nta

ge o

f Lo

st R

ate

Err

or

to T

ota

l De

man

d

Demand Rate (/hr) - On Scene Service Time (min) - Accessibility Distance (km)

Error for Experimental Network

Figure 6: The ratio of difference between the loss rates calculated by simulation and the MHQA.

25 parallel simulation instances were created with 11 batches simulating length of 50 dayseach. The mean of the last 10 batches of all 25 parallel simulations are reported. Length of thesimulations are selected in a way that calculated values have tight enough confidence intervalsto guarantee steady state.

For each instance, simulation took around 25s whereas our method took around 7s on average.The comparison showed that compared to simulation, our method gives results with acceptableerror (less than 5% error on average and 10% in the worst case) in less run time (28% ofsimulation run time). This error gets even less for increased server range, for the scenarioswith the range of 20km, average error is less than 1% and in 95% of the cases error is lessthan 2%. Furthermore, simulation needs to run longer to have accurate results if more detailedperformance measures (e.g. loss rate per number of busy servers) are needed.

5.2 Heuristics for Better Location of Servers

In this section, our exact 3n HQM (for cases with less than or equal to 8 servers) and mixaggregate hypercube (for cases with more than 8 servers) algorithms are tested inside twoheuristic approaches to identify close-to-optimum locations of servers: variable neighborhoodsearch (Mladenovic and Hansen, 1997) and simulated annealing (Kirkpatrick et al., 1983).Both methods are initialized with the maximum expected coverage location model (MEXCLP)(Daskin, 1983). MEXCLP is selected because it is efficient and gives good results. In VNSalgorithm, it is assumed that if two instances’ all but one servers are in different locations, they

11


#se

rver

s

onsc

ene

serv

.

time

dem

and

ME

XC

LP

VNS SA

value % impr. value % impr.

6

55 0.076 0.055 26.72 0.055 26.67

8 0.725 0.594 18.09 0.594 18.09

10 1.691 1.452 14.1 1.452 14.1

20

5 0.432 0.377 12.63 0.377 12.67

8 2.074 1.926 7.11 1.926 7.11

10 3.626 3.439 5.16 3.439 5.16

7

55 0.022 0.013 39.27 0.013 39.27

8 0.343 0.252 26.29 0.252 26.29

10 0.994 0.775 22 0.775 22

20

5 0.207 0.167 19.34 0.168 19.15

8 1.436 1.267 11.81 1.27 11.62

10 2.813 2.566 8.78 2.569 8.69

8

55 0.005 0.003 47.51 0.003 47.51

8 0.129 0.098 23.94 0.098 23.94

10 0.483 0.387 19.82 0.387 19.82

20

5 0.084 0.065 22.77 0.068 19.71

8 0.894 0.776 13.21 0.798 10.73

10 2.035 1.826 10.28 1.848 9.21

12

510 0.003 <1E-3 84.54 <1E-3 84.54

16 0.263 0.054 79.36 0.054 79.36

20 1.272 0.472 62.90 0.472 62.9

20

10 0.200 0.095 52.49 0.099 50.18

16 2.680 2.062 23.03 2.064 22.99

20 5.714 5.004 12.42 4.989 12.68

16

516 0.004 0.001 87.32 0.001 87.32

20 0.068 0.008 87.92 0.008 87.92

30 2.423 0.786 67.58 0.786 67.58

20

16 0.551 0.282 48.85 0.290 47.37

20 2.203 1.458 33.84 1.448 34.29

30 9.947 9.058 8.94 9.172 7.8

Table 1: The best lost rate found by MEXCLP, VNS and SA algorithms for the experimentalnetwork given in Figure 5.

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are neighbors. In every iteration, a randomly selected server’s location is changed. We use thesame neighborhood structure in SA algorithm. We set the starting “temperature” coefficient to1 and increase it by 10% in every 20 iterations. Temperature is assumed to be the division oftemperature coefficient with the average lost rate in every iteration. We have applied 3n HQMfor cases with 6,7 and 8 servers for total arrival rates of 5, 8, 10, 15 and 20 requests/hour. Forcases with 12 and 16 servers, arrival rates are doubled (i.e. 10, 16, 20, 30, 40 requests/hour),problems are solved by MHQA with two partitions of equal size. Run times for the former three(i.e. 6, 7 and 8-server) cases are set to one hour, whereas the latter two (i.e. 12 and 16-server)cases are run for four hours. We have applied two different on scene service times: 5 and 20minutes. For all cases, maximum accessibility distance is set to 30km. Found minimum lost rateand percent lost rate improvements after MEXCLP by both heuristics (i.e. VNS and SA) forcases with realistic lost ratios (ratio of lost rate to the total arrival rate) can be seen in table 1.

From the experiments, we observed that there is an improvement of more than 20% on averagefor the lost rates over MEXCLP. We have not observed any significant difference between VNSand SA. Since our primary goal in this research is to test the applicability of hypercube modelsinside search algorithms, we have not searched for parameters that may give better final resultsfor both VNS and SA. One can also observe that the lost rates dramatically increase with theincrease of on scene service time. Small increase in demand has also considerable influence onthe lost rate. Queueing systems are unpredictably complex and need custom-built algorithms tobe tested. We also observe that the performance of SA and VNS against MEXCLP get betterwith the increase in the number of servers. Last but not least, careful readers might realize thatfor the same value we have calculated different percent improvements. This is the consequenceof showing results with limited precision. We also noticed (not shown here) that even after 30minutes (instead of 4 hours) the VNS and SA methods provide similar improvement with a 4hour run.

6 Conclusions

In this short paper, we have briefly describe two new 3n hypercube models and two algorithmsthat utilize these two methods that we propose. For evaluating our methods, we first compareour approach’s results with the results of a discrete event simulation and show the accuracy ofour approach for different parameters on an experimental network. To see the applicability ofour two approximation algorithms (3n HQM and MHQA) inside an optimization framework,the two methods are implemented with variable neighborhood search and simulated annealing.Experiments have shown that, although hypercube queueing models are not optimization models,they can be utilized inside optimization frameworks.

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