PPoint queue models: a unified approach
Wen-Long Jin ∗ October 7, 2018
Abstract
In transportation and other types of facilities, various queues arise when the demandsof service are higher than the supplies, and many point and fluid queue models have beenproposed to study such queueing systems. However, there has been no unified approachto deriving such models, analyzing their relationships and properties, and extendingthem for networks. In this paper, we derive point queue models as limits of two link-based queueing model: the link transmission model and a link queue model. With twodefinitions for demand and supply of a point queue, we present four point queue models,four approximate models, and their discrete versions. We discuss the properties of thesemodels, including equivalence, well-definedness, smoothness, and queue spillback, bothanalytically and with numerical examples. We then analytically solve Vickrey’s pointqueue model and stationary states in various models. We demonstrate that all existingpoint and fluid queue models in the literature are special cases of those derived fromthe link-based queueing models. Such a unified approach leads to systematic methodsfor studying the queueing process at a point facility and will also be helpful for studieson stochastic queues as well as networks of queues.
Keywords : Point queue models; link transmission model; link queue model; demandand supply; Vickrey’s point queue model; stationary states.
When the demands of service are higher than the supplies at such facilities as road networks,security check points in airports, supply chains, water reservoirs, document processors, taskmanagers, and computer servers, there arise queues of vehicles, customers, commodities,water, documents, tasks, and programs, respectively. Many strategies have been developed to ∗ Department of Civil and Environmental Engineering, California Institute for Telecommunications andInformation Technology, Institute of Transportation Studies, 4000 Anteater Instruction and Research Bldg,University of California, Irvine, CA 92697-3600. Tel: 949-824-1672. Fax: 949-824-8385. Email: [email protected] author a r X i v : . [ m a t h . D S ] M a y ontrol, manage, plan, and design such queueing processes so as to improve their performancein safety, efficiency, environmental impacts, and so on.Since last century, many queueing models have been proposed to understand the charac-teristics of such systems, including waiting times, queue lengths, and stationary states andtheir stability (Kleinrock, 1975; Newell, 1982). These models can be represented by arrivaland service of individual customers or accumulation and dissipation of continuum fluid flows;they can be continuous or discrete in time; and the arrival and service patterns of queueingcontents can be random or deterministic. In addition, a queue can be initially empty or not,the storage capacity of a queue can be finite or infinite, there can be single or multiple servers,and the serving rule can be First-In-First-Out or priority-based. Traditionally, queueingtheories concern with random arrival and service processes of individual customers (Lindley,1952; Asmussen, 2003; Kingman, 2009).In recent years, fluid queue models have gained much attention in various disciplines(Kulkarni, 1997). In these models, the dynamics of a queueing system are described bychanges in continuum flows of queueing contents. Such models were first proposed for damprocesses in 1950’s (Moran, 1956, 1959). Since then, both discrete and continuous versions offluid queue models have been extensively discussed with random arrival and service patterns.In the transportation literature, point queue models were introduced to study the congestioneffect of a bottleneck with deterministic, dynamic arrival patterns (Vickrey, 1969). Suchmodels have been applied to study dynamic traffic assignment problems (Drissi-Ka¨ıtouni andHameda-Benchekroun, 1992; Kuwahara and Akamatsu, 1997; Li et al., 2000). In (Nie andZhang, 2005), Vickrey’s point queue model was compared with other network loading models.In (Armbruster et al., 2006b), a point queue model for a supply chain was proposed and shownto be consistent with a hyperbolic conservation law. In (Ban et al., 2012), continuous pointqueue models were presented as differential complementarity systems. In (Han et al., 2013),Vickrey’s point queue model was formulated with the help of the LWR model (Lighthill andWhitham, 1955; Richards, 1956). In existing studies on point queue models, the downstreamservice rate is usually assumed to be constant, and the storage capacity of the point queueinfinite. Note that fluid queue models can also be considered as point queue models, butwith a finite storage capacity and general service rates, since the dimension of a server or areservoir is also assumed to be infinitesimal. However, there exists no unified approach toderiving such point queue models, analyzing their relationships and properties, and extendingthem for networks.In this study we attempt to fill this gap by presenting a unified approach, which isbased on two observations. The first observation is that many traffic flow models can beviewed as queueing models, since traffic flow models describe the dynamics associated withthe accumulation and dissipation of vehicular queues. For examples, the Cell TransmissionModel (CTM) describes dynamics of cell-based queues (Daganzo, 1995; Lebacque, 1996);the Link Transmission Model (LTM) describes dynamics of link-based queues (Ypermanet al., 2006; Yperman, 2007); and the Link Queue Model (LQM) also describes dynamics oflink-based queues (Jin, 2012b). Among these models, continuous versions of LTM and CTM2re equivalent to but different formulations of network kinematic wave models, which canadmit discontinuous shock wave solutions and can be unstable in diverge-merge networks (Jin,2014), but LQM admits smooth solutions and is stable (Jin, 2012b). The second observationis that a point facility can be considered as the limit of a link: if we shrink the link length butmaintain the storage capacity, then we obtain a point queue. Therefore we can derive pointqueue models as limits of the link-based queueing models: LTM and LQM. In this approach,point queue models inherit two critical components from both LTM and LQM: we first definedemand and supply of a point queue, and then apply macroscopic junction models, whichwere originally proposed in CTM and later adopted for both LTM and LQM, to calculateboundary fluxes from upstream demands and downstream supplies. This approach enablesus to derive all existing point and fluid queue models and their generalizations, discuss theirrelationships, and analyze their properties.The rest of the paper is organized as follows. After reviewing LTM and LQM in Section2, we take the limits of the demand and supply of a link to obtain two demand formulasand two supply formulas for a point queue in Section 3. Further with different combinationsof demand and supply formulas, we derive four continuous point queue models and theirdiscrete versions. In Section 4, we present four approximate point queue models and theirdiscrete versions. In Section 5, we present some analytical solutions of these models. InSection 6, we present numerical results. Finally in Section 7, we summarize the study anddiscuss future research topics. In this section, we consider a homogeneous road link, shown in Figure 1, whose length is L and number of lanes N . Vehicles on the road segment form a link queue. We assume thatthe road has the following triangular fundamental diagram of flow-density relation (Munjalet al., 1971; Haberman, 1977; Newell, 1993): q = Q ( k ) = min { V k, ( N K − k ) W } , (1)where k ( x, t ) is the total density at x and t , q ( x, t ) the total flow-rate, K the jam densityper lane, V the free-flow speed, and − W the shock wave speed in congested traffic. Wedenote the harmonic mean of V and W by U = V WV + W . We also denote T = LV , which isthe free-flow traverse time, T = LW , which is the shock wave traverse time, and T = LU .Generally T < T < T . Thus for each lane, the critical density is ¯ K = WV + W K = UV K ,and the capacity is C = V ¯ K = U K . We denote the storage capacity of the road link byΛ =
N LK and the initial number of vehicles by Λ = N Lk , where the initial density isconstant at N k at all locations on the road. Then the total capacity is N C = Λ T .For a road link shown in Figure 1, we define the upstream and downstream cumulativeflows by F ( t ) and G ( t ), respectively, and set G (0) = 0 and F (0) = Λ . Correspondingly the3igure 1: An illustration of a homogeneous road linkupstream in- and downstream out-fluxes are denoted by f ( t ) and g ( t ). Then we havedd t F ( t ) = f ( t ) , (2a)dd t G ( t ) = g ( t ) . (2b)In addition, we denote the number of vehicles on the road by ρ ( t ) = (cid:82) L k ( x, t ) dx , whichsatisfies ρ ( t ) = F ( t ) − G ( t ) . (3)There are two formulations for a link-based queueing model with two types of statevariables: either the accumulation of vehicles, such as ρ ( t ), or the cumulative flows, F ( t ) and G ( t ), can be used as state variables. In addition, the boundary fluxes are calculated in twosteps: first, a link’s demand, d ( t ), and supply, s ( t ), are defined from state variables; second,the flux through a boundary can be calculated from the upstream demand and downstreamsupply with macroscopic junction models: f ( t ) = min { δ ( t ) , s ( t ) } , (4a) g ( t ) = min { d ( t ) , σ ( t ) } , (4b)where δ ( t ) is the origin demand, and σ ( t ) the destination supply. LTM and LQM differ inthe definitions of demand and supply functions and the corresponding state variables relatedto vehicle accumulations. In LTM, two new variables related to vehicle accumulation are the link queue size, λ ( t ), andthe link vacancy size, γ ( t ), which are given by (Jin, 2014): λ ( t ) = (cid:26) Λ T t − G ( t ) , t ≤ T F ( t − T ) − G ( t ) , t > T (5a) γ ( t ) = (cid:26) Λ − Λ T t + Λ − F ( t ) , t ≤ T G ( t − T ) + Λ − F ( t ) , t > T (5b)4ince both F ( t ) and G ( t ) are continuous, then λ ( t ) and γ ( t ) are continuous with λ (0) = 0and γ (0) = 0.The link demand and supply in LTM are defined as (Jin, 2014): d ( t ) = min (cid:110) Λ T + H ( λ ( t )) , Λ T (cid:111) , t ≤ T min (cid:110) f ( t − T ) + H ( λ ( t )) , Λ T (cid:111) , t > T (6a) s ( t ) = min (cid:110) Λ − Λ T + H ( γ ( t )) , Λ T (cid:111) , t ≤ T min (cid:110) g ( t − T ) + H ( γ ( t )) , Λ T (cid:111) , t > T (6b)where the indicator function H ( y ) for y ≥ H ( y ) = lim ∆ t → + y ∆ t = (cid:26) , y = 0 ∞ , y > λ ( t ) and γ ( t ), are the state variables, we have from(5): dd t λ ( t ) = (cid:26) Λ T − g ( t ) , t ≤ T f ( t − T ) − g ( t ) , t > T (8a)dd t γ ( t ) = (cid:26) Λ − Λ T − f ( t ) , t ≤ T g ( t − T ) − f ( t ) , t > T (8b)where f ( t ) and g ( t ) are calculated from (4) and (6). In this formulation, the rates ofchange in both λ ( t ) and γ ( t ) depend on λ ( t ), γ ( t ), f ( t − T ), and g ( t − T ).2. When cumulative flows, F ( t ) and G ( t ), are the state variables, we have from (2):dd t F ( t ) = min { δ ( t ) , s ( t ) } , (9a)dd t G ( t ) = min { d ( t ) , σ ( t ) } . (9b)where d ( t ) and s ( t ) are calculated from (5) and (6). In this formulation, the rates ofchange in F ( t ) and G ( t ) depend on F ( t ), G ( t ), F ( t − T ), G ( t − T ), f ( t − T ), and g ( t − T ).In both formulations, f ( t − T ), and g ( t − T ) can be calculated from the corresponding statevariables at t − T and t − T . In addition, the right-hand sides of both formulations arediscontinuous, and LTM is a system of discontinuous ordinary differential equations withdelays. Thus their solutions may not be differentiable or smooth.5 .2 The link queue model (LQM) In LQM, ρ ( t ) can be used as a state variable. The link demand and supply in LQM aredefined as (Jin, 2012b): d ( t ) = min { ρ ( t ) T , Λ T } , (10a) s ( t ) = min { Λ − ρ ( t ) T , Λ T } . (10b)From (2), (3), (4), and (10) we obtain the following two continuous formulations of LQM:1. When ρ ( t ) is the state variable, we have:dd t ρ ( t ) = min { δ ( t ) , Λ − ρ ( t ) T , Λ T } − min { ρ ( t ) T , Λ T , σ ( t ) } . (11)2. When F ( t ) and G ( t ) are the state variables, we have:dd t F ( t ) = min { δ ( t ) , Λ − ( F ( t ) − G ( t )) T , Λ T } , (12a)dd t G ( t ) = min { F ( t ) − G ( t ) T , Λ T , σ ( t ) } . (12b)In both formulations, the right-hand sides are continuous, and LQM are systems of continuousordinary differential equations without delays. Therefore, solutions of LQM are smooth withcontinuous derivatives of any order. We consider a point facility as a limit of a road link, shown in Figure 2, with the same storagecapacity, Λ, origin demand, δ ( t ), and destination supply, σ ( t ). The macroscopic junctionmodels are also the same as (4). But we let the link length L →
0, and the number of lanes N = Λ LK → ∞ . Parking lots, security check points, water reservoirs, document processors,task managers, and computer servers can all be approximated by point facilities. The queuein a point facility is referred to as a point queue. Based on the observation that a point facility is a limit of a link, we can derive demand andsupply of a point queue from (6) in LTM and (10) in LQM by letting L → Lemma 3.1
When L → , from the queue and vacancy sizes of LTM, (5), we obtain thequeue and vacancy sizes of a point queue as (for t ≥ + ) λ ( t ) = F ( t ) − G ( t ) , (13a) γ ( t ) = Λ − λ ( t ) . (13b) Proof . From (5a) we have λ ( T ) = Λ − G ( T ) and λ ( t ) = F ( t − T ) − G ( t ) for t > T .When L →
0, we have T → + , λ (0 + ) = Λ and λ ( t ) = F ( t ) − G ( t ) for t >
0. Thus λ ( t ) = F ( t ) − G ( t ) for t ≥ + .Similarly from (5b) we have γ ( T ) = Λ − F ( T ) and γ ( t ) = G ( t − T ) + Λ − F ( t ) for t > T . When L →
0, we have T → + , γ (0 + ) = Λ − Λ and γ ( t ) = Λ − ( F ( t ) − G ( t )) for t >
0. Thus γ ( t ) = Λ − λ ( t ) for t ≥ + . (cid:4) From Lemma 3.1, we can see that, in a point queue, λ ( t ) = ρ ( t ) = Λ − γ ( t ). Thus we canuse λ ( t ), the queue size, to uniquely represent the vehicle accumulation. Lemma 3.2
When L → , from the demand and supply in LTM, (6), we obtain the demandand supply of a point queue as (for t ≥ + ) d ( t ) = δ ( t ) + H ( λ ( t )) , (14a) s ( t ) = σ ( t ) + H (Λ − λ ( t )) . (14b) Proof . When t > T and t > T , we have from (4) and (6) d ( t ) = min { min { δ ( t − T ) , s ( t − T ) } + H ( λ ( t )) , Λ T } ,s ( t ) = min { min { d ( t − T ) , σ ( t − T ) } + H ( γ ( t )) , Λ T } . L →
0, then T , T , T → + , γ ( t ) → Λ − λ ( t ), and d ( t ) = min { δ ( t ) , s ( t ) } + H ( λ ( t )) ,s ( t ) = min { d ( t ) , σ ( t ) } + H (Λ − λ ( t )) . Therefore, when λ ( t ) < Λ for t >
0, then s ( t ) = ∞ , and d ( t ) = δ ( t ) + H ( λ ( t )); when λ ( t ) > t >
0, then d ( t ) = ∞ , and s ( t ) = σ ( t ) + H (Λ − λ ( t )). Thus we obtain (14). (cid:4) Lemma 3.3
When L → , from the demand and supply in LQM, (10), we obtain the demandand supply of a point queue as (for t ≥ + ) d ( t ) = H ( λ ( t )) , (15a) s ( t ) = H (Λ − λ ( t )) . (15b) Proof . When L → T , T , T →
0. Thus d ( t ) = 0 when ρ ( t ) = 0 and ∞ otherwise. Similarly, s ( t ) = 0 when ρ ( t ) = Λ and ∞ otherwise. Hence (15) is correct. (cid:4) Clearly the demand and supply functions derived from LTM are different from those fromLQM. They can lead to different point queue models.
To model dynamics of a point queue, we can use the queue length, λ ( t ), or the cumulativeflows, F ( t ) and G ( t ), as state variables. Once the demand and supply functions are given,from (2), (3), (4) we can derive two formulations of a point queue model:A. With λ ( t ) as the state variable, we havedd t λ ( t ) = min { δ ( t ) , s ( t ) } − min { d ( t ) , σ ( t ) } , (16)which is referred to as A-PQM.B. With F ( t ) and G ( t ) as the state variables, we havedd t F ( t ) = min { δ ( t ) , s ( t ) } , (17a)dd t G ( t ) = min { d ( t ) , σ ( t ) } , (17b)which is referred to as B-PQM.In the preceding subsection, we obtain from LTM and LQM two different definitionsfor both demand and supply of a point queue, which are functions of λ ( t ) or F ( t ) and G ( t ).Therefore, as shown in Table
1, we then obtain four combinations of demand and supplyfunctions, which lead to four point queue models. Here PQM1 is the limit of LTM, PQM2the limit of LQM, but PQM3 and PQM4 are mixtures of the limits of both LTM and LQM. Without loss of generality we assume that δ ( t ), σ ( t ), d ( t ), and s ( t ) are all continuous in time. (cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80) s ( t ) d ( t ) δ ( t ) + H ( λ ( t )) H ( λ ( t )) σ ( t ) + H (Λ − λ ( t )) PQM1 PQM4 H (Λ − λ ( t )) PQM3 PQM2Table 1: Demand and supply functions for four point queue models (PQMs): λ ( t ) = F ( t ) − G ( t ) Definition 3.4
A continuous point queue model is well-defined if and only if λ ( t ) is alwaysbetween 0 and Λ in A-PQM; or G ( t ) ≤ F ( t ) ≤ G ( t ) + Λ in B-PQM. Theorem 3.5
With demand and supply functions in
Table
1, all four point queue modelsare well-defined. That is, λ ( t ) is always between 0 and Λ in (16), and G ( t ) ≤ F ( t ) ≤ G ( t ) + Λ in (17).Proof . When λ ( t ) = 0, s ( t ) = ∞ , and dd t λ ( t ) = δ ( t ) − min { d ( t ) , σ ( t ) } ≥
0, since d ( t ) ≤ δ ( t );when λ ( t ) = Λ, d ( t ) = ∞ , and dd t λ ( t ) = min { δ ( t ) , s ( t ) } − σ ( t ) ≤
0, since s ( t ) ≤ σ ( t ).Similarly from we can see that G ( t ) ≤ F ( t ) ≤ G ( t ) + Λ in (17). (cid:4) Theorem 3.6
The four point queue models are equivalent.Proof . When 0 < λ ( t ) < Λ, we have d ( t ) = s ( t ) = ∞ in all four point queue models, anddd t λ ( t ) = δ ( t ) − σ ( t ) for all models. When λ ( t ) increases from 0 or decreases from Λ, thedemand and supply functions and, therefore, rates of change in λ ( t ) are different for thesemodels. But since the number of such instants is countable, the integrals of different rates ofchange at such instants are zero. That is, λ ( t ) is the same for all models, when they havethe same origin demand and destination supply. This leads to the equivalence among allmodels. (cid:4) If the storage capacity of a point facility is infinite; i.e., if Λ = ∞ , then its supply s ( t ) = ∞ in all four models. In this case, PQM1 and PQM3 are the same, and so are PQM2 and PM4.In particular, A-PQM1 becomesdd t λ ( t ) = max {− H ( λ ( t )) , δ ( t ) − σ ( t ) } = (cid:26) δ ( t ) − σ ( t ) , λ ( t ) > { , δ ( t ) − σ ( t ) } , λ ( t ) = 0 (18)which is referred to as A-PQM1i, and B-PQM1 becomesdd t F ( t ) = δ ( t ) (19a)dd t G ( t ) = (cid:26) min { δ ( t ) , σ ( t ) } , F ( t ) = G ( t ) σ ( t ) , F ( t ) > G ( t ) (19b)9hich is referred to as B-PQM1i. Similarly, we can also obtain the special cases of PQM2.In A-PQM1i, if we allow δ ( t ), σ ( t ), and λ ( t ) to be random variables, then (18) is thesame as the traditional fluid queue model (Kulkarni, 1997, Equation 1) with the differencebetween origin demand and destination supply, δ ( t ) − σ ( t ), as the drift function. Also inA-PQM1i, if the destination supply, σ ( t ), is constant, then (18) is the same as the continuousversion of Vickrey’s point queue model (Han et al., 2013, Equations 2.4 and 2.5 with t = 0).Therefore, PQM1 and PQM3 are generalized versions of existing continuous fluid queue andVickrey’s point queue models, since it also applies to a queue with a finite storage capacityand arbitrary origin demand and destination supply patterns. (cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88)(cid:88) s ( t )∆ t d ( t )∆ t δ ( t )∆ t + λ ( t ) λ ( t ) σ ( t )∆ t + Λ − λ ( t ) PQM1-D PQM4-DΛ − λ ( t ) PQM3-D PQM2-DTable 2: Demand and supply functions for four discrete point queue models: λ ( t ) = F ( t ) − G ( t )For four continuous point queue models, we discretize them with a time-step size of ∆ t such that H ( y )∆ t = y . Then with the discrete demand and supply functions during from t to t + ∆ t , shown in Table
2, (16) and (17) can be numerically solved.1. With λ ( t ) as the state variable, we have the following four discrete models by replacingdd t λ ( t ) by λ ( t +∆ t ) − λ ( t )∆ t in (16): λ ( t + ∆ t ) = λ ( t ) + min { δ ( t )∆ t, s ( t )∆ t } − min { d ( t )∆ t, σ ( t )∆ t } . (20)2. With F ( t ) and G ( t ) as the state variables, we have the following four discrete modelsby replacing dd t F ( t ) by F ( t +∆ t ) − F ( t )∆ t and dd t G ( t ) by G ( t +∆ t ) − G ( t )∆ t in (17): F ( t + ∆ t ) = F ( t ) + min { δ ( t )∆ t, s ( t )∆ t } , (21a) G ( t + ∆ t ) = G ( t ) + min { d ( t )∆ t, σ ( t )∆ t } . (21b)In particular, A-PQM3-D can be written as λ ( t + ∆ t ) = min { δ ( t )∆ t + λ ( t ) , Λ } − min { δ ( t )∆ t + λ ( t ) , σ ( t )∆ t } , (22)which is the discrete dam process model in (Moran, 1959, Chapter 3). In this sense, PQM3 isthe continuous version of Moran’s model. 10 efinition 3.7 A discrete point queue model, (20), is well-defined if and only if λ ( t + ∆ t ) isbetween and Λ for λ ( t ) between and Λ ; a discrete point queue model, (21), is well-definedif and only if G ( t + ∆ t ) ≤ F ( t + ∆ t ) ≤ G ( t + ∆ t ) + Λ for G ( t ) ≤ F ( t ) ≤ F ( t ) + Λ . Then we have the following theorem.
Theorem 3.8
PQM1-D and PQM2-D are always well-defined with any ∆ t ; PQM3-D iswell-defined if and only if ∆ t ≤ Λ σ ( t ) ; and PQM4-D is well-defined if and only if ∆ t ≤ Λ δ ( t ) .Thus when ∆ t ≤ Λmax t { δ ( t ) , σ ( t ) } , (23) all four discrete models are well-defined.Proof . For A-PQM1-D, we have λ ( t + ∆ t ) = min { δ ( t )∆ t + λ ( t ) , σ ( t )∆ t + Λ } − min { δ ( t )∆ t + λ ( t ) , σ ( t )∆ t } . Since σ ( t )∆ t +Λ > σ ( t )∆ t , there are three cases: (i) when δ ( t )∆ t + λ ( t ) ≤ σ ( t )∆ t , λ ( t +∆ t ) =0; (ii) when σ ( t )∆ t < δ ( t )∆ t + λ ( t ) < σ ( t )∆ t + Λ, λ ( t + ∆ t ) = δ ( t )∆ t + λ ( t ) − σ ( t )∆ t < Λ;(iii) when δ ( t )∆ t + λ ( t ) ≥ σ ( t )∆ t + Λ, λ ( t + ∆ t ) = Λ. In all three cases, λ ( t + ∆ t ) ∈ [0 , Λ],and A-PQM1-D is well-defined.For A-PQM2-D, we have λ ( t + ∆ t ) = min { δ ( t )∆ t + λ ( t ) , Λ } − min { λ ( t ) , σ ( t )∆ t } . Since λ ( t ) ≤ min { δ ( t )∆ t + λ ( t ) , Λ } ≤ Λ, there are two cases: (i) when σ ( t )∆ t ≤ λ ( t ), λ ( t + ∆ t ) = min { δ ( t )∆ t + λ ( t ) , Λ } − σ ( t )∆ t ∈ [0 , Λ]; (ii) when σ ( t )∆ t > λ ( t ), λ ( t + ∆ t ) =min { δ ( t )∆ t + λ ( t ) , Λ } − λ ( t ) ∈ [0 , Λ]. Thus A-PQM2-D is well-defined.For A-PQM3-D, we have λ ( t + ∆ t ) = min { δ ( t )∆ t + λ ( t ) , Λ } − min { δ ( t )∆ t + λ ( t ) , σ ( t )∆ t } . If Λ < σ ( t )∆ t ≤ δ ( t )∆ t + λ ( t ), then λ ( t + ∆ t ) = Λ − σ ( t )∆ t <
0, and the discrete modelis not well-defined. However, if Λ ≥ σ ( t )∆ t or ∆ t ≤ Λ σ ( t ) , there are three cases: (i) when δ ( t )∆ t + λ ( t ) ≤ σ ( t )∆ t , λ ( t + ∆ t ) = 0; (ii) when σ ( t )∆ t < δ ( t )∆ t + λ ( t ) < Λ, λ ( t + ∆ t ) = δ ( t )∆ t + λ ( t ) − σ ( t )∆ t ∈ (0 , Λ); (iii) when δ ( t )∆ t + λ ( t ) ≥ Λ, λ ( t + ∆ t ) = Λ − σ ( t )∆ t ∈ [0 , Λ].Thus A-PQM3-D is well-defined if and only if ∆ t ≤ Λ σ ( t ) .For A-PQM4-D, we have λ ( t + ∆ t ) = min { δ ( t )∆ t + λ ( t ) , σ ( t )∆ t + Λ } − min { λ ( t ) , σ ( t )∆ t } . If Λ < δ ( t )∆ t and σ ( t )∆ t is very large such that δ ( t )∆ t + λ ( t ) ≤ σ ( t )∆ t +Λ and λ ( t ) ≤ σ ( t )∆ t ,then λ ( t + ∆ t ) = δ ( t )∆ t > Λ, and the discrete model is not well-defined. However, if Λ ≥ ( t )∆ t or ∆ t ≤ Λ δ ( t ) , there are two cases: (i) when σ ( t ) ≤ λ ( t ) ≤ min { δ ( t )∆ t + λ ( t ) , σ ( t )∆ t +Λ } , λ ( t + ∆ t ) = min { δ ( t )∆ t + λ ( t ) , σ ( t )∆ t + Λ } − λ ( t ) ∈ [0 , Λ]; (ii) when σ ( t ) > λ ( t ), λ ( t + ∆ t ) =min { δ ( t )∆ t, σ ( t )∆ t + Λ − λ ( t ) } ∈ [0 , Λ]. Thus A-PQM4-D is well-defined if and only if∆ t ≤ Λ δ ( t ) .We can similarly prove that the other formulations with F ( t ) and G ( t ) as the statevariables are also well-defined under the respective conditions. Furthermore, all discretemodels are well-defined when (23) is satisfied. (cid:4) For a point queue with an infinite storage capacity with Λ = ∞ , all the discrete modelsare well-defined for any ∆ t , since (23) is always satisfied. In addition, PQM1-D and PQM3-Dare the same and can be simplified as λ ( t + ∆ t ) = max { , δ ( t )∆ t + λ ( t ) − σ ( t )∆ t } , (24)or F ( t + ∆ t ) = F ( t ) + δ ( t )∆ t, (25a) G ( t + ∆ t ) = min { F ( t + ∆ t ) , σ ( t )∆ t + G ( t ) } . (25b)(24) is the discrete version of Vickrey’s point queue model (Nie and Zhang, 2005), whichis a special case of both PQM1-D and PQM3-D, Moran’s discrete dam process model. Inaddition, (25) is another formulation of Vickrey’s point queue model. Similarly, PQM2-Dand PQM4-D are the same and can be simplified accordingly. In the four continuous point queue models derived in the preceding section, the right-handsides are continuous (assuming that both δ ( t ) and σ ( t ) are continuous) but not differentiabledue to non-differentiability of the function H ( y ). In this section we present four approximatemodels, which admit smooth solutions. (cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80) s (cid:15) ( t ) d (cid:15) ( t ) δ ( t ) + λ (cid:15) ( t ) (cid:15) λ (cid:15) ( t ) (cid:15) σ ( t ) + Λ − λ (cid:15) ( t ) (cid:15) (cid:15) -PQM1 (cid:15) -PQM4 Λ − λ (cid:15) ( t ) (cid:15) (cid:15) -PQM3 (cid:15) -PQM2Table 3: Demand and supply functions for four approximate point queue models: λ (cid:15) ( t ) = F (cid:15) ( t ) − G (cid:15) ( t ) 12f we approximate H ( y ) ≈ y(cid:15) for a very small (cid:15) , then we can obtain differentiable demandand supply functions, d (cid:15) ( t ) and s (cid:15) ( t ), shown in Table
3. Denoting the correspondingstate variables by λ (cid:15) ( t ), F (cid:15) ( t ), and G (cid:15) ( t ), we obtain the following two formulations of eachapproximate model:A. With λ (cid:15) ( t ) as the state variable, we havedd t λ (cid:15) ( t ) = min { δ ( t ) , s (cid:15) ( t ) } − min { d (cid:15) ( t ) , σ ( t ) } , (26)which is referred to as (cid:15) -A-PQM.B. With F (cid:15) ( t ) and G (cid:15) ( t ) as the state variables, we havedd t F (cid:15) ( t ) = min { δ ( t ) , s (cid:15) ( t ) } , (27a)dd t G (cid:15) ( t ) = min { d (cid:15) ( t ) , σ ( t ) } , (27b)which is referred to as (cid:15) -B-PQM. Theorem 4.1 (cid:15) -PQM1 and (cid:15) -PQM2 are always well-defined with any (cid:15) . (cid:15) -PQM3 is well-defined if and only if (cid:15) ≤ Λ σ ( t ) . (cid:15) -PQM4 is well-defined if and only if (cid:15) ≤ Λ δ ( t ) . Thus allapproximate models are well-defined, if (cid:15) satisfies (cid:15) ≤ Λmax t { δ ( t ) , σ ( t ) } . (28) Proof . This theorem can be proved by showing that dd t λ (cid:15) ( t ) ≥ λ (cid:15) = 0 and dd t λ (cid:15) ( t ) ≤ λ (cid:15) = Λ. The proof is straightforward and omitted here. (cid:4) Since the right-hand sides of these models are differentiable, their solutions are smooth(Coddington and Levinson, 1972). Clearly when (cid:15) →
0, the approximate models converge tothe original point queue models.Furthermore, when Λ = ∞ , (28) is satisfied for any (cid:15) , all the approximate point queuemodels with an infinite storage capacity are well-defined according to Theorem 4.1. Inaddition, (cid:15) -PQM1 is the same as (cid:15) -PQM3, which can be written asdd t λ (cid:15) ( t ) = max { δ ( t ) − σ ( t ) , − λ (cid:15) ( t ) (cid:15) } , (29)which is the α -model in (Ban et al., 2012; Han et al., 2013) if we define α = (cid:15) ; (cid:15) -PQM2 isthe same as (cid:15) -PQM4, which can be written asdd t λ (cid:15) ( t ) = δ ( t ) − min { σ ( t ) , λ (cid:15) ( t ) (cid:15) } , (30)which is the (cid:15) -model in (Armbruster et al., 2006a; F¨ugenschuh et al., 2008; Han et al., 2013).13 (cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80) s (cid:15) ( t ) d (cid:15) ( t ) δ ( t )∆ t + λ (cid:15) ( t ) (cid:15) ∆ t λ (cid:15) ( t ) (cid:15) ∆ tσ ( t )∆ t + Λ − λ (cid:15) ( t ) (cid:15) ∆ t (cid:15) -PQM1-D (cid:15) -PQM4-D Λ − λ (cid:15) ( t ) (cid:15) ∆ t (cid:15) -PQM3-D (cid:15) -PQM2-DTable 4: Demand and supply functions for four discrete approximate point queue models: λ (cid:15) ( t ) = F (cid:15) ( t ) − G (cid:15) ( t ) With a time-step size ∆ t , the demand and supply functions for the four discrete approximatepoint queue models are shown in Table
4. We can see that when ∆ t = (cid:15) , they are the sameas the discrete versions of the original point queue models. Theorem 4.2
All the discrete versions of four approximate point queue models are well-defined if and only if the continuous versions are well-defined and ∆ t ≤ (cid:15) .Proof . The proof is straightforward and omitted here. (cid:4) For the four point queue models and their approximate versions, the dynamics are driven bythe origin demand, δ ( t ), and the destination supply, σ ( t ). In general, as nonlinear ordinarydifferential equations, they cannot be analytically solved with arbitrary demand and supplypatterns. In this section, we first analytically solve Vickrey’s point queue model and thensolve all point queue models with constant origin demands and destination supplies. Even though the destination supply is generally constant in Vickrey’s point queue model, westill refer to (18) and (19) as Vickrey’s point queue model with a general destination supply.When Λ = 0, its analytical solution is given in the following theorem. Theorem 5.1
When Λ = 0 , Vickrey’s point queue model, (19), is solved by F ( t ) = (cid:90) t δ ( τ ) dτ, (31a) G ( t ) = min ≤ τ ≤ t { F ( τ ) − S ( τ ) } + S ( t ) , (31b) where S ( t ) = (cid:82) T σ ( τ ) dτ , and (18) is solved by λ ( t ) = F ( t ) − S ( t ) − min ≤ τ ≤ t { F ( τ ) − S ( τ ) } . (32)14 roof . Since Λ = 0, F (0) = G (0) = 0. (31a) is straightforward from (19a). Next we firstprove the corresponding discrete version of (31b) ( i ≥ G ( i ∆ t ) = min ≤ j ≤ i { F ( j ∆ t ) − S ( j ∆ t ) } + S ( i ∆ t ) , (33)where S ( i ∆ t ) = (cid:80) i − j =0 σ ( j ∆ t )∆ t .When i = 0, (33) is correct. Assuming that (33) is true for i ≥
0, then from (25) we have G (( i + 1)∆ t ) = min { F (( i + 1)∆ t ) , G ( i ∆ t ) + σ ( i ∆ t )∆ t } . Substituting (33) into this equation, we then have G (( i + 1)∆ t ) = min { F (( i + 1)∆ t ) , min ≤ j ≤ i { F ( j ∆ t ) − S ( j ∆ t ) } + S ( i ∆ t ) + σ ( i ∆ t )∆ t } = min { F (( i + 1)∆ t ) − S (( i + 1)∆ t ) , min ≤ j ≤ i { F ( j ∆ t ) − S ( j ∆ t ) }} + S (( i + 1)∆ t )= min ≤ j ≤ i +1 { F ( j ∆ t ) − S ( j ∆ t ) } + S (( i + 1)∆ t ) . Thus (33) is correct for i + 1. From the method of induction, (33) is correct for any i ≥ t ≥
0, we set i = t ∆ t and τ = j ∆ t for 0 ≤ j ≤ i . Then when ∆ t → S ( i ∆ t ) → S ( t ),and (33) leads to (31b) and (32). (cid:4) Corollary 5.2
When the destination supply is constant: σ ( t ) = σ , then S ( t ) = σt , and theout-flow is G ( t ) = min ≤ τ ≤ t { F ( τ ) − στ } + σt, (34) and the queue length is λ ( t ) = F ( t ) − σt − min ≤ τ ≤ t { F ( τ ) − στ } = max ≤ τ ≤ t { F ( t ) − F ( τ ) − ( t − τ ) σ } . (35)Note that Corollary 5.2 is the same as Theorem 4.10 in (Han et al., 2013). In addition, (35)was also obtained in Lemma 1.2.2 or Corollary 1.5.2 of (Le Boudec and Thiran, 2001). Theorem 5.3
When the destination supply is constant; i.e., when σ ( t ) = σ , then (Li et al.,2000), F ( t ) = G ( t + λ ( t ) σ ) . (36) That is, the queueing time for a vehicle entering the queue at t is π ( t ) = λ ( t ) σ . (37) In addition, we have (Iryo and Yoshii, 2007) ( g ( t ) − σ ) · π ( t ) = 0 . (38)15 roof . When λ ( t ) = 0, F ( t ) = G ( t ), and (36) is true. When λ ( t ) >
0, for τ ∈ [0 , λ ( t ) τ ), wehave from (19) G ( t + τ ) ≤ G ( t ) + στ < F ( t ) ≤ F ( t + τ ), which leads to λ ( t + τ ) > g ( t + τ ) = min { δ ( t + τ ) + H ( λ ( t + τ )) , σ } = σ . Therefore, G ( t + λ ( t ) σ ) = G ( t ) + λ ( t ) = F ( t ),which is (36). Then (37) is true.Also from g ( t ) = min { δ ( t ) + H ( λ ( t )) , σ } we can see that either λ ( t ) = 0 and g ( t ) ≤ σ or λ ( t ) > g ( t ) = σ . Combining (37), we obtain (38). (cid:4) In this subsection, we find stationary solutions, λ ( t ) = λ , for the point queue models andtheir approximations, when both the origin demand and destination supply are constant; i.e.,when δ ( t ) = δ and σ ( t ) = σ . This is the traffic statics problem for a point queue (Jin, 2012c).Clearly both d ( t ) = d and s ( t ) = s are constant in stationary states. Lemma 5.4
In stationary states of a point queue, its in- and out-fluxes are f = g = min { δ, s } = min { d, σ } = min { δ, s, d, σ } . (39) Proof . In stationary states, from (4) and (16) we have f = min { δ, s } , g = min { d, σ } , and f = g . Thus we have (39). (cid:4) Thus we have the following stationary states in the four point queue models.
Theorem 5.5
All four point queue models have the same stationary states:1. When δ > σ , λ = Λ . That is, the point queue is full.2. When δ < σ , λ = 0 . That is, the point queue is empty.3. When δ = σ , λ ∈ [0 , Λ] . That is, the point queue can be stationary at any state.Proof .1. In PQM1, the stationary states satisfy min { δ, σ + H (Λ − λ ) } = min { δ + H ( λ ) , σ } .(a) When δ > σ , λ = Λ. That is, the point queue is full.(b) When δ < σ , λ = 0. That is, the point queue is empty.(c) When δ = σ , λ ∈ [0 , Λ]. That is, the point queue can be stationary at any state.2. In PQM2, the stationary states satisfy min { δ, H (Λ − λ ) } = min { H ( λ ) , σ } .(a) When δ > σ , λ = Λ when σ = 0, and there is no solution for λ when σ > λ = Λ − σ ∆ t → Λ when ∆ t → δ < σ , λ = 0 when δ = 0, and there is no solution for λ when δ > λ = δ ∆ t → t → δ = σ , λ ∈ [0 , Λ].3. In PQM3, the stationary states satisfy min { δ, H (Λ − λ ) } = min { δ + H ( λ ) , σ } .(a) When δ > σ , λ = Λ when σ = 0, and there is no solution for λ when σ > λ = Λ − σ ∆ t → Λ when ∆ t → δ < σ , λ = 0.(c) When δ = σ , λ ∈ [0 , Λ].4. In PQM4, the stationary states satisfy min { δ, σ + H (Λ − λ ) } = min { H ( λ ) , σ } .(a) When δ > σ , λ = Λ.(b) When δ < σ , λ = 0 when δ = 0, and there is no solution for λ when δ > λ = δ ∆ t → t → δ = σ , λ ∈ [0 , Λ].Thus the theorem is proved. (cid:4)
Theorem 5.5 confirms that all point queue models are equivalent as shown in Theorem3.6.
Theorem 5.6
All four approximate point queue models, where (cid:15) satisfies (28), have thedifferent stationary states:1. When δ > σ , λ (cid:15) = Λ in (cid:15) -PQM1 and (cid:15) -PQM4, and λ (cid:15) = Λ − (cid:15)σ in (cid:15) -PQM2 and (cid:15) -PQM3.2. When δ < σ , λ (cid:15) = 0 in (cid:15) -PQM1 and (cid:15) -PQM3, and λ (cid:15) = (cid:15)δ in (cid:15) -PQM2 and (cid:15) -PQM4.3. When δ = σ , λ (cid:15) ∈ [0 , Λ] in (cid:15) -PQM1, λ (cid:15) ∈ [ (cid:15)δ, Λ − (cid:15)σ ] in (cid:15) -PQM2, λ (cid:15) ∈ [0 , Λ − (cid:15)σ ] in (cid:15) -PQM3, λ (cid:15) ∈ [ (cid:15)δ, Λ] in (cid:15) -PQM4.Proof . The proof is straightforward and omitted. (cid:4) Corollary 5.7
When (cid:15) → , the stationary states of the approximate point queue modelsconverge to those of the original models.Proof . This can be easily proved by comparing Theorems 5.5 and 5.6. (cid:4) Corollary 5.7 confirms that the approximate point queue models converge to the originalones when (cid:15) →
0. 17
Numerical results
In this section we simulate the dynamics of a point queue during a period of two hours( t ∈ [0 ,
2] hr). The origin demand is δ ( t ) = max { πt ) , } vph, and the destinationsupply σ ( t ) = 1200 vph. Unless otherwise stated, the storage capacity Λ = 200 veh. Weassume that the queue is initially empty: Λ = 0. In this subsection, we use the discrete models in Section 3.3 to numerically solve the fourmodels with ∆ t = 0 .
01 hr, which satisfies (23). Both the overall and zoom-in plots of queuelengths in these models are shown in Figure 3. From the figure we have the followingobservations. (i) From Figure 3(a), queue lengths are always between 0 and 200 veh, thestorage capacity. This confirms that all four models and their discrete versions are well-defined. (ii) From Figure 3(c), the maximum queue length between 0.6 and 0.8 hr equalsΛ − σ ∆ t = 200 −
12 = 188 veh for PQM2 and PQM3. From Figure 3(d), the minimum queuelength after 1.8 hr equals δ ∆ t = 10 veh for PQM2 and PQM4. This is consistent with thediscussions in the proof of Theorem 5.5. (iii) The four models provide different numericalsolutions: when the queue starts to accumulate but before its size reaches maximum, PQM1and PQM3 are the same, and PQM2 and PQM4 are the same; when the queue size ismaximum and the queue starts to dissipate before it is empty, PQM1 and PQM4 are thesame, and PQM2 and PQM3 are the same; after the queue size reaches minimum, PQM1and PQM3 are the same, and PQM2 and PQM4 are the same again.In Figure 4, we further compare the four models with smaller time-step sizes. ComparingFigure 3(a), Figure 4(a), and Figure 4(b), we can see that the four models converge tothe same results when ∆ t →
0. This confirms that all continuous point queue models areequivalent, as stated in Theorem 3.6. The queueing dynamics are as follows: the queue startsto build up at around 0.2 hr, reaches maximum at 0.55 hr, starts to dissipate at 0.8 hr, anddisappears at 1.8 hr. Due to the finite storage capacity, the demand that is not satisfiedis discarded. In addition, from Figure 4(b) we can see that the queue length λ ( t ) is notdifferentiable when it reaches the maximum and minimum values. Thus, as expected, thesolutions of the point queue models are not smooth, even though the origin demand anddestination supply are both continuous.Next we solve the discrete versions of four approximate point queue models in Section 4.2with (cid:15) = 0 .
001 hr, which satisfies (28). We set ∆ t = 0 . ≤ (cid:15) hr. Both the overall andzoom-in plots of the queue length are shown in Figure 5, from which we have the followingobservations: (i) Both continuous and discrete approximate models are well-defined, since thequeue length is between 0 and 200 veh; (ii) From Figure 5(c), the maximum queue length for (cid:15) -PQM2 and (cid:15) -PQM3 is Λ − (cid:15)σ = 198 . (cid:15) -PQM1 and (cid:15) -PQM4 is 200 veh,as predicted in Theorem 5.6; (iii) From Figure 5(d), the minimum queue length for (cid:15) -PQM218igure 3: Comparison of four point queue models: solid lines for PQM1, dashed lines forPQM2, dotted lines for PQM3, and dash-dotted lines for PQM419igure 4: Comparison of four point queue models: solid lines for PQM1, dashed lines forPQM2, dotted lines for PQM3, and dash-dotted lines for PQM4; (a) ∆ t = 0 .
001 hr; (b)∆ t = 0 . (cid:15) -PQM1,dashed lines for (cid:15) -PQM2, dotted lines for (cid:15) -PQM3, and dash-dotted lines for (cid:15) -PQM4and (cid:15) -PQM4 is (cid:15)δ = 1 veh, and that for (cid:15) -PQM1 and (cid:15) -PQM3 is 0, also as predicted inTheorem 5.6; (iv) Comparing Figure 5(a) and Figure 4(a), we can see that the approximatemodels converge to the original models when (cid:15) →
0; (v) From the zoom-in plots we can seethat solutions of the approximate point queue models are smooth.
In this subsection we consider a tandem of two point queues: the upstream point queue hasan infinite storage capacity, and the downstream one a finite storage capacity Λ = 200 veh.Both queues are initially empty. The origin demand and destination supply are the same asin the preceding subsection.We denote the queue sizes, demands, and supplies by λ i ( t ), d i ( t ), and s i ( t ) ( i = 1 , d ( t )∆ t = δ ( t )∆ t + λ ( t ) ,s ( t )∆ t = ∞ ,d ( t )∆ t = d ( t )∆ t + λ ( t ) ,s ( t )∆ t = σ ( t )∆ t + Λ − λ ( t ) ,λ ( t + ∆ t ) = λ ( t ) + δ ( t )∆ t − min { d ( t )∆ t, s ( t )∆ t } ,λ ( t + ∆ t ) = λ ( t ) + min { d ( t )∆ t, s ( t )∆ t } − min { d ( t )∆ t, σ ( t )∆ t } . With ∆ t = 0 . In this study we presented a unified approach for point queue models, which can be derivedas limits of two link-based queueing model: the link transmission and link queue models. Inparticular, we provided two definitions of demand and supply of a point queue. The newapproach led to the following additional contributions:22. From combinations of demand and supply definitions, we presented four point queuemodels, four approximate models, and their discrete versions. We discussed the followingproperties of these models: (i) equivalence: the four point queue models are equivalent toeach other; (ii) well-definedness: all the continuous and discrete models are well-definedunder suitable conditions; and (iii) smoothness: the solutions to the original modelsare generally not smooth, but those to the approximate models are. These resultsare obtained theoretically and also verified with numerical simulations. A numericalexample also shows that the proposed models with finite storage capacities can capturethe queue spillback effect.2. We analytically solved Vickrey’s point queue model, which has an infinite storagecapacity and is initially empty. The results are consistent with but more general thanthose in the literature. We also solved stationary states in the four point models andtheir approximate versions with constant origin demands and destination supplies. Thisdemonstrates that the new unified formulations enable analytical solutions in specialdynamical or stationary cases.3. In this study we demonstrated that all existing point and fluid queue models in theliterature, including Moran’s dam process model and Vickrey’s point queue model,are special cases of those derived from the link-based queueing models. This furtherconfirms the usefulness of the new unified approach.Even though in this study point queue models have deterministic origin demands anddestination supplies, they can be used to model stochastic queues with random origin demandsand destination supplies. For example, for a single queue with an infinite storage capacity, ifthe origin demand is random and follows a Poisson process, and the destination supply isconstant, then this becomes an M/D/1 queue, and some of the results in Section 5.1 stillapply. In the future we will be interested in studying stochastic queues with different modelsderived in this study.With demand and supply variables, point queue models have the same structure aslink-based queueing models. Therefore, they can be integrated into a network of both pointand link queues with appropriate macroscopic junction models, which calculate boundaryfluxes from upstream demands and downstream supplies (Jin, 2012a). In the future we willbe interested in studying stationary patterns and dynamics in networks of deterministic orstochastic queues, which can arise in computer, air traffic, and other systems. Furthermore,we will also be interested in control, management, scheduling, planning, and design of suchqueueing networks.