Point queue models: a unified approach

Transcription

1 Point queue moels: a unifie approach arxiv: v1 [math.ds] 29 May 2014 Wen-Long Jin October 7, 2018 Abstract In transportation an other types of facilities, various queues arise when the emans of service are higher than the supplies, an many point an flui queue moels have been propose to stuy such queueing systems. However, there has been no unifie approach to eriving such moels, analyzing their relationships an properties, an extening them for networks. In this paper, we erive point queue moels as limits of two linkbase queueing moel: the link transmission moel an a link queue moel. With two efinitions for eman an supply of a point queue, we present four point queue moels, four approximate moels, an their iscrete versions. We iscuss the properties of these moels, incluing equivalence, well-efineness, smoothness, an queue spillback, both analytically an with numerical examples. We then analytically solve Vickrey s point queue moel an stationary states in various moels. We emonstrate that all existing point an flui queue moels in the literature are special cases of those erive from the link-base queueing moels. Such a unifie approach leas to systematic methos for stuying the queueing process at a point facility an will also be helpful for stuies on stochastic queues as well as networks of queues. Keywors: Point queue moels; link transmission moel; link queue moel; eman an supply; Vickrey s point queue moel; stationary states. 1 Introuction When the emans of service are higher than the supplies at such facilities as roa networks, security check points in airports, supply chains, water reservoirs, ocument processors, task managers, an computer servers, there arise queues of vehicles, customers, commoities, water, ocuments, tasks, an programs, respectively. Many strategies have been evelope to Department of Civil an Environmental Engineering, California Institute for Telecommunications an Information Technology, Institute of Transportation Stuies, 4000 Anteater Instruction an Research Blg, University of California, Irvine, CA Tel: Fax: wjin@uci.eu. Corresponing author 1

2 control, manage, plan, an esign such queueing processes so as to improve their performance in safety, efficiency, environmental impacts, an so on. Since last century, many queueing moels have been propose to unerstan the characteristics of such systems, incluing waiting times, queue lengths, an stationary states an their stability (Kleinrock, 1975; Newell, 1982). These moels can be represente by arrival an service of iniviual customers or accumulation an issipation of continuum flui flows; they can be continuous or iscrete in time; an the arrival an service patterns of queueing contents can be ranom or eterministic. In aition, 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, an the serving rule can be First-In-First-Out or priority-base. Traitionally, queueing theories concern with ranom arrival an service processes of iniviual customers (Linley, 1952; Asmussen, 2003; Kingman, 2009). In recent years, flui queue moels have gaine much attention in various isciplines (Kulkarni, 1997). In these moels, the ynamics of a queueing system are escribe by changes in continuum flows of queueing contents. Such moels were first propose for am processes in 1950 s (Moran, 1956, 1959). Since then, both iscrete an continuous versions of flui queue moels have been extensively iscusse with ranom arrival an service patterns. In the transportation literature, point queue moels were introuce to stuy the congestion effect of a bottleneck with eterministic, ynamic arrival patterns (Vickrey, 1969). Such moels have been applie to stuy ynamic traffic assignment problems (Drissi-Kaïtouni an Hamea-Benchekroun, 1992; Kuwahara an Akamatsu, 1997; Li et al., 2000). In (Nie an Zhang, 2005), Vickrey s point queue moel was compare with other network loaing moels. In (Armbruster et al., 2006b), a point queue moel for a supply chain was propose an shown to be consistent with a hyperbolic conservation law. In (Ban et al., 2012), continuous point queue moels were presente as ifferential complementarity systems. In (Han et al., 2013), Vickrey s point queue moel was formulate with the help of the LWR moel (Lighthill an Whitham, 1955; Richars, 1956). In existing stuies on point queue moels, the ownstream service rate is usually assume to be constant, an the storage capacity of the point queue infinite. Note that flui queue moels can also be consiere as point queue moels, but with a finite storage capacity an general service rates, since the imension of a server or a reservoir is also assume to be infinitesimal. However, there exists no unifie approach to eriving such point queue moels, analyzing their relationships an properties, an extening them for networks. In this stuy we attempt to fill this gap by presenting a unifie approach, which is base on two observations. The first observation is that many traffic flow moels can be viewe as queueing moels, since traffic flow moels escribe the ynamics associate with the accumulation an issipation of vehicular queues. For examples, the Cell Transmission Moel (CTM) escribes ynamics of cell-base queues (Daganzo, 1995; Lebacque, 1996); the Link Transmission Moel (LTM) escribes ynamics of link-base queues (Yperman et al., 2006; Yperman, 2007); an the Link Queue Moel (LQM) also escribes ynamics of link-base queues (Jin, 2012b). Among these moels, continuous versions of LTM an CTM 2

3 are equivalent to but ifferent formulations of network kinematic wave moels, which can amit iscontinuous shock wave solutions an can be unstable in iverge-merge networks (Jin, 2014), but LQM amits smooth solutions an is stable (Jin, 2012b). The secon observation is that a point facility can be consiere as the limit of a link: if we shrink the link length but maintain the storage capacity, then we obtain a point queue. Therefore we can erive point queue moels as limits of the link-base queueing moels: LTM an LQM. In this approach, point queue moels inherit two critical components from both LTM an LQM: we first efine eman an supply of a point queue, an then apply macroscopic junction moels, which were originally propose in CTM an later aopte for both LTM an LQM, to calculate bounary fluxes from upstream emans an ownstream supplies. This approach enables us to erive all existing point an flui queue moels an their generalizations, iscuss their relationships, an analyze their properties. The rest of the paper is organize as follows. After reviewing LTM an LQM in Section 2, we take the limits of the eman an supply of a link to obtain two eman formulas an two supply formulas for a point queue in Section 3. Further with ifferent combinations of eman an supply formulas, we erive four continuous point queue moels an their iscrete versions. In Section 4, we present four approximate point queue moels an their iscrete versions. In Section 5, we present some analytical solutions of these moels. In Section 6, we present numerical results. Finally in Section 7, we summarize the stuy an iscuss future research topics. 2 Review of two link-base queueing moels In this section, we consier a homogeneous roa link, shown in Figure 1, whose length is L an number of lanes N. Vehicles on the roa segment form a link queue. We assume that the roa has the following triangular funamental iagram of flow-ensity relation (Munjal et al., 1971; Haberman, 1977; Newell, 1993): q = Q(k) = min{v k, (NK k)w }, (1) where k(x, t) is the total ensity at x an t, q(x, t) the total flow-rate, K the jam ensity per lane, V the free-flow spee, an W the shock wave spee in congeste traffic. We enote the harmonic mean of V an W by U = V W. We also enote T V +W 1 = L, which is V the free-flow traverse time, T 2 = L, which is the shock wave traverse time, an T W 3 = L. U Generally T 1 < T 2 < T 3. Thus for each lane, the critical ensity is K = W K = U K, V +W V an the capacity is C = V K = UK. We enote the storage capacity of the roa link by Λ = NLK an the initial number of vehicles by Λ 0 = NLk 0, where the initial ensity is constant at Nk 0 at all locations on the roa. Then the total capacity is NC = Λ T 3. For a roa link shown in Figure 1, we efine the upstream an ownstream cumulative flows by F (t) an G(t), respectively, an set G(0) = 0 an F (0) = Λ 0. Corresponingly the 3

4 Figure 1: An illustration of a homogeneous roa link upstream in- an ownstream out-fluxes are enote by f(t) an g(t). Then we have F (t) t = f(t), (2a) G(t) t = g(t). (2b) In aition, we enote the number of vehicles on the roa by ρ(t) = L k(x, t)x, which 0 satisfies ρ(t) = F (t) G(t). (3) There are two formulations for a link-base queueing moel with two types of state variables: either the accumulation of vehicles, such as ρ(t), or the cumulative flows, F (t) an G(t), can be use as state variables. In aition, the bounary fluxes are calculate in two steps: first, a link s eman, (t), an supply, s(t), are efine from state variables; secon, the flux through a bounary can be calculate from the upstream eman an ownstream supply with macroscopic junction moels: f(t) = min{δ(t), s(t)}, (4a) g(t) = min{(t), σ(t)}, (4b) where δ(t) is the origin eman, an σ(t) the estination supply. LTM an LQM iffer in the efinitions of eman an supply functions an the corresponing state variables relate to vehicle accumulations. 2.1 The link transmission moel (LTM) In LTM, two new variables relate to vehicle accumulation are the link queue size, λ(t), an the link vacancy size, γ(t), which are given by (Jin, 2014): λ(t) = γ(t) = { Λ0 T 1 t G(t), t T 1 (5a) F (t T 1 ) G(t), t > T 1 { Λ Λ0 T 2 t + Λ 0 F (t), t T 2 G(t T 2 ) + Λ F (t), t > T 2 (5b) 4

5 Since both F (t) an G(t) are continuous, then λ(t) an γ(t) are continuous with λ(0) = 0 an γ(0) = 0. The link eman an supply in LTM are efine as (Jin, 2014): { } Λ min 0 T (t) = 1 + H(λ(t)), Λ T 3, t T 1 } (6a) min {f(t T 1 ) + H(λ(t)), ΛT3, t > T 1 { } Λ Λ min 0 T s(t) = 2 + H(γ(t)), Λ T 3, t T 2 } (6b) min {g(t T 2 ) + H(γ(t)), ΛT3, t > T 2 where the inicator function H(y) for y 0 is efine as H(y) = lim t 0 + y t = { 0, y = 0, y > 0 (7) Then we obtain two continuous formulations of LTM: 1. When the queue an vacancy sizes, λ(t) an γ(t), are the state variables, we have from (5): t λ(t) = t γ(t) = { Λ0 T 1 g(t), t T 1 (8a) f(t T 1 ) g(t), t > T 1 { Λ Λ0 T 2 f(t), t T 2 (8b) g(t T 2 ) f(t), t > T 2 where f(t) an g(t) are calculate from (4) an (6). In this formulation, the rates of change in both λ(t) an γ(t) epen on λ(t), γ(t), f(t T 1 ), an g(t T 2 ). 2. When cumulative flows, F (t) an G(t), are the state variables, we have from (2): F (t) t = min{δ(t), s(t)}, (9a) G(t) t = min{(t), σ(t)}. (9b) where (t) an s(t) are calculate from (5) an (6). In this formulation, the rates of change in F (t) an G(t) epen on F (t), G(t), F (t T 1 ), G(t T 2 ), f(t T 1 ), an g(t T 2 ). In both formulations, f(t T 1 ), an g(t T 2 ) can be calculate from the corresponing state variables at t T 1 an t T 2. In aition, the right-han sies of both formulations are iscontinuous, an LTM is a system of iscontinuous orinary ifferential equations with elays. Thus their solutions may not be ifferentiable or smooth. 5

6 2.2 The link queue moel (LQM) In LQM, ρ(t) can be use as a state variable. The link eman an supply in LQM are efine as (Jin, 2012b): (t) = min{ ρ(t) T 1, Λ T 3 }, (10a) s(t) = min{ Λ ρ(t) T 2, Λ T 3 }. (10b) From (2), (3), (4), an (10) we obtain the following two continuous formulations of LQM: 1. When ρ(t) is the state variable, we have: Λ ρ(t) ρ(t) = min{δ(t),, Λ } min{ ρ(t), Λ, σ(t)}. (11) t T 2 T 3 T 1 T 3 2. When F (t) an G(t) are the state variables, we have: Λ (F (t) G(t)) F (t) = min{δ(t),, Λ }, t T 2 T 3 (12a) (t) G(t) G(t) = min{f, Λ, σ(t)}. t T 1 T 3 (12b) In both formulations, the right-han sies are continuous, an LQM are systems of continuous orinary ifferential equations without elays. Therefore, solutions of LQM are smooth with continuous erivatives of any orer. 3 Four point queue moels We consier a point facility as a limit of a roa link, shown in Figure 2, with the same storage capacity, Λ, origin eman, δ(t), an estination supply, σ(t). The macroscopic junction moels are also the same as (4). But we let the link length L 0, an the number of lanes N = Λ. Parking lots, security check points, water reservoirs, ocument processors, LK task managers, an computer servers can all be approximate by point facilities. The queue in a point facility is referre to as a point queue. 3.1 Deman an supply of a point queue Base on the observation that a point facility is a limit of a link, we can erive eman an supply of a point queue from (6) in LTM an (10) in LQM by letting L 0. 6

7 Figure 2: An illustration of a point queue Lemma 3.1 When L 0, from the queue an vacancy sizes of LTM, (5), we obtain the queue an vacancy sizes of a point queue as (for t 0 + ) λ(t) = F (t) G(t), (13a) γ(t) = Λ λ(t). (13b) Proof. From (5a) we have λ(t 1 ) = Λ 0 G(T 1 ) an λ(t) = F (t T 1 ) G(t) for t > T 1. When L 0, we have T 1 0 +, λ(0 + ) = Λ 0 an λ(t) = F (t) G(t) for t > 0. Thus λ(t) = F (t) G(t) for t 0 +. Similarly from (5b) we have γ(t 2 ) = Λ F (T 2 ) an γ(t) = G(t T 2 ) + Λ F (t) for t > T 2. When L 0, we have T 2 0 +, γ(0 + ) = Λ Λ 0 an γ(t) = Λ (F (t) G(t)) for t > 0. Thus γ(t) = Λ λ(t) for t 0 +. From Lemma 3.1, we can see that, in a point queue, λ(t) = ρ(t) = Λ γ(t). Thus we can use λ(t), the queue size, to uniquely represent the vehicle accumulation. Lemma 3.2 When L 0, from the eman an supply in LTM, (6), we obtain the eman an supply of a point queue as (for t 0 + ) Proof. When t > T 1 an t > T 2, we have from (4) an (6) (t) = δ(t) + H(λ(t)), (14a) s(t) = σ(t) + H(Λ λ(t)). (14b) (t) = min{min{δ(t T 1 ), s(t T 1 )} + H(λ(t)), Λ T 3 }, s(t) = min{min{(t T 2 ), σ(t T 2 )} + H(γ(t)), Λ T 3 }. 7

8 Let L 0, then T 1, T 2, T 3 0 +, γ(t) Λ λ(t), an 1 (t) = min{δ(t), s(t)} + H(λ(t)), s(t) = min{(t), σ(t)} + H(Λ λ(t)). Therefore, when λ(t) < Λ for t > 0, then s(t) =, an (t) = δ(t) + H(λ(t)); when λ(t) > 0 for t > 0, then (t) =, an s(t) = σ(t) + H(Λ λ(t)). Thus we obtain (14). Lemma 3.3 When L 0, from the eman an supply in LQM, (10), we obtain the eman an supply of a point queue as (for t 0 + ) (t) = H(λ(t)), (15a) s(t) = H(Λ λ(t)). (15b) Proof. When L 0, T 1, T 2, T 3 0. Thus (t) = 0 when ρ(t) = 0 an otherwise. Similarly, s(t) = 0 when ρ(t) = Λ an otherwise. Hence (15) is correct. Clearly the eman an supply functions erive from LTM are ifferent from those from LQM. They can lea to ifferent point queue moels. 3.2 Point queue moels To moel ynamics of a point queue, we can use the queue length, λ(t), or the cumulative flows, F (t) an G(t), as state variables. Once the eman an supply functions are given, from (2), (3), (4) we can erive two formulations of a point queue moel: A. With λ(t) as the state variable, we have which is referre to as A-PQM. λ(t) = min{δ(t), s(t)} min{(t), σ(t)}, (16) t B. With F (t) an G(t) as the state variables, we have which is referre to as B-PQM. F (t) t = min{δ(t), s(t)}, (17a) G(t) t = min{(t), σ(t)}, (17b) In the preceing subsection, we obtain from LTM an LQM two ifferent efinitions for both eman an supply of a point queue, which are functions of λ(t) or F (t) an G(t). Therefore, as shown in Table 1, we then obtain four combinations of eman an supply functions, which lea to four point queue moels. Here PQM1 is the limit of LTM, PQM2 the limit of LQM, but PQM3 an PQM4 are mixtures of the limits of both LTM an LQM. 1 Without loss of generality we assume that δ(t), σ(t), (t), an s(t) are all continuous in time. 8

9 (t) s(t) δ(t) + H(λ(t)) H(λ(t)) σ(t) + H(Λ λ(t)) PQM1 PQM4 H(Λ λ(t)) PQM3 PQM2 Table 1: Deman an supply functions for four point queue moels (PQMs): λ(t) = F (t) G(t) Definition 3.4 A continuous point queue moel is well-efine if an only if λ(t) is always between 0 an Λ in A-PQM; or G(t) F (t) G(t) + Λ in B-PQM. Theorem 3.5 With eman an supply functions in Table 1, all four point queue moels are well-efine. That is, λ(t) is always between 0 an Λ in (16), an G(t) F (t) G(t)+Λ in (17). Proof. When λ(t) = 0, s(t) =, an λ(t) = δ(t) min{(t), σ(t)} 0, since (t) δ(t); t when λ(t) = Λ, (t) =, an λ(t) = min{δ(t), s(t)} σ(t) 0, since s(t) σ(t). t Similarly from we can see that G(t) F (t) G(t) + Λ in (17). Theorem 3.6 The four point queue moels are equivalent. Proof. When 0 < λ(t) < Λ, we have (t) = s(t) = in all four point queue moels, an λ(t) = δ(t) σ(t) for all moels. When λ(t) increases from 0 or ecreases from Λ, the t eman an supply functions an, therefore, rates of change in λ(t) are ifferent for these moels. But since the number of such instants is countable, the integrals of ifferent rates of change at such instants are zero. That is, λ(t) is the same for all moels, when they have the same origin eman an estination supply. This leas to the equivalence among all moels. If the storage capacity of a point facility is infinite; i.e., if Λ =, then its supply s(t) = in all four moels. In this case, PQM1 an PQM3 are the same, an so are PQM2 an PM4. In particular, A-PQM1 becomes λ(t) = max{ H(λ(t)), δ(t) σ(t)} = t { δ(t) σ(t), λ(t) > 0 max{0, δ(t) σ(t)}, λ(t) = 0 (18) which is referre to as A-PQM1i, an B-PQM1 becomes F (t) = δ(t) (19a) t { min{δ(t), σ(t)}, F (t) = G(t) t G(t) = (19b) σ(t), F (t) > G(t) 9

10 which is referre to as B-PQM1i. Similarly, we can also obtain the special cases of PQM2. In A-PQM1i, if we allow δ(t), σ(t), an λ(t) to be ranom variables, then (18) is the same as the traitional flui queue moel (Kulkarni, 1997, Equation 1) with the ifference between origin eman an estination supply, δ(t) σ(t), as the rift function. Also in A-PQM1i, if the estination supply, σ(t), is constant, then (18) is the same as the continuous version of Vickrey s point queue moel (Han et al., 2013, Equations 2.4 an 2.5 with t 0 = 0). Therefore, PQM1 an PQM3 are generalize versions of existing continuous flui queue an Vickrey s point queue moels, since it also applies to a queue with a finite storage capacity an arbitrary origin eman an estination supply patterns. 3.3 Discrete versions (t) t s(t) t δ(t) t + λ(t) λ(t) σ(t) t + Λ λ(t) PQM1-D PQM4-D Λ λ(t) PQM3-D PQM2-D Table 2: Deman an supply functions for four iscrete point queue moels: λ(t) = F (t) G(t) For four continuous point queue moels, we iscretize them with a time-step size of t such that H(y) t = y. Then with the iscrete eman an supply functions uring from t to t + t, shown in Table 2, (16) an (17) can be numerically solve. 1. With λ(t) as the state variable, we have the following four iscrete moels by replacing λ(t+ t) λ(t) λ(t) by in (16): t t λ(t + t) = λ(t) + min{δ(t) t, s(t) t} min{(t) t, σ(t) t}. (20) 2. With F (t) an G(t) as the state variables, we have the following four iscrete moels by replacing F (t+ t) F (t) F (t) by an G(t+ t) G(t) G(t) by in (17): t t t t In particular, A-PQM3-D can be written as F (t + t) = F (t) + min{δ(t) t, s(t) t}, (21a) G(t + t) = G(t) + min{(t) t, σ(t) t}. (21b) λ(t + t) = min{δ(t) t + λ(t), Λ} min{δ(t) t + λ(t), σ(t) t}, (22) which is the iscrete am process moel in (Moran, 1959, Chapter 3). In this sense, PQM3 is the continuous version of Moran s moel. 10

11 Definition 3.7 A iscrete point queue moel, (20), is well-efine if an only if λ(t + t) is between 0 an Λ for λ(t) between 0 an Λ; a iscrete point queue moel, (21), is well-efine if an 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 an PQM2-D are always well-efine with any t; PQM3-D is well-efine if an only if t Λ Λ ; an PQM4-D is well-efine if an only if t. σ(t) δ(t) Thus when t all four iscrete moels are well-efine. Proof. For A-PQM1-D, we have Λ max t {δ(t), σ(t)}, (23) 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], an A-PQM1-D is well-efine. 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-efine. 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, an the iscrete moel is not well-efine. However, if Λ σ(t) t or t Λ, there are three cases: (i) when σ(t) 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-efine if an 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 an σ(t) t is very large such that δ(t) t+λ(t) σ(t) t+λ an λ(t) σ(t) t, then λ(t + t) = δ(t) t > Λ, an the iscrete moel is not well-efine. However, if Λ 11

12 δ(t) t or t Λ, there are two cases: (i) when σ(t) λ(t) min{δ(t) t+λ(t), σ(t) t+λ}, δ(t) 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-efine if an only if t Λ. δ(t) We can similarly prove that the other formulations with F (t) an G(t) as the state variables are also well-efine uner the respective conitions. Furthermore, all iscrete moels are well-efine when (23) is satisfie. For a point queue with an infinite storage capacity with Λ =, all the iscrete moels are well-efine for any t, since (23) is always satisfie. In aition, PQM1-D an PQM3-D are the same an can be simplifie as or λ(t + t) = max{0, δ(t) t + λ(t) σ(t) t}, (24) 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 iscrete version of Vickrey s point queue moel (Nie an Zhang, 2005), which is a special case of both PQM1-D an PQM3-D, Moran s iscrete am process moel. In aition, (25) is another formulation of Vickrey s point queue moel. Similarly, PQM2-D an PQM4-D are the same an can be simplifie accoringly. 4 Approximate point queue moels In the four continuous point queue moels erive in the preceing section, the right-han sies are continuous (assuming that both δ(t) an σ(t) are continuous) but not ifferentiable ue to non-ifferentiability of the function H(y). In this section we present four approximate moels, which amit smooth solutions. 4.1 Continuous versions ɛ (t) s ɛ (t) σ(t) + Λ λɛ(t) δ(t) + λɛ(t) ɛ λ ɛ(t) ɛ ɛ-pqm1 ɛ-pqm4 ɛ ɛ-pqm3 ɛ-pqm2 ɛ Λ λ ɛ(t) Table 3: Deman an supply functions for four approximate point queue moels: λ ɛ (t) = F ɛ (t) G ɛ (t) 12

13 If we approximate H(y) y for a very small ɛ, then we can obtain ifferentiable eman ɛ an supply functions, ɛ (t) an s ɛ (t), shown in Table 3. Denoting the corresponing state variables by λ ɛ (t), F ɛ (t), an G ɛ (t), we obtain the following two formulations of each approximate moel: A. With λ ɛ (t) as the state variable, we have which is referre to as ɛ-a-pqm. t λ ɛ(t) = min{δ(t), s ɛ (t)} min{ ɛ (t), σ(t)}, (26) B. With F ɛ (t) an G ɛ (t) as the state variables, we have which is referre to as ɛ-b-pqm. t F ɛ(t) = min{δ(t), s ɛ (t)}, (27a) t G ɛ(t) = min{ ɛ (t), σ(t)}, (27b) Theorem 4.1 ɛ-pqm1 an ɛ-pqm2 are always well-efine with any ɛ. ɛ-pqm3 is wellefine if an only if ɛ Λ. ɛ-pqm4 is well-efine if an only if ɛ Λ. Thus all σ(t) δ(t) approximate moels are well-efine, if ɛ satisfies ɛ Λ max t {δ(t), σ(t)}. (28) Proof. This theorem can be prove by showing that λ ɛ(t) 0 when λ ɛ = 0 an λ ɛ(t) 0 t t when λ ɛ = Λ. The proof is straightforwar an omitte here. Since the right-han sies of these moels are ifferentiable, their solutions are smooth (Coington an Levinson, 1972). Clearly when ɛ 0, the approximate moels converge to the original point queue moels. Furthermore, when Λ =, (28) is satisfie for any ɛ, all the approximate point queue moels with an infinite storage capacity are well-efine accoring to Theorem 4.1. In aition, ɛ-pqm1 is the same as ɛ-pqm3, which can be written as t λ ɛ(t) = max{δ(t) σ(t), λ ɛ(t) }, (29) ɛ which is the α-moel in (Ban et al., 2012; Han et al., 2013) if we efine α = 1 ; ɛ-pqm2 is ɛ the same as ɛ-pqm4, which can be written as t λ ɛ(t) = δ(t) min{σ(t), λ ɛ(t) }, (30) ɛ which is the ɛ-moel in (Armbruster et al., 2006a; Fügenschuh et al., 2008; Han et al., 2013). 13

14 ɛ (t) s ɛ (t) σ(t) t + Λ λɛ(t) δ(t) t + λɛ(t) t ɛ λ ɛ(t) t ɛ t ɛ-pqm1-d ɛ-pqm4-d ɛ t ɛ-pqm3-d ɛ-pqm2-d ɛ Λ λ ɛ(t) Table 4: Deman an supply functions for four iscrete approximate point queue moels: λ ɛ (t) = F ɛ (t) G ɛ (t) 4.2 Discrete versions With a time-step size t, the eman an supply functions for the four iscrete approximate point queue moels are shown in Table 4. We can see that when t = ɛ, they are the same as the iscrete versions of the original point queue moels. Theorem 4.2 All the iscrete versions of four approximate point queue moels are wellefine if an only if the continuous versions are well-efine an t ɛ. Proof. The proof is straightforwar an omitte here. 5 Analytical solutions For the four point queue moels an their approximate versions, the ynamics are riven by the origin eman, δ(t), an the estination supply, σ(t). In general, as nonlinear orinary ifferential equations, they cannot be analytically solve with arbitrary eman an supply patterns. In this section, we first analytically solve Vickrey s point queue moel an then solve all point queue moels with constant origin emans an estination supplies. 5.1 Analytical solutions of Vickrey s point queue moel Even though the estination supply is generally constant in Vickrey s point queue moel, we still refer to (18) an (19) as Vickrey s point queue moel with a general estination supply. When Λ 0 = 0, its analytical solution is given in the following theorem. Theorem 5.1 When Λ 0 = 0, Vickrey s point queue moel, (19), is solve by where S(t) = T 0 F (t) = t 0 δ(τ)τ, G(t) = min {F (τ) S(τ)} + S(t), (31b) 0 τ t σ(τ)τ, an (18) is solve by (31a) λ(t) = F (t) S(t) min {F (τ) S(τ)}. (32) 0 τ t 14

15 Proof. Since Λ 0 = 0, F (0) = G(0) = 0. (31a) is straightforwar from (19a). Next we first prove the corresponing iscrete version of (31b) (i 0) G(i t) = min {F (j t) S(j t)} + S(i t), (33) 0 j i where S(i t) = i 1 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 {F (j t) S(j t)} + S(i t) + σ(i t) t} 0 j i = min{f ((i + 1) t) S((i + 1) t), min {F (j t) S(j t)}} + S((i + 1) t) 0 j i = min {F (j t) S(j t)} + S((i + 1) t). 0 j i+1 Thus (33) is correct for i + 1. From the metho of inuction, (33) is correct for any i 0. For t 0, we set i = t an τ = j t for 0 j i. Then when t 0, S(i t) S(t), t an (33) leas to (31b) an (32). Corollary 5.2 When the estination supply is constant: σ(t) = σ, then S(t) = σt, an the out-flow is an the queue length is G(t) = min {F (τ) στ} + σt, (34) 0 τ t λ(t) = F (t) σt min {F (τ) στ} = max {F (t) F (τ) (t τ)σ}. (35) 0 τ t 0 τ t Note that Corollary 5.2 is the same as Theorem 4.10 in (Han et al., 2013). In aition, (35) was also obtaine in Lemma or Corollary of (Le Bouec an Thiran, 2001). Theorem 5.3 When the estination supply is constant; i.e., when σ(t) = σ, then (Li et al., 2000), F (t) = G(t + λ(t) ). (36) σ That is, the queueing time for a vehicle entering the queue at t is In aition, we have (Iryo an Yoshii, 2007) π(t) = λ(t) σ. (37) (g(t) σ) π(t) = 0. (38) 15

16 Proof. When λ(t) = 0, F (t) = G(t), an (36) is true. When λ(t) > 0, for τ [0, λ(t) ), we τ have from (19) G(t + τ) G(t) + στ < F (t) F (t + τ), which leas to λ(t + τ) > 0 an g(t + τ) = min{δ(t + τ) + H(λ(t + τ)), σ} = σ. Therefore, G(t + λ(t) ) = G(t) + λ(t) = F (t), σ 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 an g(t) σ or λ(t) > 0 an g(t) = σ. Combining (37), we obtain (38). 5.2 Stationary states In this subsection, we fin stationary solutions, λ(t) = λ, for the point queue moels an their approximations, when both the origin eman an estination supply are constant; i.e., when δ(t) = δ an σ(t) = σ. This is the traffic statics problem for a point queue (Jin, 2012c). Clearly both (t) = an s(t) = s are constant in stationary states. Lemma 5.4 In stationary states of a point queue, its in- an out-fluxes are f = g = min{δ, s} = min{, σ} = min{δ, s,, σ}. (39) Proof. In stationary states, from (4) an (16) we have f = min{δ, s}, g = min{, σ}, an f = g. Thus we have (39). Thus we have the following stationary states in the four point queue moels. Theorem 5.5 All four point queue moels 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, an there is no solution for λ when σ > 0. However, from its iscrete version we have λ = Λ σ t Λ when t 0. (b) When δ < σ, λ = 0 when δ = 0, an there is no solution for λ when δ > 0. However, from its iscrete version we have λ = δ t 0 when t 0. 16

17 (c) When 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, an there is no solution for λ when σ > 0. However, from its iscrete version we have λ = Λ σ t Λ when t 0. (b) When 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, an there is no solution for λ when δ > 0. However, from its iscrete version we have λ = δ t 0 when t 0. (c) When δ = σ, λ [0, Λ]. Thus the theorem is prove. Theorem 5.5 confirms that all point queue moels are equivalent as shown in Theorem 3.6. Theorem 5.6 All four approximate point queue moels, where ɛ satisfies (28), have the ifferent stationary states: 1. When δ > σ, λ ɛ = Λ in ɛ-pqm1 an ɛ-pqm4, an λ ɛ = Λ ɛσ in ɛ-pqm2 an ɛ-pqm3. 2. When δ < σ, λ ɛ = 0 in ɛ-pqm1 an ɛ-pqm3, an λ ɛ = ɛδ in ɛ-pqm2 an ɛ-pqm4. 3. When δ = σ, λ ɛ [0, Λ] in ɛ-pqm1, λ ɛ [ɛδ, Λ ɛσ] in ɛ-pqm2, λ ɛ [0, Λ ɛσ] in ɛ-pqm3, λ ɛ [ɛδ, Λ] in ɛ-pqm4. Proof. The proof is straightforwar an omitte. Corollary 5.7 When ɛ 0, the stationary states of the approximate point queue moels converge to those of the original moels. Proof. This can be easily prove by comparing Theorems 5.5 an 5.6. Corollary 5.7 confirms that the approximate point queue moels converge to the original ones when ɛ 0. 17

18 6 Numerical results In this section we simulate the ynamics of a point queue uring a perio of two hours (t [0, 2] hr). The origin eman is δ(t) = max{2000 sin(πt), 1000} vph, an the estination supply σ(t) = 1200 vph. Unless otherwise state, the storage capacity Λ = 200 veh. We assume that the queue is initially empty: Λ 0 = Comparison of the four point queue moels an their approximations In this subsection, we use the iscrete moels in Section 3.3 to numerically solve the four moels with t = 0.01 hr, which satisfies (23). Both the overall an zoom-in plots of queue lengths in these moels are shown in Figure 3. From the figure we have the following observations. (i) From Figure 3(a), queue lengths are always between 0 an 200 veh, the storage capacity. This confirms that all four moels an their iscrete versions are wellefine. (ii) From Figure 3(c), the maximum queue length between 0.6 an 0.8 hr equals Λ σ t = = 188 veh for PQM2 an PQM3. From Figure 3(), the minimum queue length after 1.8 hr equals δ t = 10 veh for PQM2 an PQM4. This is consistent with the iscussions in the proof of Theorem 5.5. (iii) The four moels provie ifferent numerical solutions: when the queue starts to accumulate but before its size reaches maximum, PQM1 an PQM3 are the same, an PQM2 an PQM4 are the same; when the queue size is maximum an the queue starts to issipate before it is empty, PQM1 an PQM4 are the same, an PQM2 an PQM3 are the same; after the queue size reaches minimum, PQM1 an PQM3 are the same, an PQM2 an PQM4 are the same again. In Figure 4, we further compare the four moels with smaller time-step sizes. Comparing Figure 3(a), Figure 4(a), an Figure 4(b), we can see that the four moels converge to the same results when t 0. This confirms that all continuous point queue moels are equivalent, as state in Theorem 3.6. The queueing ynamics are as follows: the queue starts to buil up at aroun 0.2 hr, reaches maximum at 0.55 hr, starts to issipate at 0.8 hr, an isappears at 1.8 hr. Due to the finite storage capacity, the eman that is not satisfie is iscare. In aition, from Figure 4(b) we can see that the queue length λ(t) is not ifferentiable when it reaches the maximum an minimum values. Thus, as expecte, the solutions of the point queue moels are not smooth, even though the origin eman an estination supply are both continuous. Next we solve the iscrete versions of four approximate point queue moels in Section 4.2 with ɛ = hr, which satisfies (28). We set t = ɛ hr. Both the overall an zoom-in plots of the queue length are shown in Figure 5, from which we have the following observations: (i) Both continuous an iscrete approximate moels are well-efine, since the queue length is between 0 an 200 veh; (ii) From Figure 5(c), the maximum queue length for ɛ-pqm2 an ɛ-pqm3 is Λ ɛσ = veh, an that for ɛ-pqm1 an ɛ-pqm4 is 200 veh, as preicte in Theorem 5.6; (iii) From Figure 5(), the minimum queue length for ɛ-pqm2 18

19 Figure 3: Comparison of four point queue moels: soli lines for PQM1, ashe lines for PQM2, otte lines for PQM3, an ash-otte lines for PQM4 19

20 Figure 4: Comparison of four point queue moels: soli lines for PQM1, ashe lines for PQM2, otte lines for PQM3, an ash-otte lines for PQM4; (a) t = hr; (b) t = hr 20

21 Figure 5: Comparison of four approximate point queue moels: soli lines for ɛ-pqm1, ashe lines for ɛ-pqm2, otte lines for ɛ-pqm3, an ash-otte lines for ɛ-pqm4 an ɛ-pqm4 is ɛδ = 1 veh, an that for ɛ-pqm1 an ɛ-pqm3 is 0, also as preicte in Theorem 5.6; (iv) Comparing Figure 5(a) an Figure 4(a), we can see that the approximate moels converge to the original moels when ɛ 0; (v) From the zoom-in plots we can see that solutions of the approximate point queue moels are smooth. 6.2 Queue spillback effect In this subsection we consier a tanem of two point queues: the upstream point queue has an infinite storage capacity, an the ownstream one a finite storage capacity Λ 2 = 200 veh. Both queues are initially empty. The origin eman an estination supply are the same as in the preceing subsection. We enote the queue sizes, emans, an supplies by λ i (t), i (t), an s i (t) (i = 1, 2), respectively. Then we apply the iscrete version of PQM1 in Section 3.3 to simulate the 21

22 Figure 6: Queue spillback in a tanem of point queues ynamics in such a network: 1 (t) t = δ(t) t + λ 1 (t), s 1 (t) t =, 2 (t) t = 1 (t) t + λ 2 (t), s 2 (t) t = σ(t) t + Λ 2 λ 2 (t), λ 1 (t + t) = λ 1 (t) + δ(t) t min{ 1 (t) t, s 2 (t) t}, λ 2 (t + t) = λ 2 (t) + min{ 1 (t) t, s 2 (t) t} min{ 2 (t) t, σ(t) t}. With t = hr, the simulate queue lengths for both queues are shown in Figure 6. In the tanem of two queues, queue 2 starts to buil up at aroun 0.2 hr; when it reaches its maximum storage capacity at aroun 0.55 hr, it spills back to queue 1; queue 1 isappears before 1.4 hr, an queue 2 starts to issipate; an queue 2 oes not isappear at 2 hr. Therefore the queue spillback effect can be capture in a network of point queue moels with a finite storage capacity. 7 Conclusions In this stuy we presente a unifie approach for point queue moels, which can be erive as limits of two link-base queueing moel: the link transmission an link queue moels. In particular, we provie two efinitions of eman an supply of a point queue. The new approach le to the following aitional contributions: 22

23 1. From combinations of eman an supply efinitions, we presente four point queue moels, four approximate moels, an their iscrete versions. We iscusse the following properties of these moels: (i) equivalence: the four point queue moels are equivalent to each other; (ii) well-efineness: all the continuous an iscrete moels are well-efine uner suitable conitions; an (iii) smoothness: the solutions to the original moels are generally not smooth, but those to the approximate moels are. These results are obtaine theoretically an also verifie with numerical simulations. A numerical example also shows that the propose moels with finite storage capacities can capture the queue spillback effect. 2. We analytically solve Vickrey s point queue moel, which has an infinite storage capacity an is initially empty. The results are consistent with but more general than those in the literature. We also solve stationary states in the four point moels an their approximate versions with constant origin emans an estination supplies. This emonstrates that the new unifie formulations enable analytical solutions in special ynamical or stationary cases. 3. In this stuy we emonstrate that all existing point an flui queue moels in the literature, incluing Moran s am process moel an Vickrey s point queue moel, are special cases of those erive from the link-base queueing moels. This further confirms the usefulness of the new unifie approach. Even though in this stuy point queue moels have eterministic origin emans an estination supplies, they can be use to moel stochastic queues with ranom origin emans an estination supplies. For example, for a single queue with an infinite storage capacity, if the origin eman is ranom an follows a Poisson process, an the estination supply is constant, then this becomes an M/D/1 queue, an some of the results in Section 5.1 still apply. In the future we will be intereste in stuying stochastic queues with ifferent moels erive in this stuy. With eman an supply variables, point queue moels have the same structure as link-base queueing moels. Therefore, they can be integrate into a network of both point an link queues with appropriate macroscopic junction moels, which calculate bounary fluxes from upstream emans an ownstream supplies (Jin, 2012a). In the future we will be intereste in stuying stationary patterns an ynamics in networks of eterministic or stochastic queues, which can arise in computer, air traffic, an other systems. Furthermore, we will also be intereste in control, management, scheuling, planning, an esign of such queueing networks. References Armbruster, D., e Beer, C., Freitag, M., Jagalski, T., Ringhofer, C., 2006a. Autonomous control of prouction networks using a pheromone approach. Physica A: Statistical Mechanics 23

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