
The gut microbiota (GM), as a forgotten organ, refers to the microbial community that resides in the gastrointestinal tract and plays a critical role in a variety of physiological activities in different body organs. The GM affects its targets through neurological, metabolic, immune, and endocrine pathways. The GM is a dynamic system for which exogenous and endogenous factors have negative or positive effects on its density and composition. Since the mid-twentieth century, laboratory animals are known as the major tools for preclinical research; however, each model has its own limitations. So far, two main models have been used to explore the effects of the GM under normal and abnormal conditions: the isolated germ-free and antibiotic-treated models. Both methods have strengths and weaknesses. In many fields of host-microbe interactions, research on these animal models are known as appropriate experimental subjects that enable investigators to directly assess the role of the microbiota on all features of physiology. These animal models present biological model systems to either study outcomes of the absence of microbes, or to verify the effects of colonization with specific and known microbial species. This paper reviews these current approaches and gives advantages and disadvantages of both models.
Citation: Fatemeh Aghighi, Mahmoud Salami. What we need to know about the germ-free animal models[J]. AIMS Microbiology, 2024, 10(1): 107-147. doi: 10.3934/microbiol.2024007
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The gut microbiota (GM), as a forgotten organ, refers to the microbial community that resides in the gastrointestinal tract and plays a critical role in a variety of physiological activities in different body organs. The GM affects its targets through neurological, metabolic, immune, and endocrine pathways. The GM is a dynamic system for which exogenous and endogenous factors have negative or positive effects on its density and composition. Since the mid-twentieth century, laboratory animals are known as the major tools for preclinical research; however, each model has its own limitations. So far, two main models have been used to explore the effects of the GM under normal and abnormal conditions: the isolated germ-free and antibiotic-treated models. Both methods have strengths and weaknesses. In many fields of host-microbe interactions, research on these animal models are known as appropriate experimental subjects that enable investigators to directly assess the role of the microbiota on all features of physiology. These animal models present biological model systems to either study outcomes of the absence of microbes, or to verify the effects of colonization with specific and known microbial species. This paper reviews these current approaches and gives advantages and disadvantages of both models.
Modeling and analysing the dynamics of network flows is an important problem that has applications in many different areas such as transportation planning [7,25,42], air traffic control [32,48], communication networks [1,6,12,22], processor scheduling [49] and supply chain optimization [34,17]. Flow models are crucial for understanding the response of networked systems under different boundary conditions, estimating the state of the system, measuring system performance under different tunable parameters and devising the appropriate control strategies for efficient operation of the system. For example, in transportation networks, flow models are used for traffic estimation [53], dynamic traffic assignment or demand response assessment [30], traffic signal control [28], ramp-metering control [43] and rerouting [47]. This article focuses on modeling heterogeneous (multi-path) traffic flow through a network with a single source and multiple sinks with the objective of analytically expressing the delays at each node of the network as a function of the boundary flows at the source. This is a critical requirement when solving certain control and optimization problems over the network, where flow entering the network is one of the direct or indirect control parameters [46]. We present this model in the context of a transportation network modeling, but the results can be applied to any single origin network with no merge junctions that satisfies the following properties; 1) link flows are capacity restricted, 2) the flow through each junction satisfies the first-in-first-out (FIFO) condition, and 3) there is no holding of flow, i.e. the flow through a junction is maximized subject to the FIFO condition.
There are many types of traffic dynamics models, including microsimulation [18], cellular automaton [33], first-order or higher-order macroscopic models [19,26,44,51] and point queue models [52], listed roughly in decreasing model complexity. Microsimulation models simulate the movement of individual vehicles by applying car following models and lane changing models. This approach enables the modeling of very specific and detailed characteristics of driver behavior and networks dynamics [18] such as route choice modeling and the implications of the precise network geometry (e.g. road curvature). Therefore, microscopic models are very popular in transportation planning at an urban scale, since planners have the ability to observe the behavior on the entire network due to modifications such as road closures and changes in traffic signal plans. However, these models can not be expressed in closed form, are very difficult to calibrate [20] and have an extremely high computational overhead associated with them.
Another related approach is to use cellular automaton models to simulate traffic [33]. Typically, cellular automaton models divide the roadway into cells, each of which can be empty or occupied by only one vehicle. This approach as also been extended to model heterogeneous flow [21,23], where the behavior around freeway off-ramps are considered. Cellular automation models are similar to microscopic models in the sense that they both scale very poorly with the problem size and are generally computationally intractable since they track individual vehicles.
The third and perhaps most common approach is the use of macroscopic traffic flow models. Among these models, the most well known is the continuum or fluid-dynamic model based on the Lighthill-Williams-Richards (LWR) partial differential equation [26,44,27]. This model uses first-order partial differentiation equations to describe the relationship between aggregate traffic flow variables (flux and density), under the principle of mass conservation. Due to its simplicity, the LWR model does not capture individual driver behavior and cannot be used to model heterogeneous flows where different groups of drivers exhibit different behaviors. However, there have been extensions to the LWR model that can account for such heterogeneous flow [8,9].
The most popular solution method to the LWR PDE is the Godunov scheme, also known as the cell transmission model (CTM) [7,25]. The CTM discretizes space into cells and keeps track of the vehicle density in each cell instead of individual vehicles. Another approach for solving the LWR PDE is using the Moskowitz function. The Moskowitz function numbers vehicles as they pass a reference location and it describes the evolution of this number over time and space. The Moskowitz function can be interpreted as the cumulative vehicle count at a reference location, also known as cumulative curves, which is similar to our approach. Cumulative curves have been proposed together with triangular shaped fundamental diagram (relationship between flow and density) to simplify the traffic analysis based on the LWR model [35,36,37], because the adoption of cumulative curves automatically guarantees conservation. However, while cumulative curves do provide a nice graphical interpretation of the traffic flow, they do not present the ability to describe the network behavior as a function of variable boundary flows.
The final approach, which is also the approach used in this work, is to model the traffic via a network of point queues that are typically represented using ordinary differential equations (ODE's). One of the first point queue models was presented in the seminal work of Vickrey [52] to study a roadway with a single bottleneck. This approach has since been extended to handle networks with multiple commodities [11]. One of the main limitations of this ODE-based approach is that it does not capture the spatial propagation of traffic along a roadway (unlike the PDE-based macroscopic models) and assumes that all delayed vehicles wait at the point of the bottleneck, as if vehicles were packets in a data network. As a result, point queue models are generally unable to capture queue spillback. However, some recent work [41,40] shows how point queue models can be adapted to capture queue spillback using two queues for each bottleneck.
There is a vast literature in the transportation community on network flow models based on both ordinary differential equation (ODE) based point queues and the LWR PDE based spatio-temporal propagation. These models are also commonly referred to as network loading models [2] in dynamic traffic assignment (DTA) [31]. While the existing literature provides a rich portfolio of modeling techniques, of which some recent works we summarize below, this work presents a new technique that adds to the existing literature by enabling the analytical expression of delays within the network as a function of the inflows at the origin and the bottlenecks that are active. This however comes at the cost of some restrictions in terms of the types of networks that can be analyzed, as we will explain in detail below.
There have been a number of recent works on extending point queue models for network loading in DTA. In [3,4], the authors take a similar approach to ours in terms of extending the Vickrey point queue model (with no spillback). A time discretized iterative numerical scheme is provided to obtain a dynamic user equilibrium solution and the existence of the solution is proven under the assumptions of the model. The convergence rate of the iterative scheme is not explicitly addressed. Similarly, [11] considers a point-queue model with no spillback, but is able to model congestion delay during the free-flow phase of the fundamental diagram for hyperbolic fundamental diagrams. In contrast, [29] uses a point queue network with multiple queues to capture spillback, based on the double queue technique in [41]. The existence and uniqueness of the solution to the network loading problem is not addressed, but an existence result for the dynamic system optimal solution is provided. While the work in [11] allows for merge junctions and the model in [29] captures some queue spillback, none of the works above can analytically express the delays within the network as a function of the input flows1.
1We wish to incorporate the approach in [41] for capturing queue spillback into the proposed model as future work.
A series of articles by Han et al. [14,15,16] reformulates the Vickrey model using a PDE-based approach and extends the results to general networks. This reformulation enables more rigorous mathematical analysis of the network loading model, better numerical methods for solving the network loading problem and a proof of existence for a route and departure time user equilibrium. In a follow up work, [5] shows the existence and well-posedness of this PDE-based queuing model. Furthermore, [13] presents a very important contribution to the literature by providing a continuous time network loading model with both existence and well-posedness guarantees. This model can be considered a continuous time counterpart of the Link Transmission Model [39,54]. The results on existence and well-posedness are however limited to certain junction models. In particular, the results require fixed turning ratios at junctions, which does hold in dynamic traffic assignment.
In addition, there is a rich literature on ODE-based network flow propagation for various packet networks [1,6,12,22], but a large majority of these dynamics models violate the FIFO and no holding requirements that are essential in phyisical flow networks. The models proposed for transportation network flows, as described above, can in fact satisfy these physical requirements, but cannot analytically prescribe the internal delays of the network as a function of the boundary flows. Therefore, a new framework is required for the problem that we consider.
Our approach can be summarized as follows. We assume that the traffic flow is differentiated by the destination of the flow (i.e. Lagrangian flow) and that the different flow groups satisfy the FIFO condition at each junction. The network is limited to a contain a single origin and no merge junctions. The input flows are assumed to be piecewise continuous in time and the link capacities are assumed to be piecewise constant in time. The queuing in the network is assumed to be contained at each junction node and spill-back to the previous junction is ignored2. We show that this model leads to a well-posed ordinary differential equation for computing the dynamics of the network as a function of the boundary flows and prove that the solution is unique through a rigorous mathematical derivation of the model properties. The main benefit of this framework is the ability to analytically prescribe the delays at any junction in the network and across any sub-path as a function of the the boundary flows, which can be an important requirement when solving certain control and optimization problems, such as demand allocation problems, where the flow entering the network is one of the direct or indirect control parameters. This is achieved via the creation of a time mapping operator that maps the traffic flow at a given node at a given time to the corresponding flow at the origin of the network when that flow entered the network. We also show that this model can be solved numerically using a simple and efficient forward simulation approach. Finally, we demonstrate the application of the model by applying it to two example networks, a single path of multiple bottlenecks and a diverge junction with complex junction dynamics including time varying capacities.
2This is a restrictive assumption in the case where the queue length at a given node exceeds the length of the incoming link because the queue would have interrupted the flow at the previous junction. The model allows for these cases of queue spill-back to be identified by comparing the queue length to the queue capacity, and for control actions to be taken to mitigate the spill-back. However, in general, it may not be possible to prevent spill-back under certain input flow profiles.
The article is organized as follows. Section 2 introduces the network properties and junction dynamics. Then section 3 formalizes the time mapping operator, shows the well-posedness of the problem and proves the uniqueness of the solution to this model. Section 4 demonstrates the practical application of the mathematical framework by showing that the off-ramp model posed by Newell [38] can be modeled using this framework. Section 5 concludes the article.
The traffic network with a single source is modeled as an arborescence3. The congestion at each bottleneck is modeled as a vertical queue that is located at the start of the bottleneck. Thus, the physical propagation of the queue forming at the bottleneck is not modeled. This modeling choice is only restrictive when the queue propagates upstream to the preceding junction, as the change in dynamics at the junction due to the queue is not taken into account, but the model is equivalent to a horizontal queuing model otherwise.
3An arborescence is a directed rooted tree where all edges point away from the root.
A node
The congestion-free travel time on link
The set of incoming links to node
Linv={l∈L|voutl=v}, Loutv={l∈L|vinl=v} | (1) |
A node
The set of nodes
We define a path
p(vorig,vdest)=(vorig,⋯,vdest) s.t. (πvi,vi)∈L ∀i∈p∖vorig | (2) |
There is a unique path from any source to any destination since the network is tree structured. For each sink
Pv={p|v∈Vp} ; Pl={p|vinl∈Vp and voutl∈Vp} | (3) |
Remark 1. The path sets
Pv=∪l∈LoutvPl | (4) |
The traffic flow at a node is measured by counting the number of agents that pass through the node between an arbitrary initial time
For a node
Remark 2. The arrival curve
Definition 2.1. Acceptable cumulative arrival and departure curves
Given times
The assumption that the cumulative curves are strictly increasing is made for mathematical convenience, but can be relaxed4. Cumulative curves are required to be piecewise
4We could relax the assumption that the cumulative curves are strictly increasing and allow for non-decreasing curves. However, this results in the time mapping function
The outgoing flow
λvp=Dvpt | (5) |
Remark 3. Zero congestion-free travel time
Let
Dπvp=Avp ∀l=(πv,v)∈L,p∈Pv | (6) |
This modeling choice is made purely for mathematical convenience, since the goal of this framework is to analyze delays in the network. The total travel time for each agent can be easily reconstructed a posteriori by adding the actual congestion-free travel time for each link of the path traveled by the agent.
Thus, for all links
dAvpdt=dDπvpdt=λπvp | (7) |
\begin{equation} \dfrac{dD^{v}_{p}}{dt} = \lambda^{v}_{p} . \end{equation} | (8) |
This section defines the model dynamics for queuing and the flow propagation through a junction, which will then lead to a definition of the feasible departure curves that the model admits.
The capacity
Requirement 1. Capacity constrained flows
The inflow entering a link is always no greater than the links capacity.
∑p∈Plλvinlp(t)≤μl(t) ∀t,l∈L | (9) |
If the flows arriving at a node
Definition 2.2.Queue length
We define the path queue length
nv,p(t)=Avp(t)−Dvp(t) | (10) |
The total queue length
nv(t)=∑p∈Pvnv,p(t) | (11) |
Remark 4. Let
Definition 2.3. Delay in queue
We define
δvp(t)=[Dvp]−1(Avp(t))−t=[Dvp]−1(Dπvp(t))−t | (12) |
As
Remark 5. If
Therefore,
∀t,δv,p(t)>0⇔∃p′∈Pv:nv,p′(t)>0 | (13) |
We now describe the diverge model with a graphical illstration of a one to two diverge junction given in Figure 1.
Requirement 2. First-in-first-out (FIFO) property
The model satisfies the FIFO property. The delay encountered in queue
δv(t)=δv,p(t)=[Dvp]−1(Dπvp(t))−t ∀t,∀p∈Pv | (14) |
FIFO property implies that agents exit the queue in the same order that they enter the queue regardless of which path they belong to.
t1<t2⇔[Dvp1]−1(Dπvp1(t1))<[Dvp2]−1(Dπvp2(t2)) | (15) |
Interpreting
Proposition 1. FIFO implies conservation of the ratio of flows
If
λvp1(t+δv(t))λvp2(t+δv(t))=λπvp1(t)λπvp2(t), ∀t∈(tinit,tfinal] | (16) |
Proof. Let
\begin{array} sD^{\pi_{v}}_{p}(t) &=& D^{v}_{p}(t + \delta_{v,p}(t)) \forall p \in P_v \end{array} |
Taking the derivative with respect to
\begin{equation} \dfrac{dD^{\pi_{v}}_{p}(t)}{dt} = \left(1+\dfrac{d\delta_{v}(t)}{dt}\right) \cdot \left.\dfrac{dD^{v}_{p}}{dt}\right|_{t+\delta_{v}(t)} \end{equation} |
Using equation (8) we obtain,
\begin{array} a\lambda^{\pi_{v}}_{p}(t) &=& \left(1+\dfrac{d\delta_{v}(t)}{dt}\right) \cdot \lambda^{v}_{p}{t + \delta_{v}(t)} \forall p \in P_v \end{array} | (17) |
Therefore, it follows that
λvp1t+δv(t)λvp2t+δv(t)=λπvp1(t)λπvp2(t) |
Definition 2.4. Queue state
We define queue state as the boolean valued function
ηv(t)={1if δv(t)>00otherwise | (18) |
If
If
A queue state transition happens at time
∃ϵ>0 s.t. ∀θ∈{(−ϵ,0)∪(0,ϵ)},ηv(t−θ)=1−ηv(t+θ) | (19) |
When queue
Definition 2.5. Link constraint
Let
5The dissipation rate of the point queue at the node is only governed by the capacities of the outgoing links. This model can be extended to also impose a discharge rate constraint based on the capacity of the incoming link, but increases the complexity of the notation and the proofs.
6When there is a tie, one of them is chosen arbitrarily.
cl,v(t)=∑p∈Plλπvp(t)μl(t+δv(t)) | (20) |
Definition 2.6. Active link
We define the active link
6When there is a tie, one of them is chosen arbitrarily.
γv(t)∈argmaxl∈Loutvcl,v(t) | (21) |
We define the set of active paths
Γv(t)=Pγv(t) | (22) |
Remark 1 gives
Requirement 3. Full capacity discharge property
The model satisfies the full capacity discharge property. For each node
ηv(t)=1⇒∑p∈Γv(t)λvp(t+δv(t))=μγv(t)(t+δv(t)) | (23) |
With this last property, we complete the definition of the dynamics model.
Definition 2.7. Feasible flows
A feasible flow
The definition of the initial conditions on the network completes the definition of the model.
Definition 2.8. Initial times for each non-source node
Given a set of initial delays at each node
{t0,initial=tinitial for node v0tv,initial=tπv,initial+δv(tπv,initial) | (24) |
Now that we have fully defined the model dynamics, we consider the well-posedness of the model. In other words, given a network, link capacities and the departure functions at the source, we want to know whether the dynamics of the model admits a unique solution.
Probleme 1: General network problem
Input. An arborescence
Question. Does a corresponding set of feasible flows exist for all internal nodes
Theorem 1 states that the solution to problem 1 both exists and that the solution is unique, under certain conditions on the departure curves at the origin and the link capacities of the network.
Theorem 1. Existence and uniqueness of the solution to problem 1
Problem 1 admits a unique solution under the following conditions.
1) the path flows at the origin
2) link capacities
7Neither of these assumptions are restrictive in a practical sense, because any piecewise continuous function on a closed interval can be approximated to an arbitrary accuracy by a polynomial of appropriate degree (Stone-Weierstrass theorem [50]) and link capacities do not evolve in a continuous manner. LiIf the flows arriving at a nodenk capacities are typically subject to discrete changes due to incidents such as accidents and changes in weather.
The next section is devoted to a constructive proof of Theorem 1. The general flow of the proof is as follows. Sections 3.1-3.3 first develop a set of differential equations for delays in the network. In section 3.4, we then prove that a unique solution to differential equation on delays also implies a unique solution to problem 1. Section 3.5 proves that the differential equations on the delay at each node always admit an unique solution, which finally leads to the proof of Theorem 1.
This section builds a constructive proof of Theorem 1. Throughout sections3.1-3.3, we require that the flows at the origin are acceptable departure curves as defined in definition 2.1 and that the outflows at each node satisfy the model requirements (i.e. result in feasible flows as defined in definition 2.7).
We begin by proving Proposition 2, which gives an analytical expression for the derivative of the delay at node as a function of its downstream capacities and outgoing flow at its parent nodes.
Proposition 2. Evolution law of a single queue
If queue
dδvdt|t=∑p∈Γv(t)λπvp(t)μγvt(t+δv(t))−1 | (25) |
Proof. From Equation (17),
λπvp(t)=(1+dδv(dt)t)⋅λvp(t+δv(t))∀p∈Γv(t)⇒dδv(t)dt+1=λπvp(t)λvp(t+δv(t))∀p∈Γv(t)⇒dδv(t)dt+1=∑p∈Γv(t)λπvp(t)∑p∈Γv(t)λvp(t+δv(t)) |
Using the full capacity discharge property from Requirement 3,
dδv(t)dt+1=∑p∈Γv(t)λπvp(t)μγv(t)(t+δv(t))⇒dδvdt|t=∑p∈Γv(t)λπvp(t)μγv(t)(t+δv(t))−1 |
The evolution law stated above foWe now introduce the following time mapping function r any given node
The evolution law from Proposition 2 gives a non-linear ordinary differential equation (ODE) that governs the evolution of
Let
Definition 3.1. Node time mapping function
We define the time mapping function
Tv,πv:t↦t+δv(t) | (26) |
an agent leaving node
The notation
Proposition 3.
The function
dTv,πvdt|t=∑p∈Γv(t)λπvp(t)μl(t+δv(t))>0 | (27) |
Physically, this means that the FIFO assumption is respected: i.e. an agent
Proof. Taking the derivative of equation (26) and applying equation (25) in Proposition 2 gives,
dTv,πvdt|t=∑p∈Γv(t)λπvp(t)μl(t+δv(t)). | (28) |
The departure curves at the origin are strictly increasing since they must be acceptable departure curves. The full capacity discharge property from requirement 3 requires that one outgoing link at each node discharges at full capacity. Finally, these properties combined with Proposition 1, which states that the outflows at a node are proportional to the inflows, give us the result that
Thus
8If the acceptable set of departure curves
Definition 3.2. Node time mapping function
Given an internal node
Tπv,v∘Tv,πv=1 and Tv,πv∘Tπvv=1 | (29) |
We now consider the unique path
1)
2)
3)
As
Definition 3.3. Time mapping function from and to the origin
Let
Tv0,vn=Tv0,v1∘Tv1,v2∘⋯∘Tvπn,vn | (30) |
an agent that leaves node
Tvn,v0=Tvn,vπn∘⋯∘Tv2,v1∘Tv1,v0 | (31) |
an agent that leaves the origin at time
A sample path from the origin
Definition 3.4. Time mapping function between two arbitrary nodes
We define the time mapping function
1) There exists a path between nodes
Ti,j={Ti,i+1∘Ti+1,i+2∘⋯∘Tj−2,j−1∘Tj−1,jif i≺jTi,i−1∘Ti−1,i−2∘⋯∘Tj+2,j+1∘Tj+1,jif i≻j | (32) |
Let
2) There does not exist a path between nodes
Ti,j=Ti,v0∘Tv0,j | (33) |
Let
Definition 3.5. Time mapping operator
We define the time mapping operator
Ti,j:F→Ff↦f∘Tj,i | (34) |
We now consider the physical interpretation of
This section first studies the relationship between departure curves at different nodes and the time mapping function. We then define the time mapped versions of the other quantities in the model. The time mapping operators allow for mapping any quantity from one node to the other. This definition of a time mapped quantities thus allows any quantity to be defined with respect to the source node of the network.
Proposition 4. Physical interpretation of the time mapping function
Let
Dvip=Dv0p∘Tv0,vi ∀vi∈p | (35) |
Let
Dv0p(tv0)=Dv1p(tv1)=⋯=Dvip(tvi)=⋯=Dvnp(tvn) | (36) |
Proof. Proof by induction on the length of the sequence
Dvip=Dvi−1p∘Tvi−1,vi=Dv0p∘T0,vi−1∘Tvi−1,vi=Dv0p∘Tv0,vi |
Equation (36) follows directly from equation (35)
Remark 6. As function
We can now reformulate the first equation of Proposition 4 as follows:
Proposition 5. Time mapping of departure curve
Let
Dip=Ti,j(Djp) | (37) |
Proof. Using definition 3.5 we have,
Ti,jDjp=Djp∘Tj,i=Djp∘Tj,0∘T0,i=D0p∘T0i=Dip |
Proposition 6. Time mapping and flows
Let
λip=Ti,j(λjp)⋅dTj,idt | (38) |
Proof. From the definition of flow,
Remark 7. The time mapping and derivative operators do not commute.
Definition 3.6. Time mapping of delay
Let
9An internal node is a node
δπvv≐δv | (39) |
Let
δij≐Ti,πj(δπjj)=δπjj∘Tπj,i | (40) |
Physically, if nodes
Definition 3.7. Time mapping for capacity
We define the time mapped capacity of a link
μvinll≐μl | (41) |
Let
μvl≐Tv,vinl(μvinll)=μvinll∘Tvinl,v | (42) |
Physically, if link
Proposition 7. Physical interpretation of mapped delay and mapped capacity
Let
δv0vj(tv0)=δv1vj(tv1)=⋯=δvivj(tvi)=⋯=δvnvj(tvn) | (43) |
Let
μv0l(tv0)=μv1l(tv1)=⋯=μvil(tvi)=⋯=μvnl(tvn) | (44) |
Proof. Let
δij≐δi−1j(Tj−1,i(ti))=δi−1j(tj−1) |
Therefore,
Definition 3.8. Time mapping of active link and active paths
Let
γπvv≐γv;Γπvv≐Γv | (45) |
Let
γjv=Tjπvγπvv ; Γvj=Tj,πvΓπvv | (46) |
Physically, if node
Definition 3.9. Time mapped link constraint
Let
cπvv,l(t)≐∑p∈Plλπvp(t)μl(t+δv(t)) | (47) |
=∑p∈Plλπvp(t)μvl(t+δv(t))=∑p∈Plλπvp(t)μπvl(t) | (48) |
Let
cjv,l≐Tj,πv(cπvv,l)=cπvv,l∘Tπv,j | (49) |
cjv,l(t)=∑p∈Plλjp(t)μjl(t)⋅dTπvjdt | (50) |
Physically, if node
Remark 8. The notation of the link constraint can be simplified for convenience as follows when time mapped.
cjv,l=cjl | (51) |
We use the simplified notation in the rest of the discussion.
Proposition 8. The mapping of link constraints and active links is coherent
For all non-sink nodes
γjv(t)∈argmaxl∈Loutvcjv(t) | (52) |
Proof. Let
argmaxl∈Loutvcv,ltπv=argmaxl∈Loutvcjl(tj) | (53) |
From the definition of the link constraint in equation (20) we have
cv,ltπv≐∑p∈Plλπvp(tπv)μl(tπv+δv(tπv)) | (54) |
By definition of
μl(tπv+δv(tπv))=μjl(tj) | (55) |
Moreover, using equation (38) gives
∑p∈Plλπvp(tπv)=1dTπv,jdt|tj⋅∑p∈Plλjp(tj) | (56) |
Substituting equations (55) and (56) in the right hand side of equation (54) and using the time mapped link constraint from equation (50), we obtain
cv,l(tπv)=1dTπv,jdt|tj⋅[∑p∈Plλjp(tj)μjl(tj)] | (57) |
=1dTπv,jdt|tj⋅cjl(tj) | (58) |
For all
Definition 3.10. Capacity of the active link
For notational simplicity we denote the capacity of the active link of an agent that enters queue
Qv(t)≐μπvγv(t)(t)=μvγv(t)t+δv(t) | (59) |
=μγv(t)t+δv(t) | (60) |
Definition 3.11. Time mapped capacity of the active link
Let
Qπvv≐Qv | (61) |
Let
Qjv≐Tj,πv(Qπvv)=Qπvv∘Tπv,j=Qv∘Tπv,j | (62) |
Physically, if node
Definition 3.12 Time mapping of queue state
Let
ηij≐Ti,πv(ηπvj)=ηπvj∘Tπv,i=ηj∘Tπv,i | (63) |
Physically, if queue
We now have the necessary tools to define the evolution of delays at any node of the network with respect to the flows at any upstream node in the network.
Definition 3.13. First active upstream node
Let
Υjv(t)=max≻{u|u≺v,ηju(t)=1} | (64) |
For notational convenience we also define the following:
ˆγjv(t)≐γjΥjv(t)(t) | (65) |
ˆΓjv(t)≐ΓjΥjv(t)(t) | (66) |
ˆQjv(t)≐QjΥjv(t)(t) | (67) |
ˆηjv(t)≐ηjΥjv(t)(t) | (68) |
Theorem 2. Evolution law for delay at an arbitrary internal node
Given an arbitrary internal node
dδjvdt|t={∑p∈Γjv(t)λjp(t)Qjv(t)−dT0,jdt|tif v isthefirstactivequeue ∈p∑p∈Γjv(t)λjp(t)Qjv(t)−∑p∈ˆΓjp(t)λjp(t)ˆQjv(t)otherwise | (69) |
Proof. Let
dδvdt|t=∑p∈Γv(t)λπvp(t)Qv(t)−1 | (70) |
By the definition of the time mapping functions we have,
dδπvvdt|t=∑p∈Γπvv(t)λπvp(t)Qπvv(t)−1 | (71) |
Case 1: If node
Let for an arbitrary node
tΥv(tπv)=tπv=t | (72) |
Furthermore, since
∑p∈ˆΓπvv(t)λΥv(t)p(t)=ˆQπvv(t) | (73) |
∑p∈ˆΓπvv(t)λπvp(t)=ˆQπvv(t) | (74) |
Thus:
∑p∈ˆΓπvv(t)λπvp(t)ˆQπvv(t)=1 | (75) |
By replacing the constant
dδπvvdt|t=∑p∈Γπvv(t)λπvp(t)Qπvv(t)−∑p∈ˆΓπvv(t)λπvp(t)ˆQπvv(t) | (76) |
This gives us the result for
δjv=δπvv∘Tπv,j | (77) |
Taking its derivative with respect to time, we obtain
dδjvdt=[dδπvvdt∘Tπv,j]⋅dTπv,jdt | (78) |
dδjvdt|t=[∑p∈Γπvv∘Tπv,j(t)λπvp∘Tπv,j(t)Qπvv∘Tπv,j(t)−∑p∈ΓπvΥvt∘Tπv,j(t)λπvp∘Tπv,j(t)QπvΥv(t)∘Tπv,j(t)]⋅dTπv,jdt|t | (79) |
Equation (38) on flow mapping gives
(λπvp∘Tπv,j(t))⋅dTπv,jdt|t=λjp(t) | (80) |
Substituting this result and the simple time mapping transformations of
dδjvdt|t=∑p∈Γjv(t)λjp(t)Qjv(t)−∑p∈ΓjΥv(t)(t)λjp(t)QjΥv(t)(t) | (81) |
=∑p∈Γjv(t)λjp(t)Qjv(t)−∑p∈ˆΓjv(t)λjp(t)ˆQjv(t) | (82) |
Case 2: If node
dδjvdt|t=∑p∈Γjv(t)λjp(t)Qjv(t)−dT0,jdt|t | (83) |
Applying Theorem 2 with
Definition 3.14. Time mapped delay evolution differential equation
• If
• If
\begin{align} \left.\dfrac{d\delta^{0}_{v}}{dt}\right|_t = \begin{cases} \dfrac{\sum_{p \in \Gamma^{0}_{v}(t)}\lambda^{0}_{p}(t)}{Q^{0}_{v}(t)} - 1&\\ \text{if } v \text{ is the first active queue} \in p \\ \dfrac{\sum_{p \in \Gamma^{0}_{v}(t)}\lambda^{0}_{p}(t)}{Q^{0}_{v}(t)} - \dfrac{\sum_{p \in \hat{\Gamma}^{0}_{v}(t)}\lambda^{0}_{p}(t)}{\hat{Q}^{0}_{v}(t)} \text{otherwise} \end{cases} \end{align} | (84) |
where the time mapping functions are redefined from delays as follows:
Tj0=∑0≺i≼jδ0i | (85) |
Proposition 9. Delay evolution does not depend on departure curves
All the time mapped quantities in equations (84) can be computed using only the initial delays, departure curve at the origin and the link capacities. It does not require the departure curves for any internal nodes
Proof. The time mapping function only depends on the delay functions from definition 3.1. The time mapped flows can be obtained using the time mapping function using Proposition 6. The other time mapped quantities are by definition constructed using the time mapping function as given in section 3.2.2.
We prove Theorem 1 on the existence and uniqueness of Problem 2 by first showing the equivalence between Problem 2 and Problem 2 (defined below), and then proving the existence and uniqueness of Problem 2 in the next section.
Problem 2: General delay problem
Input. An arborescence
Question. Does a solution to the time mapped delay function from definition 3.14 for each node
Theorem 3. Problem (1) and problem (2) are equivalent
Proof. The inputs to both problems are identical. Therefore, we only need to prove that the existence of a solution to one problem implies a unique and feasible corresponding solution to the other problem.
By the definition of delay,
(86) |
By the definition of time mapped delay we obtain,
(87) |
(88) |
(89) |
which can be made a function of only
Theorem 2 then ensures that the delay functions thus defined satisfy the time mapped delay evolution from definition 3.14, i.e. a feasible solution to problem 2. Furthermore, the solution is unique from equation (89), since
(90) |
The inverse departure curve
(91) |
The departure curve
We now show that the departure curves thus defined are feasible departure curves, i.e. a feasible solution to problem 1.
1.
2. The capacity constraint on links is imposed by equation (84) due to Proposition 8.
3.The FIFO condition is satisfied by construction since the delay
4.The full capacity discharge of the active queues is also imposed by by equation (84) due to Proposition 8.
This section proves Theorem 4 on the existence and uniqueness of the solution to Problem 2.
Theorem 4.Existence and uniqueness of the solution to problem (2)
The solution to problem (2) exists and is unique on the time interval of the problem
1. the path flows at the origin
2. link capacities
The proof of this theorem is fairly technical and requires several definitions and lemmas. Theorem 1 is a direct corollary of this result due to to Theorem 3 on the equivalence of the two problems.
The main goal of the proof of Theorem 4 is to show that there are a finite number of possible transitions, and to integrate equation (84) across the transitions. The next definitions and lemmas enables to establish these properties.
Definition 3.15 Depth of a node
We define the depth
Definition 3.16 Link constraint comparators
Given a node
(92) |
Given a node
(93) |
Definition 3.17. Time segment of constant link constraint
A time segment
1. for each
2. for each pair of nodes
3. for each
Lemma 10. Under the assumptions on flows and capacities, there are a finite number of segments of constant link constraint
Proof. Consider a pair of links
Lemma 11. Constant active link
If
Proof. The result comes directly from the definition of a segment of constant constraint.
Definition 3.18. Solution of depth
A solution of problem (2) for depth
Definition 3.19. Elementary time segment
Given a node
•
• If
Lemma 12. Single transition of node state on an elementary segment
If there exists a solution to problem (2) up to depth
Proof. As for each node
Let us now consider the following four cases:
1.
2.
3.
4.
Lemma 13. Unique solution on an elementary segment
Let
Proof. By Lemma 12, there can be at most one state transition of node
• If
• If
(94) |
As all the variables in equation (94) other than the flow
We have now all the ingredients to prove Theorem 4.
Proof of Theorem 4 We will prove the following proposition: The time interval of interest
The proof is done inductively over the depth of the network. If the network contains a single node
Let
Let
• for all
•
Thus,
This also completes the proof of Theorem 1.
Proof of Theorem 1 Problem 1. is equivalent to Problem 2 by Theorem 3 and Problem 2 admits a unique solution by Theorem 4.
In some applications, it is also important to be able to computing the total delay experienced by an agents that takes a particular path. A provides analytical expressions for the total delay along a path.
The solution to problem (1) models the flows in the network given the departure time functions at the origin and the initial delays by providing the departure time functions for all the other nodes in the network. The solution can be obtained by first solving problem (2), which provides the agent delay function at each node. Practically, problem (2) easier to directly solve than problem (1), because it corresponds to an explicit automaton that is easy to implement for numerical simulations.
Given a discretization time step
Algorithm 1 Calculate approximate solution of problem (2) |
The first case we will study is that of a simple single path network with multiple queues due to several capacity bottlenecks, as illustrated in figure 3. This network can be modeled as a tree with a single sink, i.e. a single path. Thus, we will remove the path index from the notation in this section. Each internal node
(95) |
Since, the link with the smallest capacity will always be the last active link
(96) |
Thus, the evolution of total delay is equivalent to the evolution law for one queue of capacity
If the capacity of the links is time varying and
(97) |
The next application is to compute the the dynamics of a congested freeway off-ramp, using the off-ramp model presented by Newell [38]. This example shows the versatility of the proposed framework, since Newell's the model includes non-FIFO dynamics at the off-ramp. This is accommodated by introducing an additional node and state dependent capacities on two links. The description of the model is as follows. As seen in figure 4(a), there are two flows
Case 1:
Case 2:
Case 3:
Case 4:
The uniqueness and existing properties hold even with the state dependent capacities, since the flows are assumed to be piecewise polynomial and therefore lead to a finite number of state transitions. This implies that the link capacities are piecewise constant. Therefore, we can solve for the delays in this network using algorithm 1. Furthermore, this subnetwork can be part of a larger network over which we wish to compute the system delays.
Figure 6 shows the flow and delay profiles for a numerical example of the off ramp network with the following link capacities:
• At
• At
• At
•At
One interesting observation is that freeway congestion caused by the exiting agent bottleneck persists well beyond the time at which the exiting agent queue disappears.
This article presented a mathematical framework for modeling traffic flow through a network with a single source and multiple sinks. The model satisfies the standard laws of flow dynamics and is shown to lead to a well-posed dynamics problem with an unique solution. The main benefit of this framework is the ability to analytically prescribe the delays at each junction as a function of the boundary flows at any other upstream junction and the delay over any sub-path with respect to the boundary flow at the source node of the sub-path. This is a critical requirement when solving control and optimization problems over the network, since solving an optimization problem over simulation models is generally intractable in terms of computational complexity. The versatility of computing the delays as a function of the inflow at any point in the network is achieved though a mathematical framework for time mapping the delays. An algorithm for computing a discrete approximation of the system numerically is also provided. The application of the framework is illustrated using two examples, a single path consisting of multiple bottlenecks and a diverge junction with complex junction dynamics. The main limitation of this framework is the requirement that the network only have a single source. The time mapping framework presented can however be generalized to any non-cyclic (tree) network. Thus, the next step would be to introduce merging dynamics into the model to obtain a more general network model.
In some applications, it is also important to be able to computing the total delay experienced by an agents that takes a particular path. In this section, we provide analytical expressions for the total delay along a path.
Definition A.1. Total delay of a path
We define the total delay
(98) |
where
(99) |
Proposition 14. Total delay
The time mapped total delay
(100) |
where
Proof. Let
\begin{eqnarray} LHS &= &\Delta^{j}_{p}(t^{j}) \nonumber\\ &= &\mathbf{T}^{j}_{0}(\Delta{0}{p}(t^{j})) \nonumber\\ &= &\Delta^{0}_{p}(T^{0}_{j}(t^{j})) \nonumber\\ &= &\Delta^{0}_{p}(t^{0}) \nonumber\\ &= &[D^{v_{p_N}}_{p}]^{-1}({D^{0}_{p}(t^{0}})) - t^{0} \nonumber\\ \nonumber\\ RHS &= &\sum\limits_{v \in V_p \backslash (\{0\} \cup S)} \delta^{j}_{v}(t^{j}) \ \ \ = \sum_{v \in V_p \backslash (\{0\} \cup S)} \delta^{\pi_{v}}_{v}(t^{\pi_{v}}) \nonumber\\ &= &\sum\limits_{v \in V_p \backslash (\{0\} \cup S)} [D^{v}_{p}]^{-1}(D^{\pi_{v}}_{p}(t^{\pi_{v}})) - t^{\pi_{v}} \nonumber\\ &= &\sum\limits_{v \in V_p \backslash (\{0\} \cup S)} [D^{v}_{p}]^{-1}(D^{\pi_{v}}_{p}(t^{\pi_{v}})) - [D^{\pi_v}_{p}]^{-1}(D^{\pi_{v}}_{p}(t^{\pi_{v}})) \nonumber\\ &= &\sum\limits_{v \in V_p \backslash (\{0\} \cup S)} [D^{v}_{p}]^{-1}(D^{0}_{p}(t^{0})) - [D^{\pi_v}_{p}]^{-1}(D^{0}_{p}(t^{{0}})) \nonumber\\ &=&[D^{v_{p_N}}_{p}]^{-1}(D^{0}_{p}(t^{0})) - [D^{0}_{p}]^{-1}(D^{0}_{p}(t^{0})) \nonumber\\ &=&[D^{v_{p_N}}_{p}]^{-1}(D^{0}_{p}(t^{0})) \nonumber \end{eqnarray} |
Definition A.2. Active link of the last active queue of a path
Let
(101) |
For notational convenience we also define the following:
(102) |
(103) |
(104) |
Theorem 5. Evolution law for total delay
Let
(105) |
Proof. Taking the dfrac of equation (100) for , we obtain
(106) |
(107) |
Note that
(108) |
Plugging this into equation (107) gives a telescopic series, since it only considers the active nodes of the path and
(109) |
If
Remark 9. Note that this theorem can be extended to any subpath
(110) |
Received June 2016; revised December 2016.
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Algorithm 1 Calculate approximate solution of problem (2) |