The Bottleneck Model

I review the famous bottleneck model that Vickrey introduced and that Arnott expanded upon.

Many papers have expanded upon the bottleneck model that Vickery introduced in his paper (Vickrey, 1969). Some of these papers include (Arnott et al., 1990) and (Laih, 1994). In this article, I review the model as well as analyze the pricing mechanism that minimizes queuing times.

The Bottleneck Model

In this model, there is a highway that has one entrance and one exit. A queue is formed when the capacity of the bottleneck is less than the traffic flow. Let the capacity of the bottleneck be denoted by ss (whose units are agent per time) and the time at which the queue begins be denoted as tqt_q. Route segments before and after the bottleneck have sufficient capacity so that no congestion occurs on them. Hence, an agent arrives at the bottleneck immediately after he departs from home and arrives at his destination immediately after passing the bottleneck.

There are NN agents who need to use the bottleneck. Each agent (which I call he) begins at home and can choose the amount of time tt that he stays at home. However, he must arrive at work at time t∗t^*. While traveling, the amount of time that he waits in a queue is denoted as TqT_q. When he arrives earlier than t∗t^*, he spends an amount of time TeT_e waiting until t∗t^*. Likewise, when he arrives later, the amount of time that he is late is denoted as TlT_l. Hence, Te={t∗−t−Tq if t+Tq<t∗0 if t+Tq≥t∗ T_e=\left\{\begin{array}{c}t^*-t-T_q \text { if } t+T_q<t^* \\ 0 \text { if } t+T_q \geq t^*\end{array}\right. and Tl={t+Tq−t∗ if t+Tq>t∗0 if t+Tq≤t∗. T_l=\left\{\begin{array}{c}t+T_q-t^* \text { if } t+T_q>t^* \\ 0 \text { if } t+T_q \leq t^*\end{array}\right.. Therefore, he arrives at time Te=Tl=0T_e=T_l=0 by leaving at t~:=t∗−Tq(t~)\tilde{t}:=t^*-T_q(\tilde{t}). For each agent, there are penalties for when he arrives relative to the starting time. α\alpha is the value penalty per minute of Tq,βT_q, \beta is the value penalty per minute of TeT_e, and γ\gamma is the value penalty per minute of TlT_l. It is assumed that the agent will encounter a queue once he leaves and that the penalty for being late is greater than the penalty for being in the queue which is greater than the penalty for being early, i.e. γ>α>β>0\gamma>\alpha>\beta>0. Suppose that the queue begins at some time tqt_q and ends at tq′t_{q^{\prime}}, i.e. Tq(tq)=0T_q\left(t_q\right)=0 and Tq(tq′)=0T_q\left(t_{q^{\prime}}\right)=0. The toll, if one exists, is denoted by τ\tau. The cost function for each agent can be expressed as C(t)=τ(t)+{β(t∗−t) if t∈[0,tq)αTq(t)+β(t∗−t−Tq(t)) if t∈[tq,t~)αTq(t) if t=t~αTq(t)+γ(t+Tq(t)−t∗) if t∈(t~,tq′]γ(t−t∗) if t>tq′. C(t)=\tau(t)+\left\{\begin{array}{c}\beta\left(t^*-t\right) \text { if } t \in\left[0, t_q\right) \\ \alpha T_q(t)+\beta\left(t^*-t-T_q(t)\right) \text { if } t \in\left[t_q, \tilde{t}\right) \\ \alpha T_q(t) \text { if } t=\tilde{t} \\ \alpha T_q(t)+\gamma\left(t+T_q(t)-t^*\right) \text { if } t \in\left(\tilde{t}, t_{q^{\prime}}\right] \\ \gamma\left(t-t^*\right) \text { if } t>t_{q^{\prime}}\end{array}\right.. Hence, the agent must choose a time tt to minimize the above cost. At equilibrium, all agents minimize costs and no agent is incentivized to unilaterally change his time.

The Non-Toll Equilibrium

This section deals with the case when no toll is imposed, i.e. τ=0\tau=0. I will show you the equilibrium time when queuing begins to form, when it dissipates, and the time at which one departs to arrive to work on time after queueing.

Proposition.\textbf{Proposition.} At equilibrium, t~=t∗−Nsγββα+γα,tq=t∗−Nsγβ+γ,tq′=t∗+Nsββ+γ\tilde{t}=t^*-\frac{N}{s} \frac{\gamma \beta}{\beta \alpha+\gamma \alpha}, t_q=t^*-\frac{N}{s} \frac{\gamma}{\beta+\gamma}, t_{q^{\prime}}=t^*+\frac{N}{s} \frac{\beta}{\beta+\gamma}, and the minimum cost for all agents is Nsβγβ+γ\frac{N}{s} \frac{\beta \gamma}{\beta+\gamma}.

Proof.\textit{Proof.} The first order necessary condition for minimizing any agent’s cost function is taking the derivative of C(t)C(t) with respect to tt, which implies dTq(t)dt={βα−β if t∈[tq,t~)−γα+γ if t∈(t~,tq′]. \frac{d T_q(t)}{d t}=\left\{\begin{array}{l}\frac{\beta}{\alpha-\beta} \text { if } t \in\left[t_q, \tilde{t}\right) \\ -\frac{\gamma}{\alpha+\gamma} \text { if } t \in\left(\tilde{t}, t_{q^{\prime}}\right]\end{array}\right.. This means that if he arrives early, the amount of time that he waits in the queue increases as he leaves later. And if he arrives late, the amount of time that he waits in the queue decreases as he leaves later. The initial condition, Tq(tq)=0T_q\left(t_q\right)=0 and Tq(tq′)=0T_q\left(t_{q^{\prime}}\right)=0, defines TqT_q as Tq(t)={βα−β(t−tq) if t∈[tq,t~)−γα+γ(t−tq′) if t∈(t~,tq′]. T_q(t)=\left\{\begin{array}{c}\frac{\beta}{\alpha-\beta}\left(t-t_q\right) \text { if } t \in\left[t_q, \tilde{t}\right) \\ -\frac{\gamma}{\alpha+\gamma}\left(t-t_{q^{\prime}}\right) \text { if } t \in\left(\tilde{t}, t_{q^{\prime}}\right]\end{array}\right.. Combined with the definition that t~=t∗−Tq(t~)\tilde{t}=t^*-T_q(\tilde{t}), we can imagine the possible times queues begin to form and dissipate. I now present an original argument (which I have not seen in the literature) for why an agent leaving on the early schedule at some time near t~\tilde{t} will wait in queue for approximately t∗−t~t^*-\tilde{t}. Suppose ϵ\epsilon is a very small positive number. When t=t~−ϵt=\tilde{t}-\epsilon, tt is close to t~\tilde{t}, either Tq(t~−ϵ)>t∗−t~T_q(\tilde{t}-\epsilon)>t^*-\tilde{t} or Tq(t~−ϵ)<t∗−t~−ϵT_q(\tilde{t}-\epsilon)<t^*-\tilde{t}-\epsilon. The former case implies that he is late, and the latter case implies that he arrives much earlier than the starting time. WLOG, the above argument applies to the case of why an agent leaving on the late schedule near t~\tilde{t} will have a queuing time of approximately t∗−t~t^*-\tilde{t}. For an easier understanding, see the graph from (Laih, 1994).

Therefore, the following constraint is a necessary condition at equilibrium. βα−β(t~−tq)=t∗−t~=γα+γ(tq′−t~).(1) \frac{\beta}{\alpha-\beta}\left(\tilde{t}-t_q\right)=t^*-\tilde{t}=\frac{\gamma}{\alpha+\gamma}\left(t_{q^{\prime}}-\tilde{t}\right). \tag{1} The above condition is equivalent to t~−α−ββ(t∗−t~)=tq\tilde{t}-\frac{\alpha-\beta}{\beta}\left(t^*-\tilde{t}\right)=t_q and t~+α+γγ(t∗−t~)=tq′\tilde{t}+\frac{\alpha+\gamma}{\gamma}\left(t^*-\tilde{t}\right)=t_{q^{\prime}}. Further, note the constraint that the total number of commuters that are able to cross the bottleneck within the queuing interval [tq,tq′]\left[t_q, t_{q^{\prime}}\right] is N=s(tq′−tq).(2) N=s\left(t_{q^{\prime}}-t_q\right). \tag{2}

Using conditions (1) and (2), we have the following equilibrium values when there is no toll. t~=t∗−Nsγββα+γαtq=t∗−Nsγβ+γ \begin{aligned} \tilde{t} & =t^*-\frac{N}{s} \frac{\gamma \beta}{\beta \alpha+\gamma \alpha} \\ t_q & =t^*-\frac{N}{s} \frac{\gamma}{\beta+\gamma}\end{aligned} and tq′=t∗+Nsββ+γ.t_{q^{\prime}}=t^*+\frac{N}{s} \frac{\beta}{\beta+\gamma}.

The cost at equilibrium is then α(t∗−t~)=Nsβγβ+γ\alpha\left(t^*-\tilde{t}\right)=\frac{N}{s} \frac{\beta \gamma}{\beta+\gamma}.

The Tolling-Equilibrium

Consider now the case of applying a toll on crossing the bottleneck in order to eliminate the queue entirely and to ensure that no agent has an incentive to deviate.

Proposition.\textbf{Proposition.} Suppose there is no queue (i.e. Tq(t)=0T_q(t)=0 for all t∈R+t \in \mathbb{R}_{+}). Each agent’s cost is Nsβγβ+γ\frac{N}{s} \frac{\beta \gamma}{\beta+\gamma} for all t∈[tq,tq′]t \in\left[t_q, t_{q^{\prime}}\right] if and only if the toll

τ(t)={0 if t∈[0,tq)Nsβγβ+γ−β(t∗−t) if t∈[tq,t∗]Nsβγβ+γ−γ(t−t∗) if t∈[t∗,tq′]0 if t>tq′. \tau(t)=\left\{\begin{array}{c} 0 \text { if } t \in\left[0, t_q\right) \\ \frac{N}{s} \frac{\beta \gamma}{\beta+\gamma}-\beta\left(t^*-t\right) \text { if } t \in\left[t_q, t^*\right] \\ \frac{N}{s} \frac{\beta \gamma}{\beta+\gamma}-\gamma\left(t-t^*\right) \text { if } t \in\left[t^*, t_{q^{\prime}}\right] \\ 0 \text { if } t>t_{q^{\prime}} \end{array} .\right.

Proof.\textit{Proof.} Suppose each commuter has no queuing time (i.e. Tq(t)=0T_q(t)=0 for all t∈R+t \in \mathbb{R}_{+}) and each commuter’s cost is Nsβγβ+γ\frac{N}{s} \frac{\beta \gamma}{\beta+\gamma} for all t∈[tq,tq′]t \in\left[t_q, t_{q^{\prime}}\right]. First, each commuter having no queuing time implies that Tq(t~)=0⇒t~=t∗T_q(\tilde{t})=0 \Rightarrow \tilde{t}=t^*. Hence, the given toll τ\tau would result in the following cost function.

C(t)={β(t∗−t) if t∈[0,tq)Nsβγβ+γ−β(t∗−t)+β(t∗−t) if t∈[tq,t∗)Nsβγβ+γ−β(t∗−t) if t=t∗Nsβγβ+γ−γ(t−t∗)+γ(t−t∗) if t∈(t∗,tq′]γ(t−t∗) if t>tq′={β(t∗−t) if t∈[0,tq)Nsβγβ+γ if t∈[tq,t∗)Nsβγβ+γ if t=t∗Nsβγβ+γ if t∈(t∗,tq′]γ(t−t∗) if t>tq′. \begin{aligned} C(t) &= \left\{\begin{array}{c}\beta\left(t^*-t\right) \text { if } t \in\left[0, t_q\right) \\ \frac{N}{s} \frac{\beta \gamma}{\beta+\gamma}-\beta\left(t^*-t\right)+\beta\left(t^*-t\right) \text { if } t \in\left[t_q, t^*\right) \\ \frac{N}{s} \frac{\beta \gamma}{\beta+\gamma}-\beta\left(t^*-t\right) \text { if } t=t^* \\ \frac{N}{s} \frac{\beta \gamma}{\beta+\gamma}-\gamma\left(t-t^*\right)+\gamma\left(t-t^*\right) \text { if } t \in\left(t^*, t_{q^{\prime}}\right] \\ \gamma\left(t-t^*\right) \text { if } t>t_{q^{\prime}}\end{array}\right. \\ &= \left\{\begin{array}{c}\beta\left(t^*-t\right) \text { if } t \in\left[0, t_q\right) \\ \frac{N}{s} \frac{\beta \gamma}{\beta+\gamma} \text { if } t \in\left[t_q, t^*\right) \\ \frac{N}{s} \frac{\beta \gamma}{\beta+\gamma} \text { if } t=t^* \\ \frac{N}{s} \frac{\beta \gamma}{\beta+\gamma} \text { if } t \in\left(t^*, t_{q^{\prime}}\right] \\ \gamma\left(t-t^*\right) \text { if } t>t_{q^{\prime}}\end{array}\right.. \end{aligned}

The converse of the statement also follows from the cost equation above.

References

  1. Vickrey, W. S. (1969). Congestion theory and transport investment. The American Economic Review, 59(2), 251–260.
  2. Arnott, R., De Palma, A., & Lindsey, R. (1990). Economics of a bottleneck. Journal of Urban Economics, 27(1), 111–130.
  3. Laih, C.-H. (1994). Queueing at a bottleneck with single-and multi-step tolls. Transportation Research Part A: Policy and Practice, 28(3), 197–208.