Braess’s Paradox and Traffic Networks

For a traffic network, we have routes and links as before, only here we specify the delay on a link as a function of the flow along that link D_j(y_j), where y_j=\sum_r A_{jr}x_r as before. Then a natural notion of an equilibrium is a distribution of flows such that no customer stands to gain by changing route. Formally, take S the set of source-destination pairs, H_{sr} the incidence matrix of pairs s and routes r, and S(r) the routes between the same source-destination pair as r.

Then define a Wardrop equilibrium to be a flow vector (x_r,r\in R) such that for any x_r>0,

\sum_j D_j(y_j)A_{jr}=\min_{r'\in S(r)}\sum_j D_j(y_j)A_{jr}

that is, the delay on this route is less than or equal to the delay on any route between that pair of vertices.

A natural assumption would be that D_j(y_j) is increasing in y_j for each j, and if we further assume that each D_j is continuously differentiable, then we can prove that a Wardrop Equilibrium exists.

Why? Consider the optimisation problem

\min\quad\sum_j \int_0^{y_j}D_j(u)du,\quad x\geq 0,Hx=f,Ax=y

where f is the flow requirement. The feasible region is convex and the objective function is convex differentiable so the optimum exists. The Lagrangian is

L(x,y,\lambda,\mu)=\sum_j \int_0^{y_j}D_j(u)du+\lambda\cdot (f-Hx)-\mu\cdot(y-Ax)

and at the minimum

0=D_j(y_j)-\mu_j,\quad 0=-\lambda_{S(r)}+\sum_j \mu_jA_{jr}

So \mu_j the delay on j, and then

\lambda_{S(r)}\left\{\begin{array}{l l}=\sum_j\mu_jA_{jr}& \quad x_r>0\\ \leq \sum_j\mu_jA_{jr} & \quad x_r=0\\ \end{array} \right.

So can interpret \lambda_{S(r)} as the minimum delay for this route, and so a solution to this problem is a Wardrop Equilibrium. If D_js are strictly increasing, then have uniqueness in (y_j), from the optimisation problem, but not in (x_r). For example, you could have two essentially identical routes from A to B, and from B to C, and it then doesn’t matter how you concatenate them to form a flow from A to C.

Note the counterintuitive Braess’s Paradox, where adding a route increasing the size of the delay at the Wardrop Equilibrium. The canonical example can be seen here, where the addition of the road across causes the delay to increase from 83 to 92. And indeed such effects have been observed in ‘real life’ as well.

Note that one explanation for why this seems counter-intuitive is that the objective function isn’t necessarily the most natural one. For a Wardrop equilibrium, the agents are acting entirely selfishly given present information. From the point of view of society, it might make more sense to minimise the average delay \min \sum_jy_jD_j(y_j). In this case the Lagrangian gives

0=D_j(y_j)+y_jD_j'(y_j)-\mu_j,\quad 0=-\lambda_{S(r)}+\sum_j\mu_jA_{jr}

So if add a congestion charge of y_jD_j'(y_j) to the delay function, the Wardrop Equilibrium becomes equal to the societal optimal. Note that this is NOT the societal optimal for (delay + charge). However, we have seen that a toll will enable selfish behaviour to produce a societal optimum. This is an example of mechanism design, which can be considered a branch of game theory, where the aim is to design systems so that agents work in a societally optimal manner.


Queues and Migration Processes

Simple Queues

A queue generally has the form of a countable state-space Markov Chain. Here we assume that customers are served in the order they arrive (often referred to as: FIFO – First In First Out). The standard Kendall notation is M/M/C/K. Here the Ms stand for Markov (or memoryless) arrivals, and service times respectively, possibly replaced by G if the process admits more general distributions. C is the number of servers, and K the capacity of the system.

The first example is a M/M/C/C queue, motivated by a telephone network. Here, there are c lines, and an arriving call is lost if all lines are busy at the arrival time. We assume arrivals are a PP(\lambda) process, and the call times are iid Exp(\mu) RVs. We record the number of busy lines, which is a continuous-time Markov chain on state-space [0,c]. As usual, we assume the system operates in equilibrium, and so in particular we must have \lambda<\mu c. It is easy to find the equilibrium distribution. The only non-zero transition probabilities are

q(i-1,i)=\lambda, \quad q(i,i-1)=\mu i\quad i=1,\ldots,c

and so can that that \pi_i=\frac{\nu_i}{i!}\pi_0 satisfies the DBEs where \nu:=\frac{\lambda}{\mu}, sometimes called the traffic intensity. This gives

\pi_c=\mathbb{P}(c\text{ lines busy})=\frac{\nu^c}{c!}\left(\sum_{k=0}^c \frac{\nu^k}{k!}\right)^{-1}

and we define this to be E(\nu,c), Erlang’s formula.

Note that if (X(t),t\in\mathbb{R}) is a MC in equilibrium, (X(-t),t\in\mathbb{R}) is a MC with the same ED, modulo style of discontinuities (ie whether transitions are left-continuous or right-continuous). Therefore, in any queue where all customers get served, eg an M/M/1 queue, for which \lambda<\mu (otherwise the MC is transient, so no ED!), the departure process is the same (in distribution) as the arrivals process.

We can check that this holds for a series of M/M/1 queues, and that in equilibrium, the sizes of the queues are independent. This is merely an extension of the observation that future arrivals for a given queue are independent of the present, and likewise past departures are independent of the future, but the argument is immediately obvious.

Migration Processes

We consider a closed migration process on J colonies, with populations described by a vector n=(n_1,\ldots,n_J). We say that the instantaneous rate of movement of an individual from colony j to colony k is q(n,T_{jk}n)=\lambda_{jk}\phi_j(n_j), for some functions \phi_j(0)=0. We can describe the ED of this system in terms of the distribution obtained when a single individual performs a random walk through the state space, with \phi_j(1) taken to be 1 for each j. We call the DBEs for this case the traffic equations, with solutions (\alpha_j,j\in J). Then, by checking the PBEs, it is clear that the equilibrium distribution of the original migration process satisfies

\pi(n)\propto \prod_{j=1}^J \frac{\alpha_j^{n_j}}{\prod_{r=1}^{n_j}\phi_j(r)}

This is important, as it would have been computationally difficult to solve the original equations for an ED.

The same result holds for an open migration process, where individuals can enter and leave the system, arriving at colony j at rate \nu_j, and leaving at rate \mu_k\phi_k(n_k). Note that this has the same form as if each colony was served by a PP(\alpha_j\lambda_j), with departures at rate \lambda_j\phi_j(n_j):=(\mu_j+\sum_k\lambda_{jk})\phi_j(n_j). But (obviously) this interpretation is not equivalent to the model

The time reversal is also an OMP. One has to check that the transition rates have the correct form, and so the exit process from each colony (in equilibrium, naturally), is PP(\alpha_j\mu_j)$. Most importantly, given a communicating class, we can think of the restriction to this class as an OMP, so in particular, rates of transition between classes are Poisson.

Little’s Law

Informally, the mean number of customers should be equal to the arrival rate multiplied by the mean sojourn time in the system of a customer. This is easiest to formalise by taking an expectation up to a regeneration time. This is T, the first time the system returns to its original state (assumed to be 0 customers), an a.s. finite stopping time.

Set L:=\mathbb{E}\frac{1}{T}\int_0^Tn(s)ds, the average number of customers, and W:=\frac{\mathbb{E}\sum_1^N W_n}{\mathbb{E}N} where N is the number of customers arriving in [0,T], and W_n is the waiting time of the n-th customer.

Little’s Law asserts that L=\lambda W. Note that for a Markov Chain in equilibrium, can define L more simply as \mathbb{E}n and similarly for W.

It is easiest proved by considering the area between the arrivals process and the departure process in two ways: integrating over height and width. Note that working up to a regeneration time is convenient because at that time the processes are equal.

The migration processes above are said to be linear if \phi_j(n_j)=n_j. This process has the form of a Markov chain, and so even in a more general state space, the distribution of points in disjoint subsets of the state-space are independent Poisson processes.

Often though, we start with no individuals in the system, but still the distribution is given by a time-inhomogenous Poisson random measure. The mean is specified by

M(t,E)=\nu\int_0^t P(u,E)du

where \nu is the net arrival rate, and P(u,E) is the probability that an individual is in E, a time interval of u after arriving.

As one would suspect, this is easiest to check through generating functions, since independence has a straightforward generating function analogue, and the expression for a Poisson RV is manageable.