Trajectory Adaptive Routing Policy – Recursive

We firstly give a new definition to trajectory-adaptive routing policy without the trajectory as follows:

Definition 0.3 (Trajectory-Adaptive Routing Policy – Recursive) A trajectory-

adaptive routing policy μ(j0, t0) departing node j0 at time t0 to a given destination d is

recursively defined as a combination of the next node k and the set of sub-policies {μi(k,

ti)}, where ti is the i-th possible arrival time at node k from the marginal distribution of . If denote the i-th support point of the marginal distribution of as , then

.

Note that this is a recursive definition. The sub-policies μi at all the possible next node-time pairs (k, ti) are also defined recursively as a combination of the next node k' and sub-policies . The recursion stops at the destination d.

Although each policy is defined over a node-time pair only, the recursive nature allows the routing decisions dependent on the trajectory.

Consider two different possible trajectories to the current node-time pair (j, t) by following a given routing policy out of the origin and departure time (j0, t0). Assume they

start to differ at (ji, ti) due to different arrival times at the next node k, and the next node- time pairs are and respectively. The sub-policies at the two node-time pairs can then be defined such that they will both reach (j, t) with a positive probability but contain different sub-policies from (j, t). This way, the decisions at (j, t) can differ for the two different trajectories.

For example, one trajectory-adaptive routing policy out of node-time pair (o, 0) in Figure 0.1 can be recursively written as follows.

At node-time pair (b, 2), and are two different routing policies. Which one of the two will be executed depends on the trajectory traveling from the origin and departure time pair (o, 0) to the current one (b, 2): if (a, 1) is on the trajectory, then is the next policy; if (a, 0), then .

Under Definition 6.3, a sub-policy by itself does not imply any trajectory information, unlike that under Definition 6.2, where the trajectory information is included in the state variable of the routing policy. For example, one cannot tell from the sub- policy which trajectory is followed from the origin and departure time pair (o, 0) to the current one (b, 2); on the other hand, can tell us that the trajectory is H1 = {(o, 0), (a, 1), (b, 2)}.

A sub-policy under Definition 6.3 treats the current node as the origin and the current time the departure time and gives all possible arrival times at the next node. For example, treats (a, 1) as the origin and departure

time pair and gives two possible arrival times at the next node b. However, when retrieving a trajectory-adaptive routing policy from the real origin and departure time, we might encounter the problem that its sub-policy introduces arrival time at downstream node that is not compatible with the trajectory and thus sub-policy that actually is not possible to be realized. For example, when μ is retrieved from the real origin and departure time pair (o, 0), it can be observed that and are never

applied, as the arrival times at node b as 3 and 1 are not compatible with the trajectory {(o, 0), (a, 1)} and {(o, 0), (a, 0) } respectively.

We term anything that is not compatible with the trajectory (information) as "phantom". We call the arrival times at downstream nodes that are not compatible with the trajectory phantom arrival times. The sub-policies that are not possible to be realized due to phantom arrival times at the next nodes are called phantom sub-policies. Note that when two trajectory-adaptive routing policies differ only in phantom sub-policies, they are actually the same. For example, in the above trajectory-adaptive routing policy μ(o, 0),

if we replace as where , or replace as where , the

trajectory-adaptive routing policy μ(o, 0) is not changed.

Moreover, note that the travel time of a sub-policy is evaluated over all support points, and so the link travel times in some support points that are not possible to be realized due to the phantom arrival times (termed phantom travel times) will still be included in the evaluation of its travel time. However, when the travel time of a trajectory-adaptive routing policy is evaluated, the phantom travel times of its sub- policies will not be considered. For example, when evaluating the travel time of , we will calculate its support point travel times in both support points. However, when the travel time of μ(o, 0) is evaluated, only the support point travel time of in support point C1 is considered as that is the compatible support point.

In the remainder of the chapter, the phantom sub-policies will not be written in the trajectory-adaptive routing policy as they will not affect what the trajectory-adaptive routing policy really is. Thus, the above trajectory-adaptive routing policy μ(o, 0) is now written as follows.

It can also be written in a tree format as follows.

Next we show that Definitions 6.2 and 6.3 are equivalent.

Proposition 0.2. Definitions 6.2 and 6.3 of Trajectory-Adaptive Routing Policy

are equivalent.

Proof. We prove this proposition by showing that any trajectory-adaptive routing

policy under Definition 6.3 can be converted to one under Definition 6.2, and vice versa. Suppose a trajectory-adaptive routing policy μ is defined under Definition 6.3, i.e., . Assume (j0, t0) is the origin node and departure time.

Therefore the trajectory-adaptive routing policy μ is recursively defined as follows (assume that phantom sub-policies are not included).

This trajectory-adaptive routing policy μ under Definition 6.3 can be converted to one under Definition 6.2, i.e., a mapping from trajectories to next nodes, by combining node-time pairs starting from (j0, t0) as a trajectory and making the next node of the sub-

policy for the last node-time pair as the next node corresponding to the trajectory. Since the phantom sub-policies are not included, all the trajectories generated are valid.

We have shown that any trajectory-adaptive routing policy under Definition 6.3 can be converted to one under Definition 6.2, and next we will show the other way around.

Suppose a trajectory-adaptive routing policy μ is defined under Definition 6.2, i.e.,

μ is a mapping from trajectories to next nodes, . The conversion can

be conducted as follows.

For each trajectory H = {(j0, t0), (j1, t1), ..., (j, t)}, choose the last node-time pair (j, t) as the node-time pair for the current routing policy, and the next node k corresponding

t). Find all trajectories H’ = {(j0, t0), (j1, t1), ..., (j, t), (k, ti)}, where ti is a possible arrival

time at the next node k from the trajectory H, and choose the last node-time pairs (k, ti) as the node-time pair for the sub-policies of the current routing policy. Thus, we have μ(j, t) = {k; {μi(k, ti)}}.

When all the trajectories are visited, the recursive definition of the trajectory- adaptive routing policy for the origin and departure time pair (j0, t0) is complete, and no

phantom sub-policy is included. Q.E.D.

An example of the equivalence is that the example trajectory-adaptive routing policies in Sections 6.2.1 and 6.2.2 are the same – both are the one represented by Figure 0.2.