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In day-to-day modelling, traffic counts from a sequence of T days are collected, this means we can no longer assume the link flows, x, to be independent. That is, traffic flows from a succession of days will lead to a certain degree of inter-day dependence. We can imagine this as akin to weather patterns, where certain conditions on one day tend to influence conditions in the days thereafter. We denote the link counts as

x= (x1, ..., xM), and now add a superscript tto index the day of observation, i.e. xt

are the link counts on dayt,t=1, . . . , T.

Again, the superscript t is used to index the day for y = (y1, ...., yN), the corre-

sponding route flow vector, whereN is the number of routes in the network.

Under the general class of day-to-day Markov traffic models studied by Cascetta

(1989) and Cantarella and Cascetta (1995), the distribution of yt _{is defined condi-}

tionally on the finite history from the past d days, yt−1_{, . . . ,y}t−d_{, and some model}

parametersθ governing travel demand and traveller behaviour. That is, we can model the evolution of route flows day-to-day using a d-step Markov process with transition probabilities

f(yt|yt−1,yt−2, ...,yt−d,θ) (3.1) where f denotes a probability distribution as defined by its arguments and θa vector of model parameters.

We model the dependence of yt for a given day t on yt−1 through the use of link cost functions. To this end we assume that the flowxlon linklgenerates a link specific

travel costc(xl). Travel costs are defined as a combination of monetary elements such

as the amount of fuel needed (e.g. on a hilly stretch as opposed to a road on a plain), congestion charges or tolls, and non-monetary factors such as the time needed to travel along the link as well as the appeal of the road environment (e.g. a scenic route in contrast to using the motorway) or ease of driving. The cost of travelling along a route is then given by

Statistical inference for models of the form seen in Equation 3.1are difficult because they require us to explore a high-dimensional space of route flows,{yt_,yt−1_{, ...,y}t−d_}.

A natural way to simplify the expression in Equation 3.1 is to assume that decisions made on day t depend on the previous states of the system only through the travel costs on the precedingddays. In this case:

f(yt|yt−1,yt−2, ...,yt−d,θ) =f(yt|ct−1,ct−2, ...,ct−d,θ)

=f(yt|xt−1,xt−2, ...,xt−d,θ). (3.3)

Logit route choice model We present a specific class of suchd-step Markov models as described in Hazelton and Watling (2004), in part to clarify some of the notation introduced as well as to provide a framework within which we perform our simulation study later in this chapter.

We suppose that there is a vector μ = (μ1, ..., μL) of average OD flow rates and

that the realised vector of OD flows on day t is given by w = (w₁t, ..., wt_L)T with wt_o following aPois(μo) distribution independently ofwos∗ for o∗=o ors=t.

We apply the logic route choice model to calculate the probability that a traveller will select router on daytfor a journey between OD pairo as follows:

t_or = e

ψut_r

r∗_∼r

eψutr∗ (3.4)

where r∗ ∼r if and only if routes r∗ and r serve the same oth OD pair and ψ is the logit model parameter.

In Equation 3.4, we model the travel utility ut _{on day t as a function of travel}

costs experienced by drivers previously, in the sense that heavy congestion on one day is liable to lead to reduced demand the next. Specifically, utr is defined by the rth

element of the linear filter

ut₌₋ d

s=1

δskt−s. (3.5)

The powers of δ provide exponentially decreasing weights, with δ > 0 chosen so thatd_s=1δs= 1. The traveller learning process is hence determined by the length of memorydand the rateδat which the travellers consider past experiences decays with time.

Thelth element ofc, the vector of link costs, could be defined here via the commonly usedBureau of Public Roads(1964)’s link cost function,

c(xl) =a∗l 1 + 0.15 xl b∗_l 4 (3.6)

wherea∗_l is the free-flow travel time andb∗_l a measure of road capacity, forl= 1, ..., M. We assume that, conditional on past costs and the realised demand on dayt, travellers choose their routes independently. It follows that the conditional distribution of route flow choices for any given OD pair on daytfollows a Multinomial distribution:

f(yt_r|wt, xt−1, xt−2, ..., xt−d,θ) =Mn(wt,t) = w t o! r∗_∼r y_rt r∗_∼r (t_or)ytr _(3.7)

where r∗ ∼rif and only if routesr∗ andr serve the same OD pair andor the route

choice probability corresponding to the oth OD pair. Our model is of the form of Equation3.3with travel demand parameter vector θ= (ψ, d, δ,μT)T.