Analytical Solution of the Single-type Truck Problem Truck lane restriction

truck trajectory , induced traffic states

laval s solution summarized follows: assuming one-lane freeway segment obeying triangular fundamental diagram defined in figure right free-flow speed u, wave velocity w , jam density kj. 1 truck type considered. in scenario, normalized capacity of freeway segment given as：

i
=


1

r
h
c





{\displaystyle i={\frac {1}{rhc}}}

where r time-mean proportion of trucks in traffic stream,c = uwnkj/(w+u)is capacity of freeway lane without trucks , h expected value of headway between 2 consecutive trucks @ location trucks begin slow down

it can shown that, approximating truck arrivals poisson processes, probability density function (pdf) of h equation below, in τ defined clearance time of queue induced slow-moving truck, λ0=rc, λ1=ru , τ=l(w+v)/wv. note λ0 , λ1 refer mean truck arrival rate @ traffic state c or u, respectively. in particular, traffic state d, corresponds downstream of moving bottleneck, assumed equal capacity of unblocked lanes.

f

h


(
h
)
=


{




λ

1



e

−
h

λ

1




,


h
≤
τ





e

τ
(

λ

0


−

λ

1


)



λ

0



e

−
h

λ

0




,


h
>
τ








{\displaystyle f_{h}(h)={\begin{cases}\lambda _{1}e^{-h\lambda _{1}},&h\leq \tau \\e^{\tau (\lambda _{0}-\lambda _{1})}\lambda _{0}e^{-h\lambda _{0}},&h>\tau \end{cases}}}

according newell s moving bottleneck theory, have:

u
=
d
+
(



w
v
k
j


w
+
v



)


{\displaystyle u=d+({\frac {wvkj}{w+v}})}

given above information, can conclude average truck headway h h=(1-e)/(λ1)+(e)/(λ0)

and above equation gives necessary information solve normalized capacity i.