4.4 Waiting at a street intersection Consider the intersection of two city
streets shown in Figure P4.4(1). Both streets are one-way. One of them is designated as
the "primary" street and vehicles on it have priority at the intersection. On
the "secondary" street there is a stop sign at the intersection.
1 . The car on the secondary street arrives at the intersection at a random time.
2. The vehicles on the primary street do not slow down or yield to vehicles on the secondary street at the intersection.
3. The headways, H, between vehicles on the primary street are independent and identically distributed random variables with pdf fH(t), expected value E[H], variance , and so on.
4. The car on the secondary street will cross the intersection as soon as a time gap greater than t0 (a constant) is perceived before arrival of the next vehicle on the primary street (assume that drivers perceive such things correctly). Note that a car on the secondary street may cross the intersection immediately upon arrival there, if the remaining time until the arrival of the next primary-street car at the intersection is greater than t0.
Figure P4.4(2) illustrates this whole situation. Let
a. Derive an expression for E[ Y1 in terms of fH(t) (and its moments) and t0.
b. Derive an expression for Y.
c. Apply your results of parts (a) and (b) to the case where the headways are negative exponentially distributed [i.e., fH(t) = e-t for t 0]. Show that
d. Apply your results of parts (a) and (b) to the case where t0 = 4 seconds and H is uniformly distributed between 2 and 10 seconds.
X = instants when cars on primary street reach the intersection