Transcript Lec13, Ch.6, pp.201-213: Gap acceptance and Queuing Theory

Lec13, Ch.6, pp.201-213: Gap acceptance
and Queuing Theory (Objectives)
Understand the availability of gaps affects your
merge, diverge, weaving, and crossing maneuvers
Know how to determine the value of critical gap
Understand the availability of gaps can be
estimated stochastically
Be able to estimate the number of vehicles in the
queue using queuing theory (single-channel,
undersaturated, infinite queues)
What we cover today in class…
Terms related to gap analysis
Time-space diagram to explain the available gap
The procedure to determine the critical gap
Stochastic method for estimating the availability
of gaps when the arrival of vehicles is randomly
distributed
Single-Channel, undersaturated, infinite queues
Terms related to gap analysis
Gap
The headway in a major stream, evaluated and used by a driver
in a minor stream
Time lag
T1
Space lag
Merging
T2
D1
T2 – T1
D2 – D1
D2
The process by which a vehicle in one traffic stream joins
another traffic stream moving in the same direction
Diverging
The process by which a vehicle in a traffic stream leaves that
traffic stream
Weaving
The process by which a vehicle first merges into a stream of
traffic then mergers into a second stream, OR the maneuver of
drivers taking place at the cloverleaf interchange
Time-space diagram and gap: why the
availability of gaps is critical?
The driver in a minor stream evaluates the availability of gaps and he
enters the main stream only (or “accept” the gap) when the available
gap is equal to greater than the gap he feels safe, i.e., his “critical
gap”.
Critical gap? What is it?
Critical gap = The minimum average gap length that
will be accepted by drivers.
Greenshields
Raff
The gap accepted by 50% of the
drivers
The gap for which the number of accepted
gaps shorter than it is equal to the number
of rejected gaps longer than it.
If we adopt Mr. Greenshilds’s concept…
50%
Determining the critical gap
t = Time increment
used for gap analysis
The gap for which the number of accepted gaps shorter than it is equal to the number
of rejected gaps longer than it. [Raff’s definition] OR, simply stated, it is the point
where the two curves intersect.
Determining the critical gap (cont)
If you want to use the method described
in pages 204 & 205…
Step 1: Get the data like Table 6.2
Step 2: Plot them like Figure 6.10
Step 3: Find the lower bound and upper bound of the time
increment t that contains the intersecting point in the plot
Step 4: Find t1, n, p, r, and m and plug them in Eq. 6.39 to
determine tc
Wait a minute! I plotted the curves
already and I can see the intersecting
point of the two curves…
Stochastic approach
(the approach discussed in the book applies to light
to medium traffic only)
When traffic is light to medium, the arrival of
vehicles is considered random and follows a Poisson
distribution. If so, the probability of x vehicles arriving
in any interval of time t sec is:
Px  
 xe
x!
For x = 0, 1, 2, …
0 1 2 3 4 5…
P(x) = the probability of x vehicles arriving in time t sec
 = average number of vehicles arriving in time t
What you have as data are V (total number of vehicles arriving in time
T). Then the average number of vehicles arriving per second is  = V/T
and the average number of vehicles arriving in t is  = t
Stochastic approach (cont)
We can write the original Poisson probability equation as (because  = t):

t x e  t
Px  
x!
Now this is an arrival probability. There are gaps between the arriving
vehicles. What would be the probability of a gap of t second? A gap of
t second means there is NO Vehicle arriving during that time t. (x = 0,
that is) So…
 t
for t  0
This is called (negative)
 t
exponential distribution.
for t  0
P0  Ph  t   e
Ph  t   1  e
Where h is time headway and t can be the gap that you are interested in. So,
if t = tc (critical gap), you are interested in the probability of time headway
equal to or greater than the critical gap in which the driver in a minor road
merges into the main traffic stream. Note that λ=1/tavg.
Stochastic approach (cont)
Once you know the probability of having gaps equal to or
greater than the critical gap, you can estimate the number
of gaps available for the vehicles from a minor road to
enter the main traffic stream.
Suppose you have an hourly volume V, then (V – 1) gaps
occur in one hour. How many gaps can be used by the
drivers from a minor road?
Frequency of h tc = (V – 1)e-t
(Review Example 6-6)
Stochastic approach (cont)
In reality, there’s no 0 second headway. Usually there is a
minimum headway that drivers want to maintain, say 0.5 to 1
second (But, be careful, there is no infinite headway either which
the exponential distribution assumes. Also, in this model,
headway = gap which is not strictly correct.). If you want to
include this minimum headway, you have to shift the exponential
distribution by the amount of minimum headway.
P0  Ph  t   e
t 

t 
where  is the amount of shift, minimum headway

(Review Example 6-7)
Introduction to queuing theory
When demand exceeds capacity for a period of time at a
specific location, a queue is formed (even if overall
demand is less than capacity). Queuing theory attempts to
analyze this phenomenon using probability theory. Note
that in queuing analysis, the vehicles are stored in a vertical
queue. Also, overall, queuing theory applies when demand
< capacity, i.e., undersaturated case only.
The following inputs are needed:
Mean arrival value
Arrival distribution (We use random arrival in this class)
Mean service value
Service distribution (We use random arrival in this class)
Queue discipline (FIFO, FILO, etc. We use only FIFO in this class.)
No. of service channel available (We use only one channel here.)
Single-channel, undersaturated infinite
queues [M/M/1(, FIFO)]
Rate of arrival, q
Rate of service, Q
Queue
Service area
System
Undersaturated  Q > q
Prob of n units in the system:
q
Pn    
Q
n
 q
1  
 Q
Expected no.of units waiting to be
served (mean queue length):
Expected no. of units in the system:
q
E n  
Qq
q2
E m 
Q(Q  q)
(See pages 208 and 209 for others & Review Example 6-8)