1. 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)

2. 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

3. Terms related to gap analysis
The headway in a major stream, evaluated and used by a driver in a minor stream
Time lag
Space lag
Merging
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 T1 T2
T2 – T1 D1
D2 – D1 D2

4. 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”.

5. Critical gap? What is it?
Critical gap = The minimum average gap length that will be accepted by drivers.
The gap accepted by 50% of the drivers
Greenshields
The gap for which the number of accepted gaps shorter than it is equal to the number of rejected gaps longer than it.
Raff

6. If we adopt Mr. Greenshilds’s concept…
50%

7. 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.

8. 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 to determine tc
Wait a minute! I plotted the curves already and I can see the intersecting point of the two curves…

9. 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:
For x = 0, 1, 2, … …
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

10. Stochastic approach (cont)
We can write the original Poisson probability equation as (because  = t):
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…
for t  0
This is called (negative) exponential distribution.
for t  0
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.

11. 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)

12. 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.
where  is the amount of shift, minimum headway 
(Review Example 6-7)

13. 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.)

14. Single-channel, undersaturated infinite queues [M/M/1(, FIFO)]
Rate of arrival, q
Rate of service, Q
Service area
Queue
Undersaturated  Q > q
System
Prob of n units in the system:
Expected no.of units waiting to be served (mean queue length):
Expected no. of units in the system:
(See pages 208 and 209 for others & Review Example 6-8)