Unformatted text preview: Introduction to Transportation Engineering
Applications of Queueing Theory to Intersection
Analysis Level of Service
Dusan Teodorovic and Antonio A. Trani
Civil and Environmental Engineering
Virginia Polytechnic Institute and State University Blacksburg, Virginia
Spring 2005 Virginia Tech Material Covered
• Application of deterministic queueing models to study
intersection level of service
• Study various types of intersection controls schemes used
in transportation engineering
• Most of the material applies to ground transportation
modes (highways) Virginia Tech 2 Basic Ideas
• Trafﬁc control represents a surveillance of the motion of
vehicles and pedestrians in order to secure maximum
efﬁciency and safety of conﬂicting trafﬁc movements.
• Trafﬁc lights or trafﬁc signals are the basic devices used
in trafﬁc control of vehicles on roads.
• They are located at road intersections and/or pedestrian
crossings.
• The ﬁrst trafﬁc light was installed even before there was
automobile trafﬁc (London on December 10, 1868). The
current trafﬁc lights were invented in USA (Salt Lake
City, (1912), Cleveland (1914), New York and Detroit
(1920)).
Virginia Tech 3 Basic Deﬁnitions
• Drivers move toward the intersection from different
approaches
• Every intersection is composed of a number of
approaches and the crossing area (see Figure)
• Each approach can have one, or more lanes. The trafﬁc
stream is composed of all drivers who cross the
intersection from the same approach
• During green time, vehicles from the observed approach
can leave the stop line and cross the intersection
• The corresponding average ﬂow rate of vehicles that
cross the stop line is known as a saturation ﬂow. Virginia Tech 4 Intersection Geometry
Approach Crossing area Figure 1. Typical Road Intersection.
Virginia Tech 5 Flow Conditions at an Intersection
• In most cases, queues of vehicles are established
exclusively during the red phases, and are terminated
during the green phases.
• Such trafﬁc conditions are known as a undersaturated
trafﬁc conditions.
• An intersection is considered an unsaturated intersection
when all of the approaches are undersaturated.
• Trafﬁc conditions in which queue of vehicles can arrive at
the upstream intersection are known as a oversaturated
trafﬁc conditions. Virginia Tech 6 Trafﬁc Control Techniques
• Trafﬁc engineers apply various trafﬁc control strategies
(see the Figure) in order to minimize the total delay at the
intersection and/or maximize the intersection capacity.
Phase 2 Phase 1 Cycle Phase 1 Phase 2 Cycle Time Figure 2. Intersection with Two Phases.
Virginia Tech 7 Trafﬁc Signal Control Strategies
• Many isolated intersections operate under the fixedtime
control strategies.
• These strategies assume existence of the signal cycle that
represents one execution of the basic sequence of signal
combinations at an intersection.
• A phase represents part of the signal cycle, during which
one set of traffic streams has right of way. Figure 2
shows twophase traffic operations for the intersection.
• The cycle contains only two phases. Phase 1 is related to
the movement of the northsouthbound vehicles through
the intersection. Phase 2 represents the movement of the
eastwestbound vehicles.
Virginia Tech 8 Trafﬁc Control Strategies
• The cycle length c represents the duration of the cycle
measured in seconds. The sum of the phase lengths
represents the cycle length.
• For example, in the case shown in Figure 2., the cycle
length could be 90 seconds, length of the Phase 1 could
be 50 seconds, while the length of the Phase 2 could be
equal to 40 seconds.
• The cycle length is a design parameter of the intersection
as well as the green times allocated to each phase. Trafﬁc
engineers can modify the settings of intersection
controllers based on demand needs at the intersection. Virginia Tech 9 Control Strategies
Phase 1 Phase 2 Phase 3 Cycle Figure 3. Intersection with Three Phases. Virginia Tech 10 Control Strategies
• Higher number of phases is usually caused by trafﬁc
engineer’s wish to protect some movements (usually leftturning vehicles)
• “Protection” assumes avoiding potential conﬂicts with
the opposing trafﬁc movement, and/or pedestrians
• There is always a certain amount of lost time (few
seconds) during phase change. For example, when the
green light changes to red there is am amber light period
to warn drivers of an impending change
• Obviously, the higher the number of phases, the better the
protection, and the higher the value of the lost time
associated with a phase change.
Virginia Tech 11 Control Strategies
• Trafﬁc signals are control devices. The typical sequence
of lights at the intersection approach could be: “Red, Red
All, Green, Amber, Red, Red All,...”.
Flow [veh/h]
Saturation
ﬂow Time
0
Red Green
Red All Amber Effective green Figure 4. Deﬁnition of Green, Amber and Red Times.
Virginia Tech 12 Control Strategies
• “Green time”, “effective green”,”red time”, and “effective
red” are linguistic expressions frequently used by trafﬁc
engineers
• In theory, all drivers should cross the intersection during
the green light. In reality, no one driver starts his/her car
exactly in a moment of the green light appearance
• Similarly, at the end of a green light, some drivers speed
up, and cross the intersection during the amber light
• “Green Time” represents the time interval within the
cycle when observed approach has green indication. On
the other hand, “Effective Green” represents the time
interval during which observed vehicles are crossing the
intersection.
Virginia Tech 13 Vehicle delays at signalized intersections: Uniform
Vehicle Arrivals
• For simplicity, let us assume for the moment that
observed signalized intersection could be treated as a D/
D/1 (deterministic) queueing system with one server
(hence the notation (D/D/1))
• We assume uniform arrivals, and uniform departure rate
(see Figure 5). Virginia Tech 14 Queueing Theory Shorthand
Nomenclature • Queues come in different ﬂavors as demonstrated so
far • Kendall developed a simple scheme to designate
queues back in the early 50s. His nomenclature has
been widely adopted • Typically 6 parameters: •
•
•
• a/b/c/d/e/f
a = interarrival time distribution (arrivals)
b = service time distribution
c = number of servers
Virginia Tech (A.A. Trani) 14a Queueing Theory Shorthand
Nomenclature • Typically 6 parameters: •
•
•
• a/b/c/d/e/f
d = service order (i.e., FIFO, LIFO, etc)
e = Max. number of customers
f =Size of the arrival population Virginia Tech (A.A. Trani) 14b Queueing Theory Shorthand
Nomenclature •
•
•
•
• Possible outcomes for (a) and (b)
M = Times are neg. exponential (i.e., Poisson arrivals)
D = Deterministic distribution
Ek = Erlang distribution
G = general distribution Virginia Tech (A.A. Trani) 14c Example Queueing Systems we Have
Studied • M/M/1/FIFO/∞/∞
• Stochastic queue with neg. exponential time
between arrivals • Neg. exponential service times
• 1 server
• First inﬁrst out
• Inﬁnite no. of customers in system
• Inﬁnite arrival population
Virginia Tech (A.A. Trani) 14d Example Queueing Systems we Have
Studied • M/M/2/FIFO/15/15
• Stochastic queue with neg. exponential time
between arrivals • Neg. exponential service times
• 2 servers (2 pavers)
• First inﬁrst out
• Up to 15 no. of trucks in system
• 15 trucks population
Virginia Tech (A.A. Trani) 14e Deﬁnition of Queueing Terms for Intersection
Analysis
Cumulative number of vehicles Cumulative arrivals
Cumulative departures D(t)
C A(t)
g0 h A r g B c
Time
Red Green Figure 5. Arrivals and Departures at an Intersection.
Virginia Tech 15 Deterministic Queueing Analysis
Let us denote by λ vehicles arrival rate, and by µ vehicles
departure rate during the green time period. In the
deterministic case, the cumulative number of arrivals A ( t )
and the cumulative number of departures D ( t ) are:
A(t) = λ ⋅ t (1) D(t) = µ ⋅ t (2) where:
c  the duration of the signal cycle r  effective red g  effective green
Virginia Tech 16 Deterministic Queueing Analysis
The duration of the signal cycle equals:
c = r+g (3) The formed queue is the longest at the beginning of
effective green. The queue decreases at the beginning of
effective green.
We denote by g the time necessary for queue to dissipate
(Figure 5). The queue must dissipate before the end of
effective green. In the opposite case, the queue would
escalate indeﬁnitely. In other words, queue dissipation
will happen in every cycle if the following relation is
satisﬁed:
0 Virginia Tech 17 Deterministic Queueing Analysis
(4) g0 ≤ g The relation (4) will be satisﬁed if the total number of
vehicle arrivals during cycle length c is less than or equal
to the total number of vehicle departures during effective
green g , i.e.:
c g (5) ∫ λ dt ≤ ∫ µdt
0 0 λ⋅t c≤µ⋅t
0 (6) g
0 (7) λ⋅c≤µ⋅g Virginia Tech 18 Deterministic Queueing Analysis
Finally, we get:
λ
 ≤ g
 µc (8) Let us note the triangle ABC (Figure 5). Vehicles arrive
during time period ( r + g ) . Vehicles depart during time
period g . The total number of vehicle arrivals equals the
total number of vehicle departures, i.e.:
0 0 λ ⋅ ( r + g0 ) = µ ⋅ g0 (9) ( µ – λ ) ⋅ g0 = λ ⋅ r (10) Virginia Tech 19 Deterministic Queueing Analysis
The time period g required for queue to dissipate equals:
0 λ⋅r
g 0 = µ–λ (11) We divide both numerator and denominator by µ. We get:
λ
 ⋅ r
µ
g 0 = λ
1 – µ (12) Deﬁne the utilization factor ( ρ ) (or trafﬁc intensity) of the
λ
intersection as ρ =  , we can write:
µ ρ⋅r
g 0 = 1–ρ (13) Virginia Tech 20 Deterministic Queueing Analysis
The area A of the triangle ABC represents the total
delay d of all vehicles arrived during the cycle. This area
equals:
∆ ABC 1
A ∆ABC =  ⋅ r ⋅ h
2 (14) where h is the height of the triangle (ABC).
The ratio h
( r + g0 ) represents the slope λ , i.e.: h
λ = ( r + g0 ) (15) Virginia Tech 21 Deterministic Queueing Analysis
The height h of the triangle ABC is:
h = λ ⋅ ( r + g0 ) The area of the triangle ABC equals:
1
λ⋅r
1
A ∆ABC =  ⋅ r ⋅ h =  ⋅ r ⋅ λ ⋅ ( r + g 0 ) =  ⋅ ( r + g 0 )
2
2
2 (16) The total delay d of all vehicles arriving during the cycle
equals:
λ⋅r
λ⋅r λ ⋅ r2 ρ
ρ ⋅ r
d =  ⋅ ( r + g 0 ) =  ⋅ r +  =  ⋅ 1 +  2
2
2
1 – ρ
1 – ρ Virginia Tech (17) 22 Deterministic Queueing Analysis
λ ⋅ r2
d = 2 ⋅ (1 – ρ) (18) The average delay per vehicle d represents the ratio
between the total delay d and the total number of vehicles
per cycle. The total number of vehicles per cycle equals
λ ⋅ c . Therefore the average delay per vehicle d is:
dd = λ⋅c (19) or
λ ⋅ r2
2 ⋅ (1 – ρ)
d = λ⋅c (20) Virginia Tech 23 Simplifying the previous expression, average delay per
vehicle is the average:
r2
d = 2 ⋅ c ⋅ (1 – ρ) (21) Virginia Tech 24 Example Problem 1
The cycle length at the signalized intersection is 90
seconds. The considered approach has the saturation ﬂow
of 2200 [veh/hr], the green time duration of 27 seconds,
and ﬂow rate of 600 [veh/hr].
Analyze trafﬁc conditions in the vicinity of the
intersection. Calculate average delay per vehicle. Assume
that the D/D/1 queueing system adequately describes
considered intersection approach. Virginia Tech 25 Problem 1  Solution
The corresponding values of the cycle length and the
green time are:
c = 90 [ s ] ;g = 27 [ s ] The ﬂow rate ( λ ) and the service rate ( µ ) are:
veh
600 veh
veh
λ = 600  =   = 0.167 hr
3600 s
s
veh
2200 veh
veh
µ = 2200  =   = 0.611 hr
3600 s
s Trafﬁc intensity ρ equals: Virginia Tech 26 Problem 1  Solution
veh
0.167 s
λ
ρ =  =  = 0.273
µ
veh
0.611 s The duration of the red light for the considered approach
is:
r = c – g = 90 – 27 = 63 [ s ] The number of arriving vehicles per cycle is:
veh
λ ⋅ c = 0.167  ⋅ 90 [ s ] = 15.03 [ veh ]
s Virginia Tech 27 Problem 1  Solution
The number of departing vehicles during green light is:
veh
µ ⋅ g = 0.611  ⋅ 27 [ s ] = 16.497 [ veh ]
s We conclude that the following relation is satisﬁed:
λ⋅c≤µ⋅g This means that the trafﬁc conditions in the vicinity of the
intersection are undersaturated trafﬁc conditions.
The average delay per vehicle is estimated using:
r2
d = 2 ⋅ c ⋅ (1 – ρ)
Virginia Tech 28 Problem 1  Solution
63 2
d =  = 30.33 [ s ]
2 ⋅ 90 ⋅ ( 1 – 0.273 ) Virginia Tech 29 Example Problem 2
The cycle length at the signalized intersection is 60
seconds. The considered approach has the saturation ﬂow
of 2200 [veh/hr], the green time duration of 15 seconds,
and ﬂow rate of 400 [veh/hr]. Analyze trafﬁc conditions in
the vicinity of the intersection. Assume that the D/D/1
queueing system adequately describes the intersection
approach considered.
Calculate: (a) the average delay per vehicle; (b) the
longest queue length; (c) percentage of stopped vehicles. Virginia Tech 30 Problem 2  Solution:
(a) The corresponding values of the cycle length and the
green time are:
c = 60 [ s ] ;g = 20 [ s ] The red time is:
r = c – g = 60 – 20 = 40 [ s ] The ﬂow rate and the service rate are:
veh
400 veh
veh
λ = 400  =   = 0.111 hr
3600 s
s Virginia Tech 31 Problem 2  Solution
veh
2200 veh
veh
µ = 2200  =   = 0.611 hr
3600 s
s The utilization factor for the queue ρ is:
veh
0.111 s
λ
ρ =  =  = 0.182
µ
veh
0.611 s The average delay per vehicle equals:
r2
d = 2 ⋅ c ⋅ (1 – ρ) Virginia Tech 32 Problem 2  Solution
40 2
d =  = 16.3 [ s ]
2 ⋅ 60 ⋅ ( 1 – 0.182 ) (b) The longest queue length L happens at the end of a
red light (Figure 5). The quantity L is calculated as
follows:
max max veh
L max = λ ⋅ r = 0.111  ⋅ 40 [ s ] = 4.44 [ vehicles ]
s (c) Vehicles arrive all the time during the cycle. The total
number vehicles arrived A during the cycle equals:
veh
A = λ ⋅ c = 0.111  ⋅ 60 [ s ] = 6.66 [ vehicles ]
s Virginia Tech 33 Problem 2  Solution
All vehicles that arrive during time interval ( r + g ) are
stopped. The total number of stopped vehicles S equal:
0 S = λ ⋅ ( r + g0 ) The time period g required for queue to dissipate is
estimated using equation:
0 λ⋅r
g 0 = µ–λ We get:
λ⋅r
0.111 ⋅ 40
S = λ ⋅ ( r + g 0 ) = λ ⋅ r +  = 0.111 ⋅ 40 +  µ – λ
0.611 – 0.111 Virginia Tech 34 S = 5.43 [ vehicles ] The percentage of stopped vehicles equal:
S
5.43
P =  ⋅ 100 =  ⋅ 100 = 81.53 [%]
A 6.66 Virginia Tech 35 Example Problem 3
A simple “T” intersection is signalized. There are two
approaches indicated in the ﬁgure. The cycle length at the
signalized intersection (Figure) is 50 seconds.
Phase 2 Phase 1 Approach 1 Approach 2
Cycle Virginia Tech 36 Example Problem 3
Approach 1 has the saturation ﬂow of 2200 [veh/hr], the
effective green time duration of 35 seconds, and the ﬂow
rate of 600 [veh/hr]. Approach 2 has the saturation ﬂow of
2000 [veh/hr], the effective green time duration of 15
seconds, and the ﬂow rate of 550 [veh/hr]. Assume that
the D/D/1 queueing system adequately describes
considered intersection approach.
Calculate: (a) the average delay per vehicle for every
approach; (b) Allocate effective red and green time among
approaches in such a way to minimize the total delay of
the “T” intersection. Virginia Tech 37 Problem 3 Solution
(a) Approach 1:
The corresponding values of the cycle length and the
green time are:
c = 50 [ s ] ;g 1 = 35 [ s ] The red time equals:
r 1 = c – g 1 = 50 – 35 = 15 [ s ] The ﬂow rate and the service rate are respectively equal:
veh
600 veh
veh
λ 1 = 600  =   = 0.167 hr
3600 s
s Virginia Tech 38 Problem 3 Solution
veh
2200 veh
veh
µ 1 = 2200  =   = 0.611 hr
3600 s
s The utilization factor for the queue in approach 1 ρ is:
1 veh
0.167 s
λ1
ρ 1 =  =  = 0.273
µ1
veh
0.611 s The average delay per vehicle equals:
r12
15 2
d 1 =  =  = 3.09 [ s ]
2 ⋅ c ⋅ ( 1 – ρ1 )
2 ⋅ 50 ⋅ ( 1 – 0.273 ) Virginia Tech 39 Problem 3 Solution
Approach 2:
The corresponding values of the cycle length and the
green time are:
c = 50 [ s ] ;g 2 = 15 [ s ] The red time is:
r 2 = c – g 2 = 50 – 15 = 35 [ s ] The ﬂow rate and the service rate are:
veh
550 veh
veh
λ 2 = 550  =   = 0.153 hr
3600 s
s Virginia Tech 40 Problem 3 Solution
veh
2000 veh
veh
µ 2 = 2000  =   = 0.555 hr
3600 s
s The utilization factor for the queue ρ equals:
2 veh
0.153 s
λ2
ρ 2 =  =  = 0.276
µ2
veh
0.555 s The average delay per vehicle equals:
r22
35 2
d 2 =  =  = 16.92 [ s ]
2 ⋅ c ⋅ ( 1 – ρ2 )
2 ⋅ 50 ⋅ ( 1 – 0.276 ) Virginia Tech 41 Problem 3 Solution
(b) The total delay per cycle of all vehicles on both
approaches is the sum of the delays of every approach:
TD = λ 1 ⋅ d 1 + λ 2 ⋅ d 2 substituting the deﬁnitions of d and d (see equation 21)
1 2 r12
r22
TD = λ 1 ⋅  + λ 2 ⋅ 2 ⋅ c ⋅ ( 1 – ρ1 )
2 ⋅ c ⋅ ( 1 – ρ2 ) Since:
r1 + r2 = c after substitution, we get: Virginia Tech 42 r12
( c – r1 )2
TD = λ 1 ⋅  + λ 2 ⋅ 2 ⋅ c ⋅ ( 1 – ρ1 )
2 ⋅ c ⋅ ( 1 – ρ2 ) The total delay is minimal when:
d[ TD ]
 = 0
dr 1 After substitution, we get:
r12
( c – r1 )2
d λ 1 ⋅  + λ 2 ⋅ 2 ⋅ c ⋅ ( 1 – ρ1 )
2 ⋅ c ⋅ ( 1 – ρ2 )
 = 0
dr 1
r1
( c – r1 )
λ 1 ⋅  – λ 2 ⋅  = 0
c ⋅ ( 1 – ρ1 )
c ⋅ ( 1 – ρ2 ) Virginia Tech 43 Problem 3 Solution
( 50 – r 1 )
r1
0.167 ⋅  – 0.153 ⋅  = 0
50 ⋅ ( 1 – 0.273 )
50 ⋅ ( 1 – 0.276 ) After solving the equation, we get:
r 1 = 24 [ s ]
g 1 = c – r 1 = 50 – 24 = 26 [ s ]
r 2 = 26 [ s ]
g 2 = 24 [ s ] These are the optimal values of green and red times to
minimize the intersection delay.
Virginia Tech 44 Problem 3 Solution
We can recalculate the average delays per vehicle:
Approach 1
r12
24 2
d 1 =  =  = 7.92 [ s ]
2 ⋅ c ⋅ ( 1 – ρ1 )
2 ⋅ 50 ⋅ ( 1 – 0.273 ) Approach 2:
r22
26 2
d 2 =  =  = 9.34 [ s ]
2 ⋅ c ⋅ ( 1 – ρ2 )
2 ⋅ 50 ⋅ ( 1 – 0.276 ) These delays compare favorably with those obtained
before (3.09 and 16.92 seconds, respectively for
approaches 1 and 2). Virginia Tech 45 Vehicle Delays at Signalized Intersections: Random
Vehicle Arrivals
• Trafﬁc ﬂows are characterized by random ﬂuctuations
• The delay that a speciﬁc vehicle experiences depends on
the probability density function of the interarrival times,
as well as on signal timings and the time of a day when
the vehicle shows up
• Obviously, individual vehicles experience at a signalized
approach various delay values. Virginia Tech 46 Intersection with Random Arrivals Cumulative
number Cumulative arrivals of vehicles
Overﬂow Delay
Uniform delay Time τ
Red Green Figure 6. Intersection with Random Arrivals. Virginia Tech 47 Intersection with Random Arrivals
• Let us calculate the delay D for the vehicle arriving at
time τ (Figure 6). The overall delay D is composed of the
uniform delay d and the overﬂow delay d , i.e.:
R D = d + dR (22) • The uniform delay d represents delay that would be
exp rienced by a vehicle when all vehicle arrive
uniformly and when trafﬁc conditions are unsaturated
(see Equations in previous sections).
• Due to the random nature of
vehicle arrivals, the arrival
rate during some time periods can go over the capacity,
causing overﬂow queues. Virginia Tech 48 Considering Random Arrivals
• The overﬂow delay d represents the delay that is caused
by shortterm overﬂow queues. This delay can be easily
calculated using
queueing theory techniques.
R Queueing System Crossing area γ Virginia Tech λ 49 Intersection with Random Arrivals
We assume that vehicle interarrival times are
exponentially distributed. The service rate is deterministic
(we denote by γ departure rate from the artiﬁcial queue
into the signal), and there is only one server.
This means that the artiﬁcial Queueing System is M/D/1
queueing system (single server with Poisson arrivals and
deterministic service times).
The average delay per customer in the M/D/1 queueing
system equals:
α2
d R = 2 ⋅ λ ⋅ (1 – α) (23) Virginia Tech 50 Intersection with Random Arrivals
where:
α  utilization ratio in the M/D/1 queueing system The utilization ratio α in the M/D/1 queueing system
equals:
α=λ
γ (24) The departure rate from the artiﬁcial queue into the signal
γ can be expressed in terms of departure rates from the
trafﬁc signal µ . The departure rate equals µ during green
time. During red time, departure rate equals zero (see
Figure 6).
Virginia Tech 51 Intersection with Random Arrivals
Service rate g [veh/h] µ r Time 0
Green Red
Cycle Figure 7. Service Rate Deﬁnition at a Trafﬁc Intersection. Virginia Tech 52 Intersection with Random Arrivals
Departure rate γ during whole cycle : γ = 0⋅r+µ⋅g
c (25) g
γ = µ ⋅ c (26) The utilization ratio α in the M/D/1 queueing system
:
λα = λ = γ
g
µ ⋅ c (27) i.e.: Virginia Tech 53 λ⋅c
α = µ⋅g (28) The quantity α is known as a volume to capacity ratio.
The average vehicle delay is:
D = d + dR (29) r2
α2
D =  + 2 ⋅ c ⋅ (1 – ρ) 2 ⋅ λ ⋅ (1 – α) (30) It has been shown by simulation that Equation (30)
overestimate the average vehicle delay. The following two
formulas for average vehicle delay calculation were
proposed as a corrections of the equation (30): Virginia Tech 54 Webster’s formula:
1
3 5 ⋅ g 2 + cr
α  ⋅ α c D =  +  – 0.65 ⋅ 2
λ 2 ⋅ c ⋅ (1 – ρ) 2 ⋅ λ ⋅ (1 – α)
2 2 (31) Allsop’s formula:
9
α2
r2
D =  ⋅  + 10 2 ⋅ c ⋅ ( 1 – ρ ) 2 ⋅ λ ⋅ ( 1 – α ) Virginia Tech (32) 55 Example Problem 4
Using data given in the Example Problem 1, calculate: (a)
average delay per vehicle using Allsop’s formula. (b)
Calculate duration of the green time necessary to achieve
average delay per vehicle of 40 seconds.
Solution:
(a) The cycle length, green time, arrival rate, departure
rate, trafﬁc intensity, volume to capacity ratio, and red
time duration are:
c = 90 [ s ]
g = 27 [ s ] Virginia Tech 56 veh
600 veh
veh
λ = 600  =   = 0.167 hr
3600 s
s
veh
2200 veh
veh
µ = 2200  =   = 0.611 hr
3600 s
s
veh
0.167 s
λ
ρ =  =  = 0.273
µ
veh
0.611 s
veh
0.167 s
veh
λ
0.611 s
µ
α =  =  = 0.273 = 0.91
g
27 [ s ]
0.3
c
90 [ s ] Virginia Tech 57 Solution  Problem 4
r = c – g = 90 – 27 = 63 [ s ] The average delay per vehicle based on Allsop’s formula
equals:
9
α2
r2
D =  ⋅  + 10 2 ⋅ c ⋅ ( 1 – ρ ) 2 ⋅ λ ⋅ ( 1 – α )
90.91 2
63 2
D =  ⋅  + 10 2 ⋅ 90 ⋅ ( 1 – 0.273 ) 2 ⋅ 0.167 ⋅ ( 1 – 0.91 )
D = 52.083 [ s ] (b) The average delay per vehicle is: Virginia Tech 58 9
α2
r2
D =  ⋅  + 10 2 ⋅ c ⋅ ( 1 – ρ ) 2 ⋅ λ ⋅ ( 1 – α )
10
r2
α2
 =  ⋅ D – 9
2 ⋅ c ⋅ (1 – ρ)
2 ⋅ λ ⋅ (1 – α)
r= 10
α2
[ 2 ⋅ c ⋅ ( 1 – ρ ) ] ⋅  ⋅ D – 9
2 ⋅ λ ⋅ (1 – α) r= 10
0.91 2
[ 2 ⋅ 90 ⋅ ( 1 – 0.273 ) ] ⋅  ⋅ 40 – 9
2 ⋅ 0.167 ⋅ ( 1 – 0.91 ) r = 47 [ s ]
g = c – r = 90 – 47
g = 43 [ s ] Green time to achieve a delay of 40 seconds per vehicle.
Virginia Tech 59 ...
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 Atlantic Coast Conference, Queueing theory, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, T. Marshall Hahn

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