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A deterministic queueing model
Transpn
Res.
Vol. 1, pp. 117-128.
Pergamon
Press 1967. Printed in Great Britain
A DETERMINISTIC
QUEUEtNG
MODELf
ADOLF D. MAY, JR. and HARTMUT E. M. KELLER Institute of Transportation
and Traffic Engineering,
(Received
6 December
University of California,
1966; in revised form
20 March
Berkeley, California,
U.S.A.
1967)
INTRODUCTION MORE and more miles of travel
are being made on the urban freeway system; and because of these significant increases in traffic demand, the capacity at critical locations for some periods of time may be exceeded. When this occurs, freeway bottlenecks are formed, and queues and associated delays to motorists result. Recognizing these operational problems, various highway organizations have established freeway operational groups for the purpose of identifying and proposing means for solving these bottleneck situations. In order to justify the expenditure needed for these improvements and because of limited available funds, estimates of the possible benefits to highway users are required in order to determine which improvements should be undertaken and to establish the priority for various proposed improvements. The purpose of this paper is to present a deterministic queueing model which can be used to estimate the effects that capacity changes at bottlenecks have on delays to road users. The first portion of the paper is devoted to discussing traffic demand patterns during peak periods. The second portion of the paper contains the derivation of equations for determining delay and other queueing characteristics. The last portion of the paper includes the application of the derived equations to two bottleneck situations. TRAFFIC
DEMAND
PATTERNS
pattern and the bottleneck capacity during the peak traffic period are required in order to utilize deterministic queueing models. It is assumed in this model that the bottleneck capacity level remains constant throughout the peak traffic period. There is some evidence that once congestion occurs, capacity is reduced. This refinement is not included in this model, but such a condition would obviously result in even greater delays to road users. The following paragraphs are devoted to traffic demand patterns. It is assumed in this model that the traffic demand pattern will either be triangular or trapezoidal in shape (see upper portion of Fig. 1). Prior to the peak period the demand rate is assumed to be constant q,,, and less than the capacity. At time TO the demand begins to increase at a constant rate B,,eventually crossing the capacity s, and reaching the maximum demand rate q2 at time T 1. This maximum demand rate q2 is maintained for zero time if the demand is triangular in shape or until time T2 if the demand is trapezoidal in shape. Then the demand begins to decrease at a constant rate B3,eventually crossing the capacity s, and reaching the constant post-peak demand of q4 at time T3. The demand of q,, is assumed constant until the queue is dissipated. The traffic
demand
t Paper prepared for presentation at the Operations Research Society of America, Twenty-ninth National Meeting, Santa Monica, California, 18-20 May 1966, and at the Fifth World Meeting of the International Road Federation, London, England, 18-24 September 1966. 117
118
ADOLF D. MAY. JR.
and HARTMUT E. M. KELLER
44
/
Y
P L n I
FIG. 1. Generalized
demand-capacity
Time
relationship.
There is evidence to support this assumption of triangularor trapezoidal-shaped demand patterns. Inspection of most graphs showing directional volume variations by time of day normally reveal this general shape of traffic demand. A few specific studies have been made of traffic demand patterns, and two are described in the following paragraphs. The Institute of Transportation and Traffic Engineering of the University of California, Berkeley, have conducted a number of traffic surveys on the San Francisco-Oakland Bay Bridge and the variation of traffic flow by 6-min time intervals for morning and afternoon peak traffic periods is shown in Fig. 2, ITTE (1965). Volume data for two weekdays are shown for the westbound morning peak traffic period and the eastbound afternoon peak traffic period. The volume patterns in the morning are triangular in shape with a fairly constant decreasing rate after the peak demand. The volume patterns in the afternoon are trapezoidal in shape with a general increase in demand prior to the peak demand, a leveling off during the peak demand, and then followed by a general decrease in demand.
A deterministic queueing model
om
FIG.
Time of Day
119
pm
2. Peak hour flow patterns.
The Port of New York Authority has conducted studies and reported on the use of triangular and trapezoid traffic demand patterns (Foote, 1963). An example is given using the trapezoid-shaped traffic demand. Traffic counts are not always indicative of traffic demand particularly in the vicinity of bottlenecks, and therefore caution should be used in estimating traffic demand. For example, upstream traffic counts may underestimate the traffic demand if (1) a bottleneck is located upstream of the count location; (2) the queue from the bottleneck under study extends through to the traffic count location; and (3) an on-ramp is located between the bottleneck and the count station and not counted. Techniques of handling these possible problems, and also the situation of off-ramps between the bottleneck and the count station by the use of density contour maps have previously been reported (May, 1962). Referring again to Fig. 2, there is some concern that the traffic counts during the afternoon peak period may not represent the actual traffic demand pattern. The reason for this concern is the existence of a bottleneck upstream of the count station. This would tend to flatten the peak count, and would be more indicative of the capacity of the upstream bottleneck than the actual traffic demand. It was decided that the developed deterministic queueing model would still be designed to handle triangular and trapezoidal traffic demand patterns. However, those who apply this model to their particular situation should use care in determining the actual traffic demand pattern, particularly if the resulting pattern is trapezoidal in shape. MATHEMATICAL ANALYSIS Equations for calculating the delay characteristics for deterministic queueing situations are relatively simple and many have been developed to handle specific traffic problems (Clayton, 1941; Wardrop, 1952; Webster, 1958; May, 1965; Blunden and Pretty, 1965). These references were reviewed but were not applicable to the situation under study, and consequently a set of equations were derived from the proposed model. A brief description of the equations developed from this model is given in the following paragraphs.
ADOLF D. MAY. JR. and HARTMUT E. M. KELLER
120
A section of highway is considered having no access and a uniform capacity rate s(T) at the bottleneck. The traffic demand pattern proposed was discussed earlier in this paper, and is graphically shown in Fig. 1. The demand q(T) and the capacity s(T) are known at all instants of time over the study interval: %V) = %Y
O
ql(T) = A,+B,T,
TO
qz(T) = 92,
Tl
q3(T) = A3+B3T,
T2
qa(T) = q4,
T3
s(T) = s,
A,, B,, A, and B3 can be derived from Fig. 1; depending
on the demand
A, = qz-(qz-qo)(TlWTl),
B, = (q4-qo)l(DTl)
A, = q2 - (q4 - qJ (T2)l(DT3),
Bs = (q4 - qJ(DT3)
The cumulative flow is graphically by integration as shown below: Qo(T)
=
I’
qo(T)
QI(T) =
shown in the lower portion
of Fig. 1, and can be obtained
di- = qo T,
(A,+B,T)dT=
pattern:
O
TO
s
QdT) = S41(T) dT = cs+a T,
Tl
es(T)
T2
= [(A,+B,T)dT=
c,+A,T+SB,T2,
J
Q4V)
=
ji4VI
S(T) =
dT =
~4 +qa
T3
T>
s(T) dT = c, + ST,
O
s The integration
constants
can be determined Cl =
- A,(TO)
to be: - 05B,(T0)2
+ qo(T0)
c2 = c,+A,(Tl)
+05B,(T1)2
-qz(Tl)
cs = c2 - A,(T2)
- 0.5B,(T2)2
+ q,(T2)
c4 = c,+ A,(T3)
+0.5B,(T3)2
- qJT3)
c, = c,+A,(TM)+O.~B,(TM)~-s(TM) From these equations it can be seen that the uniform flow represents straight lines and the regular increasing flow represents quadratic parabolas in the cumulative diagram.
A deterministic
No queue or delay until the demand rate step was to determine time the queue begins
121
queueing model
is encountered according to the deterministic queueing approach (input rate) exceeds the capacity (output rate). Therefore the first the time when the demand rate just equals the capacity, and at this and vehicles are delayed. TM = DTO + DT 1(s -qo)/(q2 - qo)
(1)
The demand q(T) will remain greater than the capacity s(T) for the period TI. During this time period the queue length is increasing, and at the end of the TI interval, the queue length and the vehicular delay will reach their maximum values. TI = (DTl)K+
DT2+(DT3)L
(2)
where and
K = (q2 - s)l(q2 -q,,) The number
of vehicles in the system during N’
L = (s - qA/(% - q2). the queueing
period (TQ) is equal to
soTMT)
s(T)1 dT
=
(3)
The duration of the queue lasts as long as there are vehicles in the system, and therefore setting N’ = 0 one can derive an equation for the duration of the queueing period (TQ). For a finite TQ two cases have to be considered. If the queue is dissipated after T3, then TQ is equal to (4) If the queue is dissipated
before T3, then TQ becomes TQP = TI-k
E
(TI+ DT2)] ’
These equations are obtained using the fact that the number of vehicles discharged is equal to the vehicles stored during TQ. The number of vehicles adversely affected by the bottleneck can be expressed as the flow rate s times the duration of the queue: N = s(TQ)
(6)
The total delay (D) in vehicle-hours is graphically shown in the lower portion as the shaded area and can be computed by integration: D= s
of Fig. 1
oTQIQW) - W-11 dT
(7)
The solution of the integral gives the total delay as a function of the flow rates q,,(T), q&T), q*(T), s(T) and the time intervals DTO, DTl, DT2 and DT3. If the queue ends after the decreasing input flow period, the total delay is D,: D,=
s
cm Ql(T) dT+
TM
TzQz(T) dT+
s Tl
IPaQa(Tl dT+
s T3
TQ4QdT) dT-
s T3
TQ4
s TM
S(T) dT
ADOLF D. MAY, JR. and HARTMUT E. M. KELLER
122
D,
= 0*5(q4-q,)
[DT3(QDT3+
DT2+
+ DT2(DT2 +05(qc, -s)
DTl(K)-TQN)
+ DTl(K)
2 - 27’QN)]
[(DT1(K))2+(TQN)2]
(8)
-0.5(q4-2s)[DTl(K)TQN] +O+(q,-s)
If the queue ends during D,=
DTl(K)
the decreasing
*’ QI(T) dT+ s TM D,
[Y”QN-$DTl(K)] input
flow period,
T2Q2(T) dT+ s Tl
TQ3Q3(T) dTs T2
= (q4-q2)Q(1/DT3) + (q2-s)+[TQP-
the total delay is D,:
[TQP-
DTl(K)-
DT213 (9)
DT1(ZC)]2
+ (q2-s)&DTl(K)[TQP-$DTl(K)] The maximum number of vehicles interval TI and is given by:
in the queue (Q,,,)
occurs at the end of the time
Qm,, = /o%(r) - s(T)1dT Q max
= 0*5(q, - s) (TZ+ DT2)
The average number of vehicles in the queue (Q) can be computed and the period of congestion (TQ).
(10)
from the total delay (D)
e = DJTQ
The maximum vehicular the end of the time interval
delay in hours (D,,,)
(11)
occurs to that vehicle which arrives at
TZ. D mex = ;(qd(TI+DT2)
The average individual vehicle delay in hours (D) can be obtained delay (D) by the number of vehicles affected (N). i) = D/N
(12)
by dividing
the total
(13)
The resulting equations presented in the previous several paragraphs are cumbersome and could be simplified by modifying the equations to handle special cases. However, it appeared desirable to keep the equations in a more general form and thereby have broader application. In order to facilitate the handling of problems with this deterministic queueing model, a computer program was written in FORTRAN language for the IBM 1620. The following two examples utilized the computer program in solving for the various queueing characteristics as shown in equations (2, 4, 5, 6 and 8-13).
A deterministic APPLICATION
OF THE
123
queueing model QUEUEING
MODEL
The first example is for a six-lane urban freeway where the capacity for one direction (three lanes) is 5500 vehicles/hr. The traffic demand before and after the peak period is 3000 vehicles/hr, and the peak demand is 6600 vehicles/hr and lasts for 1 hr. The change in flow rate from 3000 to 6600 vehicles/hr extends over 1 hr, and similarly the change from 6600 to 3000 vehicles/hr extends over 1 hr. This demand-capacity situation is shown in Fig. 3. It is desired to calculate the various queueing characteristics for the situation described above and also for the situations where the capacity is increased by 5, 10, 15 and 20 per cent.
4 8~500 : 5 6,000 i: 4,000 ,y 2,000 2 I.0
2.0
30
40
24,Z0
t
Norm01 OV
0
I /
LO
1
2.0
I
3.0
FIG. 3. Example one-demand-capacity
4.0
I
-
5.0 t tours
relationship.
The results obtained from the computer computations are shown graphically in the upper portion of Fig. 4. The effect of the capacity increases on reducing the various queueing characteristics is shown in the lower portion of Fig. 4. The normal condition denoted on the left end of the horizontal scale is when the capacity is 5500 vehicles per hour. The 5, 10, I5 and 20 per cent increases in capacity are shown along the horizontal scale and represent capacity levels of 5780, 6060, 6330 and 6600 vehicles/hr respectively. For the normal condition (capacity 5500 vehicles/hr) the duration of queue was 2.53 hr, the maximum queue length was 1436 vehicles and the total delay was 2017 vehicle-hours. A 2.5 per cent increase in capacity would reduce the duration of queue by 7 per cent, the
ADOLF
124
D. MAY, JR. and HARTMUT E. M. KELLER
Copocify
hcreose - percent
copacify
Increase
FIG. 4. Effect of increasing
- percent
capacity
on queueing
characteristics.
TABLE 1. EFFECT OF INCREASING CAPACITY ON QUEUISNG CHARACTERISTICS
Normal
+2.5%
+ 5.0 3;
Capacity
C =5500
C ~5640
C ~5780
TI (hr) TQP or
1.61 2.53
1.53 2.36
+ 10.0%
+12.5%
+15.0x
+17.5x
+
C ~5910
C =6050
C =6190
C =6320
C =6460
C
1.46 2.21
1.38 2.06
1.31 I.90
1.23 1.73
1.15 1.55
1.08 1.36
13,930
13,324
12,756
12,158
11,489
10,733
9856
8776
1436
1220
1014
819
634
460
296
143
797 0.26
679 0.22
564 0.18
454 0.14
350 0.10
252 0.07
161 0.05
76 0.02
0.14 2018
0.12 1606
0.10 1247
0.08 934
0.06 665
0.04 437
0.03 251
0.01 104
TQN (W {ehicles) max (vehicles) Q (vehicles) $; B (hr) geh.-hr)
+ 7.5 “i,
A deterministic TABLE
2. EFFECT OF INCREASING
Normal Capacity
C =5500
TI
-
TQP or
--
+ 2.5 “/;,
CAPACITY
12
queueing model
ON QUEUEINC
(expressed
CHARACTERISTICS
as a percentage)
+5.0%
+7.5%
+10.0x
+12.5%
+15.oz
+17.5z
+.
C =5780
C =5910
C =6050
C =6190
C =6320
C =6440
C
5 I
9 13
14 18
19 25
24 32
28 39
33 46
4 15 15 15 14 20
8 29 29 31 29 38
12 43 43 46 33 54
18 56 56 61 57 67
23 68 68 73 71 78
29 79 80 81 79 88
37 90 90 92 93 95
C =5640
TQN
N
Fax D max i3 D
maximum queue length by 15 per cent and the total delay by 20 per cent. A 10 per ten increase in capacity would reduce the duration of queue by 25 per cent, the maximum queu length by 56 per cent, and the total delay by 67 per cent. A tabular summary of the corn puted queueing characteristics for the various capacity levels and the effect that thes incremental capacity increases have on the reduction of delays and other queuein characteristics is given in Tables 1 and 2.
8,000
t
I,000 0
/
I2 om
2
4
6
8
IO/2 2 om N pm fime of Day
FIG. 5. Example two-peak
I
I
I
/
4
6
8
IO pm
$
hour flow demand.
The second example is for the westbound San Francisco-Oakland Bay Bridge during th morning peak traffic period. The traffic demand pattern as obtained from traffic counts i shown in Fig. 5 as the dashed line. The assumed traffic demand pattern is shown in Fig. as the solid line. The demand levels are taken to be: q,, = 750, q2 = 8200 and q4 = 3750 while the time intervals are taken to be: DTI = 2 hr; DT2 = 0 and DT3 = 2.75 hr. Th capacity of this section of highway is actually greater than the traffic demand, and therefor’
126
ADOLF
D. MAY, JR. and HARTMUTE. M. KELLER
S,AN f9040
FRANCISCO OCT.
6,
BAY ,965
BRIDGE
f
om
8 om I0 :
f2 2
om
Time of Day
FIG. 6. Example two-
SAN
demand-capacity
FRANCISCO OCT.
Pm
relationship.
BAY 6,
2
BRIDGE
1965
s - f5,OOO 2 > 5 - l0,000
2 8
-
-40
-30
-20
-IO
5.000
0
Percent of Reduced Capacity FIG.
7. Effect of reduced capacity on queueing characteristics.
2 ?i b.
Aldeterministic queueing model
12
queueing would only occur if the capacity was reduced due to vehicle disabilities, accident! maintenance and the like. This second example then will be different than the first in tha instead of calculating the effect of capacity increases, the effect of capacity reductions on th queueing characteristics will be made. The capacity will vary from 4920 vehicles/hr tS 8200 vehicles/hr in 2.5 per cent increments. This demand-capacity situation is shown i Fig. 6. The results obtained from the computer computations are shown graphically in Fig. ; The greatest capacity reduction is depicted at the left end of the horizontal scale anI represents a capacity of 4920 vehicles/hr (40 per cent less than peak traffic demand). Th 30, 20, 10 and 0 per cent reductions in capacity are shown along the horizontal scale anI represent capacity levels of 5740, 6560, 7380 and 8200 vehicles/hr respectively. A capacit reduction of 10 per cent (from 8200 to 7380 vehicles/hr) would result in the duration c queue being 1.33 hr, a maximum queue length of 298 vehicles and a total delay of 24 vehicle-hours. A capacity reduction of 20 per cent (from 8200 to 6560 vehicles/hr) wou11 result in the duration of queue being 2.67 hr, a maximum queue length of 729 vehicles an a total delay of 1945 vehicle-hours. A tabular summary of the computed queuein characteristics for the various reduced capacity levels is given in Table 3.
TABLE3. EFFECTOF REDUCEDCAPACITYON QUEUEING CHARACTERISTICS -40%
-35%
-30%
-25%
-20%
-15%
-10%
-5%
Capacity
C =4920
C =5330
C =5740
C =6150
C =6560
C =6970
C =7380
C =7790
TI (hr) TQP or
2.90 7.34
2.54 5.34
2.18 414
1.82 3.33
1.45 2.67
1.10 2.00
0.73 1.33
0.36 0.67
36,134
28,477
23,783
20,509
17,499
13,945
9843
5194
QIllLX
4768
3651
2682
1862
1192
671
298
75
Q (vehicles) D max (hr) 6 (hr) :eh.-hr)
2622
2095
1602
1139
729
410
182
46
TQN (hr) N (vehicles) (vehicles)
0.97
0.68
0.53 19,258
0.39 11,191
0.47
0.30
0.18
0.10
0.04
0.01
0.28 6640
0.19 3799
0.11 1945
0.06 821
0.02 243
0.01 30
N C
SUMMARY
A summary of the three major portions of this paper is given in the followin] paragraphs. There is evidence that the triangular or trapezoidal-shaped traffic demand pattern i representative of actual measured traffic demands. The use of upstream traffic counts ma: underestimate the actual traffic demand, and special attention should be given to thi matter. A number of equations have been developed from this deterministic queueing mode for the purpose of determining delay and other queueing characteristics. An attempt ha been made to keep the equations in a general form in order to provide for broade application.
128
ADOLF D. MAY, JR. and HARTMLJTE. M. KELLER
The deterministic queueing model was applied to two specific traffic situations, one involving the effect of capacity increase on reducing the queueing characteristics and the other concerning the effect of reduced capacity on increasing the queueing characteristics. A trapezoidal demand pattern was used in the first example, and a triangular demand pattern was used in the second example. The results indicated that a small increase in capacity in the order of 10 per cent, reduced various queueing characteristics from 25 to 67 per cent. Similarly in the second example, a small decrease in capacity of the same magnitude due to some disturbance to the traffic stream would affect some 9800 vehicles for an overall delay of 240 vehicle-hours. These examples conclusively show that a small incremental change in capacity can have a drastic effect on the delays to the motorists.
REFERENCES BLUNDEN W. R. and PRETTY R. L. (1965). Theory of deterministic cyclic traffic flows in networks. World Traffic Engineering Conference. Institute of Traffic Engineers, Boston, Massachusetts. CLAYTONA. J. H. (1941). Road traffic calculations. J. Inst. civil Engrs 16, 247-284. FOOTE R. S. (1963). Single lane traffic flow control. 2nd ht. Symp. Theory of Traffic Flow, London. Institute of Transportation and TralTic Engineering (1965). University of California, Berkeley. Traffic Survey Saries k-23, Bay Bridge Toll Plaza. MAY A. D. JR. (1962). California freeway operations study. California Division of Highways. MAY A. D. JR. (1965). Traffic flow theory-The traffic engineer’s challenge. World Trafic Engineering Conference. Institute of Traffic Engineers, Boston, Massachusetts. WARDROPJ. G. (1952). Some theoretical aspects of road traffic research. Proc. Inst. civil Engrs II, l(2), 325-364.
WEBSTERF. V. (1958). Traffic Signal Settings. Department Research Technical Paper, No. 39. H.M.S.O., London.
of Scientific and Industrial
Research,
Road
Res.
Vol. 1, pp. 117-128.
Pergamon
Press 1967. Printed in Great Britain
A DETERMINISTIC
QUEUEtNG
MODELf
ADOLF D. MAY, JR. and HARTMUT E. M. KELLER Institute of Transportation
and Traffic Engineering,
(Received
6 December
University of California,
1966; in revised form
20 March
Berkeley, California,
U.S.A.
1967)
INTRODUCTION MORE and more miles of travel
are being made on the urban freeway system; and because of these significant increases in traffic demand, the capacity at critical locations for some periods of time may be exceeded. When this occurs, freeway bottlenecks are formed, and queues and associated delays to motorists result. Recognizing these operational problems, various highway organizations have established freeway operational groups for the purpose of identifying and proposing means for solving these bottleneck situations. In order to justify the expenditure needed for these improvements and because of limited available funds, estimates of the possible benefits to highway users are required in order to determine which improvements should be undertaken and to establish the priority for various proposed improvements. The purpose of this paper is to present a deterministic queueing model which can be used to estimate the effects that capacity changes at bottlenecks have on delays to road users. The first portion of the paper is devoted to discussing traffic demand patterns during peak periods. The second portion of the paper contains the derivation of equations for determining delay and other queueing characteristics. The last portion of the paper includes the application of the derived equations to two bottleneck situations. TRAFFIC
DEMAND
PATTERNS
pattern and the bottleneck capacity during the peak traffic period are required in order to utilize deterministic queueing models. It is assumed in this model that the bottleneck capacity level remains constant throughout the peak traffic period. There is some evidence that once congestion occurs, capacity is reduced. This refinement is not included in this model, but such a condition would obviously result in even greater delays to road users. The following paragraphs are devoted to traffic demand patterns. It is assumed in this model that the traffic demand pattern will either be triangular or trapezoidal in shape (see upper portion of Fig. 1). Prior to the peak period the demand rate is assumed to be constant q,,, and less than the capacity. At time TO the demand begins to increase at a constant rate B,,eventually crossing the capacity s, and reaching the maximum demand rate q2 at time T 1. This maximum demand rate q2 is maintained for zero time if the demand is triangular in shape or until time T2 if the demand is trapezoidal in shape. Then the demand begins to decrease at a constant rate B3,eventually crossing the capacity s, and reaching the constant post-peak demand of q4 at time T3. The demand of q,, is assumed constant until the queue is dissipated. The traffic
demand
t Paper prepared for presentation at the Operations Research Society of America, Twenty-ninth National Meeting, Santa Monica, California, 18-20 May 1966, and at the Fifth World Meeting of the International Road Federation, London, England, 18-24 September 1966. 117
118
ADOLF D. MAY. JR.
and HARTMUT E. M. KELLER
44
/
Y
P L n I
FIG. 1. Generalized
demand-capacity
Time
relationship.
There is evidence to support this assumption of triangularor trapezoidal-shaped demand patterns. Inspection of most graphs showing directional volume variations by time of day normally reveal this general shape of traffic demand. A few specific studies have been made of traffic demand patterns, and two are described in the following paragraphs. The Institute of Transportation and Traffic Engineering of the University of California, Berkeley, have conducted a number of traffic surveys on the San Francisco-Oakland Bay Bridge and the variation of traffic flow by 6-min time intervals for morning and afternoon peak traffic periods is shown in Fig. 2, ITTE (1965). Volume data for two weekdays are shown for the westbound morning peak traffic period and the eastbound afternoon peak traffic period. The volume patterns in the morning are triangular in shape with a fairly constant decreasing rate after the peak demand. The volume patterns in the afternoon are trapezoidal in shape with a general increase in demand prior to the peak demand, a leveling off during the peak demand, and then followed by a general decrease in demand.
A deterministic queueing model
om
FIG.
Time of Day
119
pm
2. Peak hour flow patterns.
The Port of New York Authority has conducted studies and reported on the use of triangular and trapezoid traffic demand patterns (Foote, 1963). An example is given using the trapezoid-shaped traffic demand. Traffic counts are not always indicative of traffic demand particularly in the vicinity of bottlenecks, and therefore caution should be used in estimating traffic demand. For example, upstream traffic counts may underestimate the traffic demand if (1) a bottleneck is located upstream of the count location; (2) the queue from the bottleneck under study extends through to the traffic count location; and (3) an on-ramp is located between the bottleneck and the count station and not counted. Techniques of handling these possible problems, and also the situation of off-ramps between the bottleneck and the count station by the use of density contour maps have previously been reported (May, 1962). Referring again to Fig. 2, there is some concern that the traffic counts during the afternoon peak period may not represent the actual traffic demand pattern. The reason for this concern is the existence of a bottleneck upstream of the count station. This would tend to flatten the peak count, and would be more indicative of the capacity of the upstream bottleneck than the actual traffic demand. It was decided that the developed deterministic queueing model would still be designed to handle triangular and trapezoidal traffic demand patterns. However, those who apply this model to their particular situation should use care in determining the actual traffic demand pattern, particularly if the resulting pattern is trapezoidal in shape. MATHEMATICAL ANALYSIS Equations for calculating the delay characteristics for deterministic queueing situations are relatively simple and many have been developed to handle specific traffic problems (Clayton, 1941; Wardrop, 1952; Webster, 1958; May, 1965; Blunden and Pretty, 1965). These references were reviewed but were not applicable to the situation under study, and consequently a set of equations were derived from the proposed model. A brief description of the equations developed from this model is given in the following paragraphs.
ADOLF D. MAY. JR. and HARTMUT E. M. KELLER
120
A section of highway is considered having no access and a uniform capacity rate s(T) at the bottleneck. The traffic demand pattern proposed was discussed earlier in this paper, and is graphically shown in Fig. 1. The demand q(T) and the capacity s(T) are known at all instants of time over the study interval: %V) = %Y
O
ql(T) = A,+B,T,
TO
qz(T) = 92,
Tl
q3(T) = A3+B3T,
T2
qa(T) = q4,
T3
s(T) = s,
A,, B,, A, and B3 can be derived from Fig. 1; depending
on the demand
A, = qz-(qz-qo)(TlWTl),
B, = (q4-qo)l(DTl)
A, = q2 - (q4 - qJ (T2)l(DT3),
Bs = (q4 - qJ(DT3)
The cumulative flow is graphically by integration as shown below: Qo(T)
=
I’
qo(T)
QI(T) =
shown in the lower portion
of Fig. 1, and can be obtained
di- = qo T,
(A,+B,T)dT=
pattern:
O
TO
s
QdT) = S41(T) dT = cs+a T,
Tl
es(T)
T2
= [(A,+B,T)dT=
c,+A,T+SB,T2,
J
Q4V)
=
ji4VI
S(T) =
dT =
~4 +qa
T3
T>
s(T) dT = c, + ST,
O
s The integration
constants
can be determined Cl =
- A,(TO)
to be: - 05B,(T0)2
+ qo(T0)
c2 = c,+A,(Tl)
+05B,(T1)2
-qz(Tl)
cs = c2 - A,(T2)
- 0.5B,(T2)2
+ q,(T2)
c4 = c,+ A,(T3)
+0.5B,(T3)2
- qJT3)
c, = c,+A,(TM)+O.~B,(TM)~-s(TM) From these equations it can be seen that the uniform flow represents straight lines and the regular increasing flow represents quadratic parabolas in the cumulative diagram.
A deterministic
No queue or delay until the demand rate step was to determine time the queue begins
121
queueing model
is encountered according to the deterministic queueing approach (input rate) exceeds the capacity (output rate). Therefore the first the time when the demand rate just equals the capacity, and at this and vehicles are delayed. TM = DTO + DT 1(s -qo)/(q2 - qo)
(1)
The demand q(T) will remain greater than the capacity s(T) for the period TI. During this time period the queue length is increasing, and at the end of the TI interval, the queue length and the vehicular delay will reach their maximum values. TI = (DTl)K+
DT2+(DT3)L
(2)
where and
K = (q2 - s)l(q2 -q,,) The number
of vehicles in the system during N’
L = (s - qA/(% - q2). the queueing
period (TQ) is equal to
soTMT)
s(T)1 dT
=
(3)
The duration of the queue lasts as long as there are vehicles in the system, and therefore setting N’ = 0 one can derive an equation for the duration of the queueing period (TQ). For a finite TQ two cases have to be considered. If the queue is dissipated after T3, then TQ is equal to (4) If the queue is dissipated
before T3, then TQ becomes TQP = TI-k
E
(TI+ DT2)] ’
These equations are obtained using the fact that the number of vehicles discharged is equal to the vehicles stored during TQ. The number of vehicles adversely affected by the bottleneck can be expressed as the flow rate s times the duration of the queue: N = s(TQ)
(6)
The total delay (D) in vehicle-hours is graphically shown in the lower portion as the shaded area and can be computed by integration: D= s
of Fig. 1
oTQIQW) - W-11 dT
(7)
The solution of the integral gives the total delay as a function of the flow rates q,,(T), q&T), q*(T), s(T) and the time intervals DTO, DTl, DT2 and DT3. If the queue ends after the decreasing input flow period, the total delay is D,: D,=
s
cm Ql(T) dT+
TM
TzQz(T) dT+
s Tl
IPaQa(Tl dT+
s T3
TQ4QdT) dT-
s T3
TQ4
s TM
S(T) dT
ADOLF D. MAY, JR. and HARTMUT E. M. KELLER
122
D,
= 0*5(q4-q,)
[DT3(QDT3+
DT2+
+ DT2(DT2 +05(qc, -s)
DTl(K)-TQN)
+ DTl(K)
2 - 27’QN)]
[(DT1(K))2+(TQN)2]
(8)
-0.5(q4-2s)[DTl(K)TQN] +O+(q,-s)
If the queue ends during D,=
DTl(K)
the decreasing
*’ QI(T) dT+ s TM D,
[Y”QN-$DTl(K)] input
flow period,
T2Q2(T) dT+ s Tl
TQ3Q3(T) dTs T2
= (q4-q2)Q(1/DT3) + (q2-s)+[TQP-
the total delay is D,:
[TQP-
DTl(K)-
DT213 (9)
DT1(ZC)]2
+ (q2-s)&DTl(K)[TQP-$DTl(K)] The maximum number of vehicles interval TI and is given by:
in the queue (Q,,,)
occurs at the end of the time
Qm,, = /o%(r) - s(T)1dT Q max
= 0*5(q, - s) (TZ+ DT2)
The average number of vehicles in the queue (Q) can be computed and the period of congestion (TQ).
(10)
from the total delay (D)
e = DJTQ
The maximum vehicular the end of the time interval
delay in hours (D,,,)
(11)
occurs to that vehicle which arrives at
TZ. D mex = ;(qd(TI+DT2)
The average individual vehicle delay in hours (D) can be obtained delay (D) by the number of vehicles affected (N). i) = D/N
(12)
by dividing
the total
(13)
The resulting equations presented in the previous several paragraphs are cumbersome and could be simplified by modifying the equations to handle special cases. However, it appeared desirable to keep the equations in a more general form and thereby have broader application. In order to facilitate the handling of problems with this deterministic queueing model, a computer program was written in FORTRAN language for the IBM 1620. The following two examples utilized the computer program in solving for the various queueing characteristics as shown in equations (2, 4, 5, 6 and 8-13).
A deterministic APPLICATION
OF THE
123
queueing model QUEUEING
MODEL
The first example is for a six-lane urban freeway where the capacity for one direction (three lanes) is 5500 vehicles/hr. The traffic demand before and after the peak period is 3000 vehicles/hr, and the peak demand is 6600 vehicles/hr and lasts for 1 hr. The change in flow rate from 3000 to 6600 vehicles/hr extends over 1 hr, and similarly the change from 6600 to 3000 vehicles/hr extends over 1 hr. This demand-capacity situation is shown in Fig. 3. It is desired to calculate the various queueing characteristics for the situation described above and also for the situations where the capacity is increased by 5, 10, 15 and 20 per cent.
4 8~500 : 5 6,000 i: 4,000 ,y 2,000 2 I.0
2.0
30
40
24,Z0
t
Norm01 OV
0
I /
LO
1
2.0
I
3.0
FIG. 3. Example one-demand-capacity
4.0
I
-
5.0 t tours
relationship.
The results obtained from the computer computations are shown graphically in the upper portion of Fig. 4. The effect of the capacity increases on reducing the various queueing characteristics is shown in the lower portion of Fig. 4. The normal condition denoted on the left end of the horizontal scale is when the capacity is 5500 vehicles per hour. The 5, 10, I5 and 20 per cent increases in capacity are shown along the horizontal scale and represent capacity levels of 5780, 6060, 6330 and 6600 vehicles/hr respectively. For the normal condition (capacity 5500 vehicles/hr) the duration of queue was 2.53 hr, the maximum queue length was 1436 vehicles and the total delay was 2017 vehicle-hours. A 2.5 per cent increase in capacity would reduce the duration of queue by 7 per cent, the
ADOLF
124
D. MAY, JR. and HARTMUT E. M. KELLER
Copocify
hcreose - percent
copacify
Increase
FIG. 4. Effect of increasing
- percent
capacity
on queueing
characteristics.
TABLE 1. EFFECT OF INCREASING CAPACITY ON QUEUISNG CHARACTERISTICS
Normal
+2.5%
+ 5.0 3;
Capacity
C =5500
C ~5640
C ~5780
TI (hr) TQP or
1.61 2.53
1.53 2.36
+ 10.0%
+12.5%
+15.0x
+17.5x
+
C ~5910
C =6050
C =6190
C =6320
C =6460
C
1.46 2.21
1.38 2.06
1.31 I.90
1.23 1.73
1.15 1.55
1.08 1.36
13,930
13,324
12,756
12,158
11,489
10,733
9856
8776
1436
1220
1014
819
634
460
296
143
797 0.26
679 0.22
564 0.18
454 0.14
350 0.10
252 0.07
161 0.05
76 0.02
0.14 2018
0.12 1606
0.10 1247
0.08 934
0.06 665
0.04 437
0.03 251
0.01 104
TQN (W {ehicles) max (vehicles) Q (vehicles) $; B (hr) geh.-hr)
+ 7.5 “i,
A deterministic TABLE
2. EFFECT OF INCREASING
Normal Capacity
C =5500
TI
-
TQP or
--
+ 2.5 “/;,
CAPACITY
12
queueing model
ON QUEUEINC
(expressed
CHARACTERISTICS
as a percentage)
+5.0%
+7.5%
+10.0x
+12.5%
+15.oz
+17.5z
+.
C =5780
C =5910
C =6050
C =6190
C =6320
C =6440
C
5 I
9 13
14 18
19 25
24 32
28 39
33 46
4 15 15 15 14 20
8 29 29 31 29 38
12 43 43 46 33 54
18 56 56 61 57 67
23 68 68 73 71 78
29 79 80 81 79 88
37 90 90 92 93 95
C =5640
TQN
N
Fax D max i3 D
maximum queue length by 15 per cent and the total delay by 20 per cent. A 10 per ten increase in capacity would reduce the duration of queue by 25 per cent, the maximum queu length by 56 per cent, and the total delay by 67 per cent. A tabular summary of the corn puted queueing characteristics for the various capacity levels and the effect that thes incremental capacity increases have on the reduction of delays and other queuein characteristics is given in Tables 1 and 2.
8,000
t
I,000 0
/
I2 om
2
4
6
8
IO/2 2 om N pm fime of Day
FIG. 5. Example two-peak
I
I
I
/
4
6
8
IO pm
$
hour flow demand.
The second example is for the westbound San Francisco-Oakland Bay Bridge during th morning peak traffic period. The traffic demand pattern as obtained from traffic counts i shown in Fig. 5 as the dashed line. The assumed traffic demand pattern is shown in Fig. as the solid line. The demand levels are taken to be: q,, = 750, q2 = 8200 and q4 = 3750 while the time intervals are taken to be: DTI = 2 hr; DT2 = 0 and DT3 = 2.75 hr. Th capacity of this section of highway is actually greater than the traffic demand, and therefor’
126
ADOLF
D. MAY, JR. and HARTMUTE. M. KELLER
S,AN f9040
FRANCISCO OCT.
6,
BAY ,965
BRIDGE
f
om
8 om I0 :
f2 2
om
Time of Day
FIG. 6. Example two-
SAN
demand-capacity
FRANCISCO OCT.
Pm
relationship.
BAY 6,
2
BRIDGE
1965
s - f5,OOO 2 > 5 - l0,000
2 8
-
-40
-30
-20
-IO
5.000
0
Percent of Reduced Capacity FIG.
7. Effect of reduced capacity on queueing characteristics.
2 ?i b.
Aldeterministic queueing model
12
queueing would only occur if the capacity was reduced due to vehicle disabilities, accident! maintenance and the like. This second example then will be different than the first in tha instead of calculating the effect of capacity increases, the effect of capacity reductions on th queueing characteristics will be made. The capacity will vary from 4920 vehicles/hr tS 8200 vehicles/hr in 2.5 per cent increments. This demand-capacity situation is shown i Fig. 6. The results obtained from the computer computations are shown graphically in Fig. ; The greatest capacity reduction is depicted at the left end of the horizontal scale anI represents a capacity of 4920 vehicles/hr (40 per cent less than peak traffic demand). Th 30, 20, 10 and 0 per cent reductions in capacity are shown along the horizontal scale anI represent capacity levels of 5740, 6560, 7380 and 8200 vehicles/hr respectively. A capacit reduction of 10 per cent (from 8200 to 7380 vehicles/hr) would result in the duration c queue being 1.33 hr, a maximum queue length of 298 vehicles and a total delay of 24 vehicle-hours. A capacity reduction of 20 per cent (from 8200 to 6560 vehicles/hr) wou11 result in the duration of queue being 2.67 hr, a maximum queue length of 729 vehicles an a total delay of 1945 vehicle-hours. A tabular summary of the computed queuein characteristics for the various reduced capacity levels is given in Table 3.
TABLE3. EFFECTOF REDUCEDCAPACITYON QUEUEING CHARACTERISTICS -40%
-35%
-30%
-25%
-20%
-15%
-10%
-5%
Capacity
C =4920
C =5330
C =5740
C =6150
C =6560
C =6970
C =7380
C =7790
TI (hr) TQP or
2.90 7.34
2.54 5.34
2.18 414
1.82 3.33
1.45 2.67
1.10 2.00
0.73 1.33
0.36 0.67
36,134
28,477
23,783
20,509
17,499
13,945
9843
5194
QIllLX
4768
3651
2682
1862
1192
671
298
75
Q (vehicles) D max (hr) 6 (hr) :eh.-hr)
2622
2095
1602
1139
729
410
182
46
TQN (hr) N (vehicles) (vehicles)
0.97
0.68
0.53 19,258
0.39 11,191
0.47
0.30
0.18
0.10
0.04
0.01
0.28 6640
0.19 3799
0.11 1945
0.06 821
0.02 243
0.01 30
N C
SUMMARY
A summary of the three major portions of this paper is given in the followin] paragraphs. There is evidence that the triangular or trapezoidal-shaped traffic demand pattern i representative of actual measured traffic demands. The use of upstream traffic counts ma: underestimate the actual traffic demand, and special attention should be given to thi matter. A number of equations have been developed from this deterministic queueing mode for the purpose of determining delay and other queueing characteristics. An attempt ha been made to keep the equations in a general form in order to provide for broade application.
128
ADOLF D. MAY, JR. and HARTMLJTE. M. KELLER
The deterministic queueing model was applied to two specific traffic situations, one involving the effect of capacity increase on reducing the queueing characteristics and the other concerning the effect of reduced capacity on increasing the queueing characteristics. A trapezoidal demand pattern was used in the first example, and a triangular demand pattern was used in the second example. The results indicated that a small increase in capacity in the order of 10 per cent, reduced various queueing characteristics from 25 to 67 per cent. Similarly in the second example, a small decrease in capacity of the same magnitude due to some disturbance to the traffic stream would affect some 9800 vehicles for an overall delay of 240 vehicle-hours. These examples conclusively show that a small incremental change in capacity can have a drastic effect on the delays to the motorists.
REFERENCES BLUNDEN W. R. and PRETTY R. L. (1965). Theory of deterministic cyclic traffic flows in networks. World Traffic Engineering Conference. Institute of Traffic Engineers, Boston, Massachusetts. CLAYTONA. J. H. (1941). Road traffic calculations. J. Inst. civil Engrs 16, 247-284. FOOTE R. S. (1963). Single lane traffic flow control. 2nd ht. Symp. Theory of Traffic Flow, London. Institute of Transportation and TralTic Engineering (1965). University of California, Berkeley. Traffic Survey Saries k-23, Bay Bridge Toll Plaza. MAY A. D. JR. (1962). California freeway operations study. California Division of Highways. MAY A. D. JR. (1965). Traffic flow theory-The traffic engineer’s challenge. World Trafic Engineering Conference. Institute of Traffic Engineers, Boston, Massachusetts. WARDROPJ. G. (1952). Some theoretical aspects of road traffic research. Proc. Inst. civil Engrs II, l(2), 325-364.
WEBSTERF. V. (1958). Traffic Signal Settings. Department Research Technical Paper, No. 39. H.M.S.O., London.
of Scientific and Industrial
Research,
Road