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The single lane merging problem with mixed cars and trucks
7ranspn Res.
Vol. 3, pp. 195-199. Pergamon
THE SINGLE
LANE
Press 1969. Printed in Great Britain
MERGING AND
PROBLEM
WITH
MIXED
CARS
TRUCKS
GEORGEH. WEISS Division of Computer
Research and Technology, National Institutes of Health, Department Education and Welfare, Bethesda, Maryland 20014 (Received
19 February
1968; in revised form 20 November
of Health,
1968)
THEREappears
to be some evidence that the presence of trucks on a highway changes the statistical characteristics of vehicles considerably (Gwynn, 1967). It is therefore of interest to investigate how the presence of long queues generated by a few trucks in a stream of traffic will change the statistics of merging delay. We will consider a simple model for computing moments of delay, based partially on the work of Miller (1962), who considers a stream of traffic to be made up, not of independent vehicles, but of moving queues. We consider a single main stream of traffic consisting of trucks and cars, each truck generating a queue of cars. A principal approximation in the present work is that there is no more than a single truck to a queue. We will assume that traffic is light throughout, so that headways have a negative exponential distribution. A more complicated theory can be developed allowing for queues with several trucks, but the present theory will probably suffice in furnishing results on the order of magnitude of the effect of queues on delay characteristics. T-T
I
T -
C DIRECTION
INTERVAL
c
c
c
T
C
OF MOTION
FIG. 1. Schematic diagram of the present model of mixed traffic.
Let us picture a stream of traffic as shown in Fig. 1. The symbols T and C stand for truck and car, and the shaded portion of the lines denote a queue following a truck. Individual cars in a queue will not be accounted for, but we will assume that a queue ends with a car and that any waiting cars on a feeder road are unable to merge with the main stream while a queue is passing. We will assume that the p.d.f. for the headway between two successive cars not in queue or between a car and a following truck will be the same and will be denoted by 4(t) = (l/p) exp (- t/p) with mean headway p while the p.d.f. for a queue length (measured in time) will be denoted by p(t) with mean duration po. Furthermore, the probability that a vehicle following a given car is a truck will be denoted by 8. Let a car arrive at a stop sign at a feeder road at f = 0 and let the driver’s gap acceptance function be a(t). We first calculate the probability of zero delay. The probability that a car is able to merge is equal to the probability that a main stream queue is not passing the feeder road at t = 0, multiplied by the probability that the residual gap is acceptable. The latter 195
I’%
Ciro~cit
probability
will be denoted
by ct and is _
H.
I
-
P
0
WFISS
xc
-.I
a(t) c u/’ dr
The probability that a queue is not passing the feeder road at a random time can be written in terms of I*~~and the mean time between the passage of the last car in a queue and the next succeeding truck, call this V, as V/(V+ ,u,J, by an elementary result in renewal theory (Cox, 1962). The parameter v can be calculated in terms of I_Land B. For this purpose let 4*(s) = l/(1 +ps). Then the p.d.f. for the headway between the last car in a queue and the next succeeding truck is
p(s) = e+*(s)+ 8(1- 0) [+*(s)]2+ 8(1- e)2[4*(s)]3+ . . . = @?+)/[I
-(I
-e)+*(s)]
(2)
= B/(0+@) Since v can be written
in terms of the first derivative
of p*(s) we find (3)
so that the probability
of zero delay is PO where
(4) where a is defined by this equation as a = p/(p+ tip,). Let us now calculate the p.d.f. of waiting time, to be denoted by Q(t). We can begin by noting that the passage of a truck on the main road can be taken as a regeneration point. The interval between t = 0 and the first truck will be termed the first interval (as opposed to a gap), the interval between the first and second trucks will be denoted by the second interval, and so forth. Then if Q,,(t) represents the p.d.f. of waiting time for a merge during interval n, we have
The calculation of Q(t) (or its Laplace For this purpose we define the following
transform) quantities
Y(t) = f$(t) [1 -a(t)],
will be effected and functions:
PO(t) = !‘
I”~(-4 dx/wQ
by calculating
Q,(t).
(6)
The function Y(r) is the p.d.f. of a rejected gap (between a car and a following vehicle), and p,,(t) is the p.d.f. of the interval between t = 0 and the passage of the first queue, provided that t = 0 occurs during the passing of a queue. We shall now calculate an expression for Q,*(s) [ = J”Fexp (-st) Cl,(t) dt], the Laplace transform of waiting time conditional on the merge being made during the first interval. The expression for this quantity is Q,*(s) = G[a+(l
-a)
p,“(s)]{1 +(I - s)Y*(S)+(l-
+(1 - Q3[Y”(s)]“+ =
4a+(l-4po*(41 I -(l-
e)Y*(s)
8)2[Y*(s)]2+
. ..} (7)
The single lane merging problem with mixed cars and trucks
197
in which the Laplace transform of a function is denoted by that same function with an asterisk and argument S. Let us consider each term in turn in this expression. The first, aG is the transform of P, 8(t), i.e. the probability of zero delay. The general term a%(1 - Qn[Y*(s)]” is the Laplace transform of the p.d.f. of the time to the first acceptable gap when t = 0 occurs between the passage of two cars (not both in a queue) and on the acceptable gap being after the passage of n cars on the main road without an intervening of the truck. The general term (1 -a) p,,*(s)(l@[Y*(s)]~ 01 - is the Laplace transform time of the first acceptable gap conditional on t = 0 occurring during the passage of a queue. Hence Q*(s) is the sum of all of the foregoing terms and has the expression given in the last line of equation (7). A derivation of an expression for Q,*(s) follows similar reasoning as that for CLl*(s). We shall first find a general expression for Q,*(s) in terms of several auxiliary functions. Let S,*(S) be the Laplace transform of the p.d.f. of the first interval provided that it is rejected, let 6*(s) be the Laplace transform of the p.d.f. of any rejected interval but the first, and let q*(s) be the Laplace transform of the p.d.f. of an interval (not the first) in which a merge occurs. Then Q,*(s) is given by Q,“(s)
= s,*(S) [6*(s)]fi--“n*(s),
n> 2
since the first n- 1 intervals are rejected and the nth contains equation (5), takes the relatively simple form Q”(s) = Q,“(s)+
a merge.
(8) Hence Q*(s), by
g Q,“(s) ?&=a
=
Q
*ls)
1
+
so*(s) rl”(s) 1-6*(s)
To complete the calculation of 0*(s) we need to furnish expressions for 6,*(s) and 6*(s). These can be derived by the same sequence of steps leading to the expression for Q,*(s). The final results are *
*(s)
=
0
aey*(s>+ Nl-4 1 -(I
T”@) = 1 -(I_ s*(s) =
Po*~s)~*(s~
- @Y*(s)
P *cd 3 Qy*(s)
ef*(s)y*(s) i -(I - eyrys)
which, together with equation (9), contributes a formal solution to the problem. It can be verified that Q*(O) = 1 as is required. It is now a simple matter to calculate moments of delay by using equation (9) as a moment-generating function. Thus, the expected delay is
198
H. WEISS
GEORGE
Detailed use of equations in terms of quantities
(7). (9) and (10) leads to an expression
for i that can be written
r/3
PC_? =
~p&)dr,
I‘
a =
tY’(t)dt
(12)
.o
(13) The first term is, of course, the expected delay in the absence of travelling queues, second and third terms give the effect of trucks. Finally, we consider the question of relating 0 to the actual percentage of trucks main road. Let this percentage be denoted by p. Let the average number of cars in be rig. Let us consider a time interval between two trucks. The expected number not m queue (that is, who are following the last car in the queue) is I - B)‘b = (1 - eye
0&(
so that the total expected number of cars in an interval Hence the fraction of trucks on the main road is 1 r?,+(l-
on the a queue of cars
(14)
between two trucks is (1 - Q/8 + Cu.
e
=-=
1+&i,
e>/e+ 1
and the
(15)
p
so that 6’ is given by (16)
H = P/(1 - p$) Although we have restricted our considerations to the small p one can see that if is, is large, B is not necessarily small.7 As an example of the foregoing theory let us consider a average gap between successive cars in a queue is TQ set so that acceptance function will be taken as a step function a(t) = H(tand H(x) = 1 for x > 0. With this set of functions the formula
+de2’1~-
1) T,
case (relatively
traffic stream in which the pQ = TQii, (set). The gap T) where H(x) = 0 for x < 0 for the mean delay is
fjo (17)
l-p?ia For an idea of the meaning T = 4 set and TCa= 2 sec.
of this formula
few trucks),
let us take p = 3 set (a flow of 1200 cars/hr),
Figure 2 shows a graph of I as a function of Es for these parameters and for p = 0.01 and 0.02. It is clear that if passing on the main highway is at all difficult so that queues tend to build up, the waiting time can be increased considerably. The effects of mixed traffic are likely to be much more pronounced when the queues themselves can contain trucks in addition to cars, since then the queues tend to be much longer. t Note
that
in this theory
the maximum
value
that
p can have
is prnnx = (1 +&-‘.
The single lane merging problem with mixed cars and trucks
L 0
FIG. 2.
1
1
/
1
I
5
IO
15
20
25
Curves of expected delay with a small admixture
1
30
of trucks.
REFERENCES Cox D. R. (1962). Renewal Theory. Methuen, London. GWYNN D. W. (1967). Truck equivalency. Highw. Res. Rec. 199, 79. MILLER A. J. (1962). A queueing model for road traffic flow. JI R. statist. Sot. 23, 6475. WEISS G. H. and MARADUDIN A. A. (1962). Some problems in traffic delay. Ops Res. 10, 74-104.
199
Vol. 3, pp. 195-199. Pergamon
THE SINGLE
LANE
Press 1969. Printed in Great Britain
MERGING AND
PROBLEM
WITH
MIXED
CARS
TRUCKS
GEORGEH. WEISS Division of Computer
Research and Technology, National Institutes of Health, Department Education and Welfare, Bethesda, Maryland 20014 (Received
19 February
1968; in revised form 20 November
of Health,
1968)
THEREappears
to be some evidence that the presence of trucks on a highway changes the statistical characteristics of vehicles considerably (Gwynn, 1967). It is therefore of interest to investigate how the presence of long queues generated by a few trucks in a stream of traffic will change the statistics of merging delay. We will consider a simple model for computing moments of delay, based partially on the work of Miller (1962), who considers a stream of traffic to be made up, not of independent vehicles, but of moving queues. We consider a single main stream of traffic consisting of trucks and cars, each truck generating a queue of cars. A principal approximation in the present work is that there is no more than a single truck to a queue. We will assume that traffic is light throughout, so that headways have a negative exponential distribution. A more complicated theory can be developed allowing for queues with several trucks, but the present theory will probably suffice in furnishing results on the order of magnitude of the effect of queues on delay characteristics. T-T
I
T -
C DIRECTION
INTERVAL
c
c
c
T
C
OF MOTION
FIG. 1. Schematic diagram of the present model of mixed traffic.
Let us picture a stream of traffic as shown in Fig. 1. The symbols T and C stand for truck and car, and the shaded portion of the lines denote a queue following a truck. Individual cars in a queue will not be accounted for, but we will assume that a queue ends with a car and that any waiting cars on a feeder road are unable to merge with the main stream while a queue is passing. We will assume that the p.d.f. for the headway between two successive cars not in queue or between a car and a following truck will be the same and will be denoted by 4(t) = (l/p) exp (- t/p) with mean headway p while the p.d.f. for a queue length (measured in time) will be denoted by p(t) with mean duration po. Furthermore, the probability that a vehicle following a given car is a truck will be denoted by 8. Let a car arrive at a stop sign at a feeder road at f = 0 and let the driver’s gap acceptance function be a(t). We first calculate the probability of zero delay. The probability that a car is able to merge is equal to the probability that a main stream queue is not passing the feeder road at t = 0, multiplied by the probability that the residual gap is acceptable. The latter 195
I’%
Ciro~cit
probability
will be denoted
by ct and is _
H.
I
-
P
0
WFISS
xc
-.I
a(t) c u/’ dr
The probability that a queue is not passing the feeder road at a random time can be written in terms of I*~~and the mean time between the passage of the last car in a queue and the next succeeding truck, call this V, as V/(V+ ,u,J, by an elementary result in renewal theory (Cox, 1962). The parameter v can be calculated in terms of I_Land B. For this purpose let 4*(s) = l/(1 +ps). Then the p.d.f. for the headway between the last car in a queue and the next succeeding truck is
p(s) = e+*(s)+ 8(1- 0) [+*(s)]2+ 8(1- e)2[4*(s)]3+ . . . = @?+)/[I
-(I
-e)+*(s)]
(2)
= B/(0+@) Since v can be written
in terms of the first derivative
of p*(s) we find (3)
so that the probability
of zero delay is PO where
(4) where a is defined by this equation as a = p/(p+ tip,). Let us now calculate the p.d.f. of waiting time, to be denoted by Q(t). We can begin by noting that the passage of a truck on the main road can be taken as a regeneration point. The interval between t = 0 and the first truck will be termed the first interval (as opposed to a gap), the interval between the first and second trucks will be denoted by the second interval, and so forth. Then if Q,,(t) represents the p.d.f. of waiting time for a merge during interval n, we have
The calculation of Q(t) (or its Laplace For this purpose we define the following
transform) quantities
Y(t) = f$(t) [1 -a(t)],
will be effected and functions:
PO(t) = !‘
I”~(-4 dx/wQ
by calculating
Q,(t).
(6)
The function Y(r) is the p.d.f. of a rejected gap (between a car and a following vehicle), and p,,(t) is the p.d.f. of the interval between t = 0 and the passage of the first queue, provided that t = 0 occurs during the passing of a queue. We shall now calculate an expression for Q,*(s) [ = J”Fexp (-st) Cl,(t) dt], the Laplace transform of waiting time conditional on the merge being made during the first interval. The expression for this quantity is Q,*(s) = G[a+(l
-a)
p,“(s)]{1 +(I - s)Y*(S)+(l-
+(1 - Q3[Y”(s)]“+ =
4a+(l-4po*(41 I -(l-
e)Y*(s)
8)2[Y*(s)]2+
. ..} (7)
The single lane merging problem with mixed cars and trucks
197
in which the Laplace transform of a function is denoted by that same function with an asterisk and argument S. Let us consider each term in turn in this expression. The first, aG is the transform of P, 8(t), i.e. the probability of zero delay. The general term a%(1 - Qn[Y*(s)]” is the Laplace transform of the p.d.f. of the time to the first acceptable gap when t = 0 occurs between the passage of two cars (not both in a queue) and on the acceptable gap being after the passage of n cars on the main road without an intervening of the truck. The general term (1 -a) p,,*(s)(l@[Y*(s)]~ 01 - is the Laplace transform time of the first acceptable gap conditional on t = 0 occurring during the passage of a queue. Hence Q*(s) is the sum of all of the foregoing terms and has the expression given in the last line of equation (7). A derivation of an expression for Q,*(s) follows similar reasoning as that for CLl*(s). We shall first find a general expression for Q,*(s) in terms of several auxiliary functions. Let S,*(S) be the Laplace transform of the p.d.f. of the first interval provided that it is rejected, let 6*(s) be the Laplace transform of the p.d.f. of any rejected interval but the first, and let q*(s) be the Laplace transform of the p.d.f. of an interval (not the first) in which a merge occurs. Then Q,*(s) is given by Q,“(s)
= s,*(S) [6*(s)]fi--“n*(s),
n> 2
since the first n- 1 intervals are rejected and the nth contains equation (5), takes the relatively simple form Q”(s) = Q,“(s)+
a merge.
(8) Hence Q*(s), by
g Q,“(s) ?&=a
=
Q
*ls)
1
+
so*(s) rl”(s) 1-6*(s)
To complete the calculation of 0*(s) we need to furnish expressions for 6,*(s) and 6*(s). These can be derived by the same sequence of steps leading to the expression for Q,*(s). The final results are *
*(s)
=
0
aey*(s>+ Nl-4 1 -(I
T”@) = 1 -(I_ s*(s) =
Po*~s)~*(s~
- @Y*(s)
P *cd 3 Qy*(s)
ef*(s)y*(s) i -(I - eyrys)
which, together with equation (9), contributes a formal solution to the problem. It can be verified that Q*(O) = 1 as is required. It is now a simple matter to calculate moments of delay by using equation (9) as a moment-generating function. Thus, the expected delay is
198
H. WEISS
GEORGE
Detailed use of equations in terms of quantities
(7). (9) and (10) leads to an expression
for i that can be written
r/3
PC_? =
~p&)dr,
I‘
a =
tY’(t)dt
(12)
.o
(13) The first term is, of course, the expected delay in the absence of travelling queues, second and third terms give the effect of trucks. Finally, we consider the question of relating 0 to the actual percentage of trucks main road. Let this percentage be denoted by p. Let the average number of cars in be rig. Let us consider a time interval between two trucks. The expected number not m queue (that is, who are following the last car in the queue) is I - B)‘b = (1 - eye
0&(
so that the total expected number of cars in an interval Hence the fraction of trucks on the main road is 1 r?,+(l-
on the a queue of cars
(14)
between two trucks is (1 - Q/8 + Cu.
e
=-=
1+&i,
e>/e+ 1
and the
(15)
p
so that 6’ is given by (16)
H = P/(1 - p$) Although we have restricted our considerations to the small p one can see that if is, is large, B is not necessarily small.7 As an example of the foregoing theory let us consider a average gap between successive cars in a queue is TQ set so that acceptance function will be taken as a step function a(t) = H(tand H(x) = 1 for x > 0. With this set of functions the formula
+de2’1~-
1) T,
case (relatively
traffic stream in which the pQ = TQii, (set). The gap T) where H(x) = 0 for x < 0 for the mean delay is
fjo (17)
l-p?ia For an idea of the meaning T = 4 set and TCa= 2 sec.
of this formula
few trucks),
let us take p = 3 set (a flow of 1200 cars/hr),
Figure 2 shows a graph of I as a function of Es for these parameters and for p = 0.01 and 0.02. It is clear that if passing on the main highway is at all difficult so that queues tend to build up, the waiting time can be increased considerably. The effects of mixed traffic are likely to be much more pronounced when the queues themselves can contain trucks in addition to cars, since then the queues tend to be much longer. t Note
that
in this theory
the maximum
value
that
p can have
is prnnx = (1 +&-‘.
The single lane merging problem with mixed cars and trucks
L 0
FIG. 2.
1
1
/
1
I
5
IO
15
20
25
Curves of expected delay with a small admixture
1
30
of trucks.
REFERENCES Cox D. R. (1962). Renewal Theory. Methuen, London. GWYNN D. W. (1967). Truck equivalency. Highw. Res. Rec. 199, 79. MILLER A. J. (1962). A queueing model for road traffic flow. JI R. statist. Sot. 23, 6475. WEISS G. H. and MARADUDIN A. A. (1962). Some problems in traffic delay. Ops Res. 10, 74-104.
199