Transportation Planning and Technology, 2014
Vol. 37, No. 1, 3–19, http://dx.doi.org/10.1080/03081060.2013.844907
The effect of green time on stochastic queues at traffic signals
Nicholas B. Taylor* and Benjamin G. Heydecker
Centre for Transport Studies, University College London, Gower Street, London WC1E 6BT, UK
(Received 10 March 2013; accepted 25 July 2013)
Many analyses of traffic signal queues use Webster and Cobbe’s formula, which
combines the net effect of the red/green cycle with a term representing stochastic
effects, idealised as an M/D/1 queue process having random arrivals and uniform
service. Several authors have noted that this component should depend not only on
demand intensity but also on throughput capacity in each green period, although an
extra empirical term may partially allow for this. Extending the service interval in M/
D/1 (M = Markovian, i.e. random, D = deterministic, i.e. uniform, 1 = one server)
enables the effect to be reproduced, but no exact expressions for its moments are
found. Approximate formulae for the extended mean exist but are accurate only near
saturation. The paper derives novel approximations for the equilibrium mean and also
variance and utilisation, using functions linking traffic intensity with green period
capacity. With three moments, equilibrium probability distributions can be estimated
for which a method based on a doubly nested geometric distribution is described.
Keywords: signal; queue; stochastic; M/D/1; variance; probability distribution
Introduction and background
Real signalised junctions are complicated by demand-responsive timings, conflicting
movements and coordination. Microscopic simulation, which need only specify short-term
individual behaviour, is used increasingly. Nevertheless, the formula of Webster and
Cobbe (1966) for queue size or delay at an isolated signal is still widely regarded. In
addition to red and green phase component, it contains a term representing stochastic
effects, including transient overload. This is equivalent to an idealised queuing process
where customers arrive randomly and are served at uniform intervals. Several authors,
including Olszewski (1990), have pointed out that this overestimates the true stochastic
queue component, which simulation shows falls, albeit slowly, with increasing throughput
capacity in the green period. Webster’s formula has an empirical term that may compensate
for this effect and can be related to a more exact formula. However, it is difficult to
integrate these with computationally efficient time-dependent approximate methods.
After describing the basic stochastic process, an extension is developed to take
account of green period capacity. Simulation results based on it are used to derive
approximations to the queue’s equilibrium utilisation, mean and variance in forms
compatible with time-dependent queuing methods, which are shown to be more
consistent than earlier empirical approximations. A novel approach makes use of link
functions between traffic intensity and green period capacity, allowing a broad physical
*Corresponding author. Email:
[email protected]
© 2013 The Author(s). Published by Taylor & Francis.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.
org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited. The moral rights of the named author(s) have been asserted.
4
N.B. Taylor and B.G. Heydecker
interpretation. Comparison with simulation shows that the results are accurate over a
range of traffic intensities and green period capacities. A method of estimating a
probability distribution from the three moments is described that may have wider
application. Finally, the theoretical nature of the analysis is justified, and various
conceptual and practical issues are discussed.
The mean queue size at a signal
Allsop and Hutchinson (1972) trace the origins of signal queue analysis back to A J H
Clayton in 1940, as well as discuss the impact of different assumptions about arrival
patterns. A well-established expression for the mean queue size L at a fixed-time traffic
signal is Equation (1), ascribed to Webster (Webster and Cobbe 1966):
L ¼ L P þ LV þ L W ¼
xlcð1 KÞ2
x2
þ
2ð 1 x Þ
2ð1 xKÞ
1
0.65ðxlcÞ3 xð2þ5KÞ
ð1Þ
The queue size in vehicles is composed of a phase term LP visualised as growing linearly
during the red phase and discharging linearly and fully during green, a stochastic and
overload term LV, and an empirical adjustment LW. Other variables are defined below as
follows:
c
g
r
Λ
s
µ
x
signal cycle time;
green time within each cycle;
red time within cycle;
green cycle time ratio = g/c (avoiding possible confusion with demand rate λ);
saturation flow, the maximum flow rate across the stop-line;
= gs/c, capacity of the movement taking account of the green/cycle ratio;
average degree of saturation or utilisation of service at the stop-line.
Webster’s formula strictly applies to Poisson arrivals at an isolated signal, so excludes
factors like minimum headway, vehicle actuation, platooning and coordination. However,
the concern here is solely with the effect of green capacity on the stochastic term and its
form in relation to existing time-dependent methods. Other variables used in the paper are
the following:
G = gs = µc, the number of vehicles1 that can pass in a single green period
ρ = λ/µ, demand intensity, relative to capacity
Note that G is present effectively as µc in the first and last terms of Equation (1) but not
the stochastic term. The degree of saturation x cannot exceed 1, so only the stochastic
term can accommodate indefinite queue growth and then in principle only by being timedependent. A more integrated approach is the derivation from first principles of
Heidemann (1994), using a generating function and the results of Meissl (1963).
Heidemann shows this gives results imperceptibly different from Webster and Cobbe’s for
values of G up to 45. If equilibrium conditions are imposed, the queue formula can be
reduced to a form analogous to Equation (1) containing the identical phase term LP:
Kð1 xÞ 2LV ½G þ Kx þ Kx
ð2Þ
L ¼ LP þ LV ½G þ LH where LH ¼
2ð1 KxÞ
Transportation Planning and Technology
5
The ‘exact’ stochastic term LV[G] here is a complicated expression that requires numerical
evaluation of the roots of a function of degree G. However, when G = 1 it reduces to LV.
The third term, LH, is free of empirical constants, and test calculations suggest it is
generally small. However, Equation (2) is again inconvenient for time-dependent traffic
modelling because of the intractable stochastic term, so a simpler expression is sought.
Dependence of the stochastic queue on green period capacity
Olszewski (1990) uses Markov simulation based on transition probabilities, allowing for
a general arrival distribution and variable cycle time, to confirm observations by Newell
(1965) and Miller (1969) that the mean size of the stochastic queue at a signal decreases
systematically with increasing green period capacity2 G. Although the decline is gradual,
it can be substantial for long green times, and this trend appears to continue indefinitely.
For example, at 90% saturation (x = 0.9), the mean queue with 40 second green is half
that with a short (1–2 second) green. The US Highway Capacity Manual (HCM)
(Rouphail, Tarko, and Li 1996) and Australian time-dependent formulae (Akçelik 1998)
also contain empirical terms depending on green period capacity. Olszewski’s Figure 33
shows that his EVOL simulation results compare well with Newell’s formula, though less
well with Akçelik’s.
Some fundamental properties of queues and time-dependent approximation
All queues obey the time-dependent deterministic Equation (3) representing conservation
of units (customers, vehicles etc.), where L0 is the initial queue and x is the average
utilisation or degree of saturation over the period of development [0,t]. If the demand
intensity ρ < 1, the mean queue tends to an equilibrium value given by the Pollaczek–
Khinchin (P–K) mean queue formula (4) (e.g. Kleinrock 1975):
Ld ðx,t Þ ¼ L0 þ ðq
Le ð xÞ ¼ Ix þ
xÞlt
Cx2
1 x
ð3Þ
ð4Þ
For equilibrium to exist, Equation (4) must be finite, so Equation (3) must also remain
finite at equilibrium, implying that ρ < 1 and x → ρ. Service occurs only when a queue
is present, so at equilibrium, the average probability that the queue is zero is the
complement of the utilisation:
p0e ¼ 1
q
ð5Þ
Utilisation x represents the proportion of time that a queue is present at the stop-line or in
service. The forms of Equations (3–4) show that it plays a crucial role in queue
development, being the only quantity on the RHS that must vary with time and is capable
in principle of producing any finite size of queue.
The coefficient I in Equation (4) reflects unavoidable service time, which conventionally applies only to priority junctions, while C depends on the coefficient of variation of
service time.4 For a signal I = 0, while theoretically C = 0.5, giving LV as in Equation (1).
6
N.B. Taylor and B.G. Heydecker
Empirically, C is found to be in the range of 0.5–0.6 (Burrow 1987). When Equations (3)
and (4) are equated and solved for x or L, the result is the quasi-static ‘sheared’
approximation to time-dependent queuing, including stochastic effects, being defined for
all ρ, since x never exceeds 1 (e.g. Kimber and Hollis 1979). This has some accuracy issues
but is convenient for use in dynamic network assignment programs such as CONTRAM
(Taylor 2003). Shearing the entire signal queue formula is problematic, as found by Han
(1996). However, the methods described so far do not provide all the moments needed to
determine queue size probability distributions, enabling the risk of long tailbacks, or
spillback across upstream facilities, to be estimated. The rest of the paper, therefore, aims
to obtain variance along with mean.
The M/D/1 process as an idealisation of the stochastic queue at a signal
The M/D/1 queue (M = Markovian, i.e. random, D = deterministic, i.e. uniform, 1 = one
server) represents an idealised system where customers are served singly, but more than
one random arrival can take place in each fixed average service time interval 1/µ. The
effect of overflow from the red/green signal cycle is averaged out. In the M/M/1 process,
by contrast, arrivals and service both occur randomly at given mean rates, which is more
typical of a priority junction. Each process can be described by recurrence relations
between queue state probabilities, which can be animated as Markov processes (Kendall
1951). Both yield closed-form equilibrium moments, including means in the P–K form
Equation (4), making them convenient for use in time-dependent traffic modelling.
Arrivals at exponentially distributed random intervals result in the number of arrivals in
each service period being Poisson distributed. For M/D/1, the probability of i customers
being in the queue after µt + 1 service periods (i.e. that amount of throughput capacity) is
the sum of the probabilities of i + 1 being in the queue at service point µt and no arrivals in
the interval [µt, µt +1), i at µt and one arrival, i − 1 at µt with two arrivals, and so on.
Hence, the probabilities of queue states {pi} where i={0,1,2,…} evolve according to:
pi ðlt þ 1Þ ¼
iþ1 j
X
qe
q
j!
j¼0
piþ1 j ðlt Þ
ð6Þ
pi ðlt Þ
ð7Þ
Dpi ðlt Þ ¼ pi ðlt þ 1Þ
At equilibrium, by definition, Δpi(µt) = 0 for all i, so rearranging Equation (6) and allowing
for the ‘absorbing barrier’ at i = 0, which corresponds to periods when the queue is zero and
service is not utilised, the following equilibrium recurrence relations are obtained:
p1 ¼ ð e q
pi ¼ ð e q
qÞpi
1
q
1Þp0
i
X
qj
j¼2
j!
pi
ð8Þ
j
ði > 1Þ
ð9Þ
The first two terms in Equation (8) come from Equation (6) with i = 0, while the third can
be obtained from Equation (6) by setting i = −1 (notional ‘negative queue states’ feature
Transportation Planning and Technology
7
prominently later, and are discussed at the end of the paper). Moments of the equilibrium
distribution can now be got from Equations (8–9) by evaluating the next highest timedependent moment in each case:
1
X
Probability of zero queue by evaluating
ipi : p0e ¼ eq ð1 qÞ
ð10Þ
i¼0
Mean queue by evaluating
1
X
i 2 pi :
i¼0
Variance by evaluating
1
X
i¼0
i 3 pi :
Ve ¼
Le ¼
q2
2ð1 qÞ
q2 ð6 2q q2 Þ
12ð1 qÞ
ð11Þ
ð12Þ
The form of Equation (11) is that of the stochastic queue LV in Equation (1) and is a
particular case of Equation (4). However, Equation (10) is inconsistent with Equation (5).
This is because it applies at the end of the green period, at which time, p0 will be greatest
since it excludes transient queues that have come and gone during the green period. That
the end-of-green p0e is always greater than the average-over-green p0e (since eρ ≥ 1) is
consistent with this interpretation. This distinction does not arise where a queuing process
can be formulated using infinitesimal service periods, as in the case of M/M/1.
Extending the M/D/1 queue process to general green period capacities
In this paper, we use a common subscript notation for queue moments, where e indicates
equilibrium, [G] is added to indicate the dependence on green period capacity and if
omitted G = 1 is assumed, and a final letter, e.g. M, distinguishes a particular origin, for
example, the initial of an author, while an overscore indicates an average value and
notional negative queue state indices are placed in brackets.
The basic M/D/1 process describes a situation where only one customer can be served
in each green period (like a ramp-metre with a fixed cycle time). A more realistic model
Table 1. Conditions for specified final queue in green period with capacity G.
Initial queue state
Arrivals in green period
Zero queue i = 0 at end of green period
0
1
J
>G
Up to G
Up to G−1
Up to G−j
Not possible
Non-zero queue i > 0 at end of green period
0
1
J
>G+i
Exactly G + i
Exactly G−1 + i
Exactly G−j + i
Not possible
N.B. Taylor and B.G. Heydecker
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Le_M/D/1[G]
p0
8
r
0.25
0.4
0.5
0.6
0.7
0.8
0.9
r increasing
0
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
10 20 30 40 50 60 70 80 90 100
r
r increasing
0.25
0.4
0.5
0.6
0.7
0.8
0.9
10 20 30 40 50 60 70 80 90 100
0
G
G
Figure 1. Dependence of M/D/1[G] equilibrium p0 (left) and mean queue (right) on ρ, G from
simulation (imax = 10,000). Points measured from Olszewski (1990) are shown (EVOL).
should allow up to G customers to be serviced in each green period in which case the
conditions for final queues of specified sizes are as given in Table 1.
Table 1 suggests that while the development of the queue size probability distribution
{pi} for i > > 0 will be broadly similar to Equation (6), the expression for p0 will be more
complicated, involving several components. Cronjé (1983b) introduces negative net
contributions in each cycle, from −G to 0, representing actual arrivals minus green period
capacity and sums these terms to give the value of p0 at the end of the cycle. This
amounts effectively to employing notional negative queue states down to −G. By a
similar analysis to that for M/D/1, recurrence relations corresponding to Equations (6–9)
are obtained:
pi ðlt þ GÞ ¼
iþG
X
ðGqÞj e
j¼0
Gq
piþG j ðlt Þ
j!
ði
GÞ
ð13Þ
The presence of G on the LHS and the substitution of Gρ for ρ on the RHS reflect the fact
that all calculations relate to an idealised service period of G/µ in place of 1/µ. Equation
(13) is valid for the notional states provided these include a notional zero state. The real
(absorbing) zero state probability is got by summing all the notional probabilities,
including (0), Equation (14) and states I > G can be expressed in terms of real states
alone:
i¼0
X
p0 ¼
pð iÞ
ð14Þ
i¼ G
ðeÞ
G
pi ¼ eGq pi
i
X
ðGqÞj
j¼1
j!
pi
j
ði>GÞ
ð15Þ
However, there appears to be no equivalent formula for real states in the range 0 < i<G.
Nevertheless, all states can be simulated incrementally using Equations (13–14). Figure 1
shows how Markov simulated5 equilibrium p0 (left) and mean queue size (right) vary
with traffic intensity ρ and green period capacity G. These results are consistent with
Transportation Planning and Technology
9
Table 2. Representation of recurrence relations for G = 5 lower queue states.
G=5
Initial state
1
2
3 4
5
6
7
8
9
Final 0
state
0
(–5)
(–4)
1
0
(–3)
2
1
0
(–2)
3
2
1
0
(–1)
4
3
2
1 0
5
4
2 1
0
(0)
3
1
6
5
4
3 2
1 0
2
7
6
5
4 3
2 1
0
3
8
7
6
5 4
3 2
1
0
4
9
8
7
6 5
4 3
2
1
0
5
10 9
7 6
5 4
3
2
1
8
6
11 10 9
8 7
6 5
4
3
2
Notes: Numbers in cells = numbers of arrivals in green period;
mean arrival rate = Gr = 5r; departures in period = G exactly.
10
11
0
1
0
there being no positive lower limit on the mean queue size as G is increased (in practice,
of course, delay on conflicting streams facing long red times might outweigh this!).
Table 2 visualises the raw recurrence relations for G = 5, where notional states appear
in the final state (leftmost) column as bracketed indices, and the real initial states on
which they depend are shaded. The figures in interior cells are numbers of arrivals, which
translate into Poisson coefficients of the initial probabilities (columns) as in Equation (13).
The sum of terms in the column for an initial state k in Table 2 is given by:
Kk ¼
"
1
X
ði þ k
ipð
iÞ
i¼0
ðGqÞi e
GÞ
i!
Gq
#
pk ¼ ½Gq þ k
G pk
ð16Þ
The total of these columns must equal that of the final state (leftmost) column, so:
1
X
ipi
1
G
X
0
¼ Le½G
G
X
0
ipð
iÞ
¼
1
X
0
Kk ¼ Le½G
Gð1
qÞ
ð17Þ
In Equation (17), Le[G] is the mean steady-state queue. Therefore, the mean of the
notional probability terms, which sum to the real p0, satisfies Equation (18).
Lð Þ ¼
G
X
0
ipð
iÞ
¼ Gð1
qÞ or
p0e ¼
G
1 X
ipð
G 0
iÞ
¼ ð1
qÞ
ð18Þ
Thus the average probability of zero queue during the service period is consistent with
the deterministic queue Formula (3) and Equation (5). For G = 1, the simplicity of the
10
N.B. Taylor and B.G. Heydecker
formula for p0e, Equation (10), is the happy result of there being two equations to solve
for two unknowns. For G > 1, a simple formula for p0e appears not to exist, and
calculating the variance of notional states does not give a value for p0e either, although
the following hold:
pð
var pð
GÞ
¼e
iÞ ½G !
Gq
ð19Þ
p0e
Gq ðp0e ! 1Þ
ð20Þ
Earlier empirical approximations to the mean stochastic queue
Given the computational cost of simulation, Miller (1969) proposed the following
empirical approximation to the stochastic mean queue (with notation modified to be
consistent with that used throughout this paper):
Le½GM ¼
1
2ð 1
qÞ
exp
4y
3q
where y ¼ ð1
pffiffiffiffi
qÞ G
ð21Þ
Newell (1960) also devised an approximation, which Miller considered ‘too complicated’. Cronjé (1983a) offers (without further explanation) a ‘suggested modification to
Newell’:
Le½GC ¼
Ia q exp y 12 y2
where y is as defined in Equation.
2ð1 qÞ
ð22Þ
In Equation (22), Ia represents dispersal of arrivals, but the P–K Formula (4) does not
normally allow for this, and it seems a somewhat arbitrary addition. We, therefore, drop
the factor Ia in what follows. The Equations (21–22) appear to be optimised for ‘heavy
traffic’, i.e. ρ≈1 but not exceeding 1, as is often the case in the literature, possibly
because queue modelling is of most practical interest around saturation. In both cases, the
exponential terms, which in principle ought to yield the M/D/1 mean when G = 1, do so
approximately only for ρ≈1. However, the Cronjé–Newell method is accurate for higher
values of ρ, making it attractive as the basis of an approximation adjusted to give
improved results for smaller values of ρ. Rewriting Equation (22) as the basic mean
queue multiplied by a factor fC, as in Equation (23), an adjustment can be explored
empirically by examining the behaviour of its error, Figure 2.
Le½GC ¼ fC ðq,GÞLe½G¼1
where fC ðq,GÞ ¼
exp
y
q
1
2
y2
ð23Þ
Since logarithm of the factor error escalates rapidly but linearly for values of ρ <~ 0.5, the
simplest correction is a bi-linear term embedded in an exponential function (24):
Transportation Planning and Technology
10
11
0.25
Slope
0.2
r
FiƩed
0.15
Slope
r
Factor error
ing
as
cre
In
0.25
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0.05
0
1
0
5
10
G
15
20
Model
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
r
–0.05
25
1
Figure 2. Errors in Cronjé–Newell factor (left) and relationship of slopes v. G to ρ (right).
1
0.9
0.8
0.7
Model
0.6
0.5
0.4
0.3
Cronje-Newell
0.2
Adjusted
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SimulaƟon data
Figure 3. Performance of normalised original and adjusted Cronjé–Newell methods.
Le½Gest ¼ Le½G¼1 min expðmaxð0.4
0.75q,0ÞGÞ
exp
y
q
1
2
y2
,1
ð24Þ
Figure 3 shows that the adjusted approximation of Equation (24) outperforms the original
Equation (23), but while queues produced by low traffic intensities ρ < ~0.5 could be
12
N.B. Taylor and B.G. Heydecker
considered negligible for practical purposes, it is clear that some underlying structure has
not been captured.
Link-function approach
The adjustment to Cronjé’s method in Equation (24) is unsatisfying because apart from its
one change of gradient, it conveys the message that no practical smooth function can
represent the errors in a way that gives insight into an underlying structure.
Experimentation reveals that the trends of p0e[G], Le[G] and Ve[G] for various values of ρ
and G can be made to overlap by transforming them using ‘link functions’ of the form of
Equations (25–26).
z ð hÞ ¼
where sre ¼ l
1
Gþh
Gþh
lsre ðqÞ sre1 ðqÞ
pffiffiffi
q
1
2
ð25Þ
ð26Þ
is a relaxation time.
For the sake of clarity, the dependence of z on ρ and G is, henceforth, ‘understood’. The
quantity τre is frequently cited as the stochastic relaxation time of a queue tending
towards equilibrium. The variant τre1 = µτre is a dimensionless quantity which depends
only on the demand intensity ρ. Hence the function z is also dimensionless and is
independent of µ.
Figure 4 plots link-transformed p0e[G] against z(2) for a range of values of ρ and G,
showing how closely the points lie on a common parametric curve. For G = 1, p0e and ρ
are already related by Equation (10). Because those points are dispersed along the graph,
extension to G > 1 is immediate by defining an ‘effective ρ′, η0:
p0½Gest ¼ eg0 ð1
1
1
0.9
0.9
0.7
0
0.6
0.5
0.4
0.3
0.2
0.1
ð27Þ
where
0.8
r
0.7
0.6
Expanded
horizontal
scale
0
r
0.25
0.4
0.5
0.6
0.7
0.8
0.9
Model
p
0.8
p
g0 Þ
0.5
0.4
0.3
0.2
0.25
0.4
0.5
0.6
0.7
0.8
0.9
Model
0.1
0
0
0
1
2
3
4
5
6
Z
7
8
9
10
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Z
Figure 4. Plots of Markov-simulated p0e against link function z which depends on ρ and G.
1
Transportation Planning and Technology
1
1
0.9
0.6
0.5
0.4
0.3
r
0.25
0.4
0.5
0.6
0.7
0.8
0.9
Model
0.8
0.7
Vel/Ve(1)
0.7
h/r
0.9
r
0.25
0.4
0.5
0.6
0.7
0.8
0.9
Model
0.8
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
13
0
0
0.5
1
1.5
2
0
2.5
0.5
1
Z (0)
1.5
2
2.5
Z (1)
Figure 5. Plots of link-function mapping for simulated Le[G] (left) and Ve[G] (right).
g0 ¼
rffiffiffiffiffiffiffiffiffiffiffiffi
G þ 2
1
3
max 1
pffiffiffi
q ,0
!!2
ð28Þ
To estimate the stochastic mean queue, an ‘effective ρ′ for each simulated queue value is
estimated by inverting Equation (11). This quantity can be considered to represent the
effective utilisation, distinct from the end-of-period p0 given by Equation (27). For G > 1,
as Figure 5 (left) shows, values of its ratio to ρ as a function of z(0) fall roughly onto a
common exponential curve, leading to Equations (29–30). The normalised equilibrium
variance (right) also has an exponential appearance and can be approximated directly by
Equation (31), an ‘effective ρ′ not being required. Upper limits ensure nominally correct
results in the case G = 1. Note that both η0 and η1→1 when ρ→1, so they behave like real
traffic intensities.
Le½G ¼
g21
2ð 1
g1 ¼ q min exp
Ve½G Ve½1 min exp
G
ð29Þ
where
g1 Þ
,1
and also
sre1
Gþ1
,1
3
sre1
ðfor G > 1Þ
ð30Þ
ð31Þ
Figure 6, comparing Markov simulated and estimated moments, shows that that the
family of approximations described earlier gives good results over the range of parameter
values tested, namely, ρ∊[0.25,0.9] and G∊[1,100].
Estimating equilibrium queue size probability distributions
Based on his simulation results, Olszewski (1990) proposes Gamma or Negative
Binomial distributions to represent the time-averaged queue size probability distribution
over a peak. The Gamma distribution has been proposed elsewhere for describing queue
14
N.B. Taylor and B.G. Heydecker
Le (based on η)
p
Ve
5
25
4
20
Model estimated
0
Model estimated
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
3
2
1
10
5
Markov simulated
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
15
0
1
2
3
0
4
5
0
5
10
15
20
25
Figure 6. M/D/1[G] simulation and estimation compared for 25 pairs of ρ and G.
size distributions during oversaturated peaks (Halcrow Fox and Associates, under
contract to TRL, unpublished report). It has several advantages including having two
parameters for fitting and the exponential distribution (continuous equivalent of
geometric) as a special case. The extended equilibrium probability distributions of
M/D/1[G], including the notional terms, have a Gamma-like appearance, and manual
fitting demonstrates that a close fit can be obtained for higher values of G, as shown by
Figure 7. Poisson or LogNormal functions are unsatisfactory because they retain too
much skewness, and their asymptotic behaviour is not exponential.
A simpler approach can be used for the real probabilities only by noting that,
sufficiently far from the zero state, any equilibrium distribution is geometric, with a
constant ratio between successive discrete state probabilities. This can be ascribed to a
need for local invariance of the form of the distribution with change of viewpoint.
Certain equilibrium distributions in the literature have a nested form, involving
two parameters instead of just ρ, where the geometric sequence applies only to queue
sizes >0. However, three moments, p0, L and V are needed to estimate queue size
0.3
1
2
5
10
20
Gamma 1
Gamma 2
Gamma 5
Gamma 10
Gamma 20
M/D/1 Probability
0.25
0.2
0.15
0.1
0.05
0
0
5
10
15
20
25
30
i+G
Figure 7. Gamma fits to M/D/1[G] extended distributions for ρ = 0.8, G up to 20.
Transportation Planning and Technology
15
probability distributions (Taylor 2005), and three parameters are required to fit three
known moments. With three parameters as in Equation (32), the distribution becomes
doubly nested geometric, whose first and second moments are given by Equation (33):
p0e ¼ 1
Le ¼
q
p1 ¼ q ð1
pi ¼ q q^ð1 ~qÞ~
qi
q^Þ
2
ði 2Þ
q ð1 þ q^ ~
qÞ
q ð1 þ 3^
q ~
qð2 þ q^
Ve þ L2e ¼
1 ~
q
ð1 ~
qÞ 2
ð32Þ
~
qÞÞ
ð33Þ
By inverting the moments in Equation (33), the ‘effective ρs’ are obtained as
Equations (34):
q ¼ 1
p0e q^ ¼
ðVe þ Le ðLe 1ÞÞð1
2q
~
qÞ2
~
q¼
Ve þ Le ðLe 3Þ þ 2q
Ve þ Le ðLe 1Þ
ð34Þ
This will generally not work for time-dependent queue probability distributions, which
can have Normal or bi-modal shape. An indicator of failure is that q^ > 1. However,
fitting a non-monotonic distribution where p1 > p0 is still possible, though not relevant
to the present topic. Examples of fitted distributions for two values of G are given in
Figure 8, showing that while p1 is not fitted exactly, the shape of the distribution is fairly
reproduced.
In principle, a fourth nesting level would allow p1 to be fitted precisely. However,
unlike p0, which is related to the utilisation of service, p1 cannot be calculated by
analytical approximations such as shearing. Apart from this, a fourth moment, as required
for a fully specified problem, is unlikely to be available from analytical approximations.
So there would seem to be little benefit from making the method more complicated.
The doubly nested geometric distribution is limited to unimodal cases with mode ≤1.
However, being based on general principles, it can be applied to any equilibrium
distribution where all three ‘effective ρs’ are less than 1, such as those arising from
Erlang-k bunched or staged arrivals or service. It cannot be assumed without justification,
as is sometimes done, that equilibrium results can be applied to transient behaviour
r =0.9, G=2
r =0.9, G=10
0.3
Nested
M/D/1[G]
0.25
M/M/1
pi
pi
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
i
6
7
8
9
10
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
Nested
M/D/1[G]
M/M/1
0
1
2
3
4
5
6
7
8
9
10
i
Figure 8. Fits between doubly nested geometric and Markov simulated M/D/1[G] distributions,
with M/M/1 for comparison (lower graphs).
16
N.B. Taylor and B.G. Heydecker
(Sharma 1990). One of us (Taylor) has also been investigating ways of fitting functions
representing instantaneous probability distributions to dynamic queue moments.
Discussion
No new work has been found relevant to the topic of the effect of green period capacity
on signal queues since the derivation by Heidemann (1994), possibly because the
problem has been considered solved or of limited practical importance compared to the
effects of optimisation, adaptive timing, platooning, coordination, etc. State-of-the-art
reviews by Rouphail, Tarko, and Li (1996) and Akçelik (1998) quote Newell’s
approximation, and the empirically corrected HCM and Australian formulae developed
in the 1980s, but do not propose any new methods, possibly because existing ones are
considered sufficient.
While acknowledging this, the motivation for this paper lies in widening the range of
queue types that can be handled by computationally efficient time-dependent methods
like shearing, and in predicting variance and probability distributions along with means.
That the green period effect emerges naturally from Heidemann’s derivation indicates that
it is more than just a convenient idealisation. Given that M/D/1 is taken to be a model of a
stochastic signal queue, it would seem remiss to ignore the effect of something so
essential.
Aside from computational efficiency, the need for simplifying approximations in
queue calculation arises from the difficulty of describing even the simplest queue
processes. Morse (1958) gives an exact formula for the time development of probabilities
of the simplest M/M/1 random queue as a potentially infinite series of exponential or
Bessel functions. Even this only describes development from a precise initial size. In
realistic calculations, convolution with an initial probability distribution is required.
Kleinrock (1975) calls this situation ‘disheartening. While common methods exist for
analysing queuing, such as the P–K transform approach (e.g. Kleinrock 1975), recent
research tends to be microscopic and complex (e.g. Mirchandani and Zou 2007).
A feature of queuing, as of nature generally, is that simple relationships may emerge
from a complex process through general principles of conservation and symmetry.
Equilibrium results like that in Equation (4) are ‘emergent’ in the sense that exact
formulae need not exist for the equilibrium mean of a general queuing process, since it is
the limiting outcome of an infinite sequence of random events repeated infinitely many
times. Yet all queue processes must obey deterministic conservation Equation (3), which,
therefore, cannot say anything about any result that depends on the details of the process.
An equilibrium queue is defined only for traffic intensities 0 ≤ ρ < 1 and is unbounded
as ρ → 1. Thus it should be possible to normalise equilibrium moments by factoring by
finite expressions in ρ and G since this maps an infinite range into an infinite range. A
physical interpretation is that random arrivals in green periods of different duration
should on average have a similar impact on the stochastic queue at the end of a period,
provided that the relationship between green time and the characteristic stochastic time
scale is the same. The link-function approach exploits this symmetry. But constant factors
are not necessarily appropriate for time-dependent queues. Expressing results in terms of
generic formulae (3) and (4) with modified parameters – in the present case ‘effective ρs’
– allows the use of existing computationally efficient time-dependent methods.
On what basis can it be claimed that ‘effective ρs’ can be substituted directly into the
time-dependent methods? Physically, the longer the green period, the more traffic can
Transportation Planning and Technology
17
‘disappear without trace’ before the queue is assessed at the end of the green period. This
translates into a reduction in the effective demand intensity and degree of saturation,
which are the critical variables in time-dependent queuing. The quasi-static principle
relies on the rate at which information propagates through a queue being much greater
than the rate of change of the queue itself so that the static relationship between queue
size and degree of saturation can be assumed to apply approximately to dynamic cases.
As long as these assumptions hold, it should be possible to apply methods like shearing
with some confidence. Some observational validation of shearing was done by Kimber
and Daly (1986). Validation of the extensions proposed here is part of a larger question
concerning time-dependent methods, which involves considering dynamic probability
distributions or at least their moments, currently being addressed by one of us (e.g.
Taylor 2005).
The primary impact of the work described here is, therefore, greater consistency. The
extent to which approximate time-dependent methods can accommodate realistic factors
such as platooning, coordination and finite capacity queues has yet to be determined,
though pursuance of research may depend on whether macroscopic modelling is
considered likely to have a role as against microscopic simulation, an issue which is
still disputed (Wood 2012). An ad hoc method of accounting for green waves was used in
CONTRAM (Taylor 2003), but manipulation of parameters in an extended P–K mean
formula representing arrival and service statistics would be preferable. On the other hand,
a purely statistical model like the P–K formula may be inappropriate to cases where
variability of demand is no longer simply random (Chow 2013). Whether efficient timedependent methods can continue to absorb real-life complexities through the implementation of general principles may be a question for further research.
Conclusion
This paper responds to a perceived need to: (1) broaden the range of queue processes that
can be modelled using computationally efficient time-dependent methods based on closed
formulae; (2) incorporate variance and in particular (3) take account of the observation
that the size of a stochastic signal queue depends on the green period capacity and not
just the ratio of green to cycle time through the demand intensity.
An idealised stochastic signal queue process that takes account of green period
capacity has been formulated by extending the basic M/D/1 model embodied in some
existing signal queue formulae. Simulation using Markov methods has been used to
generate test cases to compare with existing empirical approximations and to verify novel
formulae for equilibrium values of the probability of zero queue, mean queue and
variance in a form compatible with computationally efficient macroscopic time-dependent
methods.
It is thought on structural grounds that the realism of transient behaviour using timedependent methods should be similar to that for the basic process. Three moments enable
probability distributions to be estimated and a simple doubly nested geometric
approximation has been defined and demonstrated.
Acknowledgements
The work described in this paper forms part of the first author’s research towards Ph.D. under the
supervision of the second author. An early version of the paper was presented at the Universities
Transport Studies Group (UTSG) Conference at Oxford, January 2013. Helpful comments by the
18
N.B. Taylor and B.G. Heydecker
editors and two anonymous reviewers and the general support of Dr Alan Stevens Transportation
Chief Scientist at TRL are gratefully acknowledged.
Notes
1. Strictly, capacity in vehicles should vary with traffic composition.
2. G as used here corresponds to Olszewski’s B.
3. We were unable to obtain permission to reproduce this, but some points taken from it are shown
in Figure 1 later.
4. Some authors make C include the dispersion of arrivals also, but this does not emerge from the
derivation.
5. The Markov simulation program used, which evaluates recurrence relations using small time
steps, was developed by the first author with algorithm design and programming assistance by
Neil H Spencer, then a sandwich student at TRL.
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