=Paper=
{{Paper
|id=Vol-2393/paper_310
|storemode=property
|title=Reliability of Adaptive Traffic Lights Ensured by Warm Standby with Estimation of its Use
|pdfUrl=https://ceur-ws.org/Vol-2393/paper_310.pdf
|volume=Vol-2393
|authors=Oleksandr Makarichev,Peter Horbachov,Oleksandr Voronkov,Stanislav Svichynskyi
|dblpUrl=https://dblp.org/rec/conf/icteri/MakarichevHVS19
}}
==Reliability of Adaptive Traffic Lights Ensured by Warm Standby with Estimation of its Use==
https://ceur-ws.org/Vol-2393/paper_310.pdf
Reliability of Adaptive Traffic Lights Ensured by Warm
Standby With Estimation of its Use
Oleksandr Makarichev1[0000-0003-2442-1208], Peter Horbachov2[0000-0002-8180-4072],
Oleksandr Voronkov3[0000-0003-2744-7948] and Stanislav Svichynskyi4[0000-0002-8549-1712]
1, 2, 3, 4Kharkiv National Automobile and Highway University, Kharkiv 61002, Ukraine
gorbachov.pf@gmail.com
Abstract. For the case when quantitative characteristics of traffic flows and the
failure rates of traffic lights’ elements are known, quantitative relationships for
the feasibility analysis of warm standby of the adaptive traffic lights at intersec-
tions have been found. The equations and recurrence relations of their probabil-
ity distributions are obtained as the Laplace transform of sequences of times of
limited permissive phases of alternative movement directions. The intelligent
system of traffic control with the adaptive traffic lights as the main control unit
is considered as a single-line queuing system with the FIFO (first in, first out)
service discipline for the Markov flow of recoverable components’ failures.
As a result, two-way estimate for the failure probability of the adaptive traf-
fic lights is obtained and the effect of warm standby use for the traffic lights at
two-way stop-controlled intersection is assessed. In the presented example wait-
ing time for the vehicles at the intersection with adaptive controlled traffic ap-
peared to be considerably less than in the case when there is no adaptive traffic
control. It allows to highlight advantages of warm standby use as a mean to en-
sure the reliability of the adaptive traffic lights.
Keywords: Warm Standby, Reliability, Markov Flow, Adaptive Traffic Lights,
Laplace Transform.
1 Introduction
Adaptive traffic lights are the most common and most powerful components of con-
trol in intelligent traffic management systems (ITMS). It allows actuating of time
phases depending on actual traffic demand and, compared to pre-timed control, it
significantly reduces traffic delays at intersections when properly configured. Though
the advantages of ITMSs are obvious, ensuring the reliability of their work is not
explored enough due to the absence of critical consequences when traffic lights fail,
since this situation is foreseen by the traffic codes and is not a direct cause of traffic
accidents. The failure of the traffic lights leads to a deterioration in the passage condi-
tions at the intersection and an increase in vehicle delays. Pedestrians do not suffer
from the failure as they receive a preferential right to cross the street via unregulated
crosswalks.
2
Ensuring the reliability of light signaling systems is of considerable attention in
railway transport, because railway signaling systems are used to ensure the safe op-
eration of railway traffic [1, 2]. As a result, a lot of effort has gone towards ensuring
the reliable operation of railway signaling systems that have to be designed to avoid
single-point failures [1].
A new impetus to ensure reliability of traffic lights is added by LED (Light-
Emitting Diode) arrays as a color signal emitter. LED emitters have allowed to mini-
mize power requirements for the functioning of traffic lights, while expanding the list
of standbys for their functioning with the ideal switching arrangement (warm
standby), which makes it possible to increase the reliability of adaptive traffic lights
in urban ITMSs.
A redundant warm standby is a very promising means of ensuring the reliability of
ITMS comparing to a cold standby since traffic lights are always dispersed over fairly
large territories. This leads to randomness of the recovery time of failed items, which
leads to high expenditures due to the need of the repair crew to get to the failed traffic
lights. Police and signal maintenance crews can often be stretched too thin when re-
sponding to power failure situations, especially in cases of concurrent failures at a
large amount of intersections [3]. Throughout this time, the ITMS does not work, that
causes undue delays of vehicles. At the same time, the warm reservation of traffic
lights’ elements allows to postpone the process of restoring its failed items until a off-
peak traffic period, when it does not lead to negative consequences for road users.
By now two warm standbys for traffic lights are known: redundant LED groups to
allow indication in degraded mode [4, 5] and alternative power sources as backup
power to maintain normal signal operations during power outages [3]. The list can be
expanded if a metric to quantify results of warm standby is defined since a cost of
new units is always known and both variants of reservation are expensive enough.
Transportation agencies, facing limited budgets, need decision-making support when
solving the question whether the traffic operation can be improved by installing warm
standbys for traffic lights – and if so, by how much.
Statistical models of road accidents [3] and a point system to score each site [6] are
not a sufficient basis for making decisions in this area. The number of reported road
accidents is relatively low and when traffic lights are fault there is no way to statisti-
cally quantify the expected number of accidents [6]. The point system includes traffic
volume, frequency of injury accidents, proximity to a school zone, speed of approach
traffic, and availability of pedestrian pre‐emption controls and provides prioritization
sites only [6]. The Markov technique is the most suitable for modelling road signaling
systems in which the level of redundancy varies with time due to component failure
and repair [1].
2 Laplace transform of the distribution function of the
intersection busy period
It is possible to evaluate the feasibility of warm standby use to ensure the reliability of
adaptive traffic lights through obtaining quantitative estimates for both the probability
3
of failure of adaptive traffic lights with a different number of standbys for one operat-
ing item, and for the negative consequences of failure from the road user’s point of
view.
The first quantitative estimates should be obtained for the most common intersec-
tion of two nearly perpendicular roads. When traffic is controlled by traffic lights, the
times of permissive phases in alternative intersecting directions are limited. Adaptive
traffic control at the intersection allows switching of the permissive signal between
alternative directions depending on current circumstances at the intersection. That is,
when, with the permissive signal turned on in one direction I , there are no vehicles
for moving in it, and the queue of vehicles is on the other direction II . Vehicles can
pass through the intersection as long as they are at a given spacing or if there are no
vehicles coming from the other direction, but the permissive phase should not exceed
the time limit for the corresponding direction.
We have:
─ BI
I
t is the distribution function for the time of passing over the intersection
from the start of the permissive signal in direction I;
I s exp st dBI t is the Laplace transform of this variable;
I I
t 0
I
─ I t is the distribution function for the intersection’s busy period as a single-
line queuing system, which begins with the I vehicles’ departure and the flow
rate I of the vehicles arriving in the direction I of the intersection;
─ I s exp st d I t is the Laplace transform of this variable.
I I
t 0
We use the additional event method according to the total probability rule. Provid-
ed that passing-over I vehicles from direction I take the time t , and during this
time t exactly n cars have arrived at the intersection in direction II , the conditional
probability that the additional event, the rate of which equals s , does not occur during
this busy period t and during the n busy periods, generated by the arrival of these n
cars, is equal to I1 s exp st . Then we multiply it by the probability of the
n
arrival of exactly n cars in the direction I during this busy period t
I t
n
exp I t .
n!
I t
n
exp I t I1 s exp st for all
n
The sum of all these multiplications
n!
non-negative integer numbers of incoming cars n 0 is equal to
I t
n
exp I t I1 s exp st
n
(1)
n0 n!
4
Then we integrate the obtained expression over all t 0 values with the distribution
dBI
I
t for the time of passing over I vehicles from the moment of the ena-
bling signal in the direction I :
I t
n
exp I t I1 s exp st dBI
I
t .
n
(2)
t 0 n0
n!
As a result, we obtain the equation
I t
n
I
s exp I t I1 s exp st dBI t .
n I
I (3)
t 0 n0
n!
After identical transformations we come to the following:
I t
n
exp I t I1 s exp st dBI t
I n
t 0 n0
n!
I t I1 s
n
exp t exp st dB I t
I I
t 0 n0 n !
I t I1 s
n
exp I t exp st dB I t
I (4)
t 0 n0 n!
exp I t exp st exp I t I1 s dBI t
I
t 0
t 0
exp s I 1 I1 s dBI
I
t .
exp s 1 s dB
I
By definition, the integral I
1
I I t is equal to the
t 0
Laplace transform I I s I 1 I1 s at the point s 1 s : I I
1
exp s 1 s dB t I I s I 1 I1 s .
1 I
I I I (5)
t 0
Then we have a functional equation for the Laplace transform as a result:
I s I s I 1 I1 s .
I I
(6)
In particular, with I 1 ,
I1 s I1 s I 1 I1 s . (7)
5
By definition, it is considered that the time of the permissive phase in the direction I
I
is limited by the value TI . Therefore, a random variable with a distribution I t
of the intersection busy period limited by the value TI to the segment 0; TI , which
began with vehicles passing over the intersection in the direction I with the flow rate
I , will have distribution function as follows:
I
I
t
I
I
TI t I
I 0 t TI I t TI (8)
I TI
and the Laplace transform
TI
I
I
TI
I
exp st d I t
I TI s exp st d I TI t 0
. (9)
I
I
0 TI
During the intersection busy period limited by the value TI , which began with I
vehicles passing over the intersection in the direction I with the flow rate I , that is,
during this permissive phase for vehicle direction I , some vehicles may arrive from
the perpendicular direction II .
Next, we find the Laplace transform of the intersection busy period, which is
formed by movement in direction II generated at the intersection by the vehicles
arriving in this direction with flow rate II during the previous permissive phase for
direction I , using the total probability rule according to the method of the additional
event, the rate of which equals s .
Provided that passing-over I vehicles from direction I take the time t , and
during this time t exactly n cars have arrived at the intersection in direction II , the
conditional probability that the additional event, the rate of which equals s , does not
occur during this busy period t and during the n busy periods, generated by the arri-
val of these n cars, is equal to I1 s exp st . Then we multiply it by the prob-
n
ability of arrival of exactly n cars in direction II during this busy period t
II t
n
exp II t .
n!
II t
n
exp II t II1 s exp st for all
n
The sum of all these multiplications
n!
non-negative integer numbers of incoming cars n 0 is equal to
II t
n
exp II t II1 s exp st .
n
(10)
n0 n!
6
Then we integrate the obtained expression over all t 0 with the distribution
d I
I
TI t for the time of passing over I vehicles from the moment of the
enabling signal in direction I .
After that we integrate the obtained expression over all d I
I
TI t of the busy
period, which began with the I vehicles’ departure and the flow rate I of arriv-
ing vehicles from the direction I of the intersection and limited by the value TI :
II t
n
exp II t II1 s exp st d I
I
TI t .
n
(11)
t 0 n0
n!
As a result, we obtain the equation
II t
n
II s exp II t II1 s exp st d I TI t
I n I
(12)
t 0 n0
n!
for the Laplace transform II s exp st d II t of the distribution of the
I I
t 0
intersection busy period in direction II , generated by vehicles arriving at the inter-
section in this direction with the flow rate II during the previous permissive phase
for direction I .
After similar identity transformations, we have
II t
n
exp II t II1 s exp st d I
I
TI t
n
t 0 n0
n! (13)
I I
TI s II 1 II s .
1
We obtain an expression of the Laplace transform:
II s I TI s II 1 II1 s .
I I
(14)
The equation for the function II1 s will be written below when studying the distri-
butions of the sequence of permissive times that begins with direction II .
By the definition, it is considered that the time of the permissive phase in direction
II is limited by the value TII . Therefore, a random variable with distribution
II t of the intersection busy period is limited by the value TII to the segment
I
0; TII , which begins with vehicles passing over the intersection in direction II with
the flow rate II have following distribution function:
II TII t II t II TII I 0 t TII I t TII
I I I
(15)
7
and the Laplace transform equals
TII
I
I
TII
I
exp st d II t
II TII s exp st d II TII t 0
. (16)
II TII
I
0
Further we use the same set of formulas to obtain other related values.
If the process of passing over the intersection begins with cars from direction II ,
then the entire sequence of the distributions of limited periods of permissive phases is
constructed similarly, but this time starting from direction II .
We have:
II
─ BII t is the distribution function for the time of passing over the intersection
from the start of the permissive signal in direction II ;
II II
II s exp st dBI t is Laplace transform for this random variable;
t 0
II
─ II t is the distribution function for the intersection busy period as a single-
line queuing system, which began with the II vehicles’ departure and the flow
rate II of arriving vehicles in direction II of the intersection;
─ II s exp st d II t is Laplace transform for this random variable.
II II
t 0
Using the method of the additional event according to the formula of full probabil-
ity, we have a functional equation for the Laplace transform:
II s II s II 1 II1 s .
II II
(17)
In particular, with II 1
II1 s II1 s II 1 II1 s . (18)
By the definition, it is considered that the time of the permissive phase in direction II
I
is limited by the value TII . Therefore, a random variable with distribution IIt
of the intersection busy period is limited by the value TII to the segment 0; TII ,
which began with vehicles passing over the intersection in direction II with the flow
rate
II , have following distribution function:
I
II
t
I
II
TI t II
I 0 t TI I t TI (19)
I TI
8
and the Laplace transform
TI
II
II
TI
II
exp st d I t
I TI s exp st d I TI t 0
. (20)
I
II
0 TI
The obtained equations and recurrent expressions of Laplace transform of the se-
quences of durations of the permissive phases in alternative traffic directions ultimate-
ly allow paying attention to the direction of increased intensity of movement and
speed of passing over the intersection, and thereby reconcile the restrictions on the
times of the permission signals with the loads
I I tdBI t I m I
I
(21)
t 0
and
II II tdBII t II m II
II
(22)
t 0
of the corresponding directions I and II using equations (7) and (18). Differentia-
tion of equation (7) with the opposite sign with respect to s at zero gives
d 1
ds s 0
d
ds
I s I1 s I 1 I1 s s 0
(23)
using the derivative of a complex function for u s I 1 I s and expression 1
of first moments
d 1
zI I s (24)
ds s 0
and
d 1
mI I u (25)
du u 0
we have
d 1 d
zI I u 1 I I1 s , (26)
du u 0 ds s 0
or
z I m I 1 I z I . (27)
9
From this equation the expectation z I of the period of vehicle service in direction I
in the absence of restrictions TI equals
mI
zI . (28)
1 I
Similarly,
m II
z II . (29)
1 II
The obtained expressions for the mathematical expectations of vehicles’ passing times
in the absence of restrictions can be used to obtain relations for constraints TI and TII .
Namely,
TI zI
II , (30)
TII z
or
mI m II
. (31)
1 I TI 1 II TII
The obtained Laplace transforms for travel time (7) and (18), in the absence of re-
strictions on the duration of the permissive phases, make it possible to determine the
ratio of the values of these restrictions from a practical point of view. The absence of
such restrictions, with a high total intensity of traffic flows on competing directions,
will lead to undue delays for a secondary direction. Failure of the adaptive traffic
lights means the removal of restrictions TI and TII .
In this regard, the study of the reliability of the adaptive traffic lights is of great in-
terest when they are considered as recoverable systems with redundancy, which is
provided by warm standby for their components.
3 An estimate of the probability of system failure during the
regeneration period
Some elements of adaptive traffic lights may fail over time. The restoration of system
elements is provided by repair facility (RF), which is a single-line queue with a FIFO
service discipline. The repair times for failed elements are independent and identically
distributed with distribution function G x . The flow of system’s element failures
complies with Markov chains.
If all the elements are in order, that is, the random process of servicing the failed
elements is in the state 0 , then the failure rate of at least one element in the system
10
is 0 . If the system contains failed elements, then the failure rate of an element in
the system is . After recovery, the element returns to where it came from. A ran-
dom regeneration process of maintenance at a time t is defined by the number of
serviced elements in the RF. The moments of regeneration are the times of transition
of a random process to the state 0 when there are no requirements in RF. At the
moment of transition of this random process from the state n to the state n 1 , a
failure occurs ( n 1, 2,... ). Let the probability of failure on the regeneration period of
this random maintenance process be denoted by q . Let
G x 1 G x (32)
and
x
n 1
bn 1 exp x G x dx . (33)
0 n 1!
Lemma 1. Let for numbers aij 0 , bi 0 , x j 0 , i 1, 2,..., n , j 1, 2,..., n it is
n
known that xi bi aij x j for all i 1, 2,..., n . Then for all i 1, 2,..., n the inequal-
j 1
n a b
bi
ity xi , where max
ij j
, is fair.
1 1 i n
j 1 bi
Proof. From the lemma’s condition for all j 1, 2,..., n we have 0 x j b j . From
bj bj
here, for all j 1, 2,..., n it can be established that x j , where j 1 . If
1 j xj
we let k max j , the chain of relations from the condition and the last equality is
1 j n
fair:
n
b n akj x j
bk 1 k k xk bk akj x j bk 1
j 1
1 k 1 k j 1 bk
(34)
n bj
akj
n
1
j
akj b j
j 1 j 1
bk 1 bk 1 b 1 .
bk
k k
Comparing the left and right sides of these relations, we see that
11
n
akj b j
bk 1 k bk 1 ,
j 1
(35)
k 1 k
1 k
b
hence the inequalities
n a b n ab
j k kj j max ij j . (36)
bk 1 i n bi
j 1 j 1
are fair.
Therefore, for all j 1, 2,..., n it is true that
bj bj
xj . (37)
1 j 1
Lemma 2. For any non-negative integers i and j the inequality bi b j Cii j b0bi j is
fair.
Proof. We denote
exp x G x
f x (38)
exp x G x dx
0
and
M i xi f x dx . (39)
0
i !bi
It needs to be noted that M i .
i b0
Hence, a chain of relationships follow from the inequality for the moments
M i M j M i j [7]:
b02 i !bi j !b j i j i j b0
2
i j b0
2
bi b j M M M i j
i ! j ! i b0 j b0
i j
i! j! i! j!
(40)
b02 i j !bi j
i j
Cii j b0 bi j ,
i ! j ! i j b0
i.e. bi b j Cii j b0bi j .
12
We denote the conditional probability of failure during the regeneration period as
qr n 1 , provided that at its beginning in RF there are exactly r complete require-
ments for the element repair, r 1, 2,..., n 1 .
Theorem 1. For all natural numbers n, the inequality
bn 1
q q1 n 1 is true.
1 b0 2n 1 1
Proof. Let j denote the number of failed elements during the recovery of the first
failed element in the RF busy period. Using the total probability rule, we have
n 1
q1 n 1 bn 1 a j q j n 1 . (41)
j 1
According to the total probability rule we create the expression for the probability of
failure qr n 1 , when at the beginning of the busy period in RF there are exactly r
(at least two) full requirements:
nr
qr n 1 bn r a j qr 1 j n 1, 2 r n . (42)
j 0
Note that a0 1 b0 and a j b j 1 b j , j 1 .
These equalities and the Abel transform make it possible to write out and estimate,
from the above formula, the second terms on the right-hand sides of the last two series
of expressions, respectively, in the form
n 1 n 1
a q n 1 b
j 1
j j
j 1
j 1 b j q j n 1
n 1
b j 1 q j n 1 q j 1 n 1 bn 1qn 1 n 1 (43)
j 1
n 1
b j 1 q j n 1 q j 1 n 1
j 1
Here by definition we consider q0 n 1 0 ,
nr nr
a q j r 1 j n 1 b j 1 b j qr 1 j n 1 1 b0 qr 1 n 1
j 0 j 1
nr
b j 1 qr 1 j n 1 qr 2 j n 1 bn r qn 1 n 1 qr 1 n 1 (44)
j 1
nr
b j 1 qr 1 j n 1 qr 2 j n 1 qr 1 n 1
j 1
for 2 r n .
13
We denote n j 1 n 1 q j n 1 q j 1 n 1 , 2 j n 1 .
By definition n n 1 q1 n 1 . Substituting the obtained upper estimates in-
stead of the secondary terms in the right-hand sides of the above expressions and
transferring the last terms (for r equal to at least two) from right to left, we transform
expressions of the probability of failures having the form of equalities into inequali-
ties for them and their mathematical differences, respectively
n 1
n n 1 bn 1 b j 1 n j 1 n 1 for r 1 (45)
j 1
and
nr
n 1 r n 1 bn r b j 1 n 1 r j 1 n 1 for 2 r n 1 . (46)
j 1
For this system of inequalities under the conditions of Lemma 1, a square matrix of
A aij of n 1 -th order is
bn 2 bn 3bn 4 ...b1b0
bn 3bn 4 bn 5 ...b0 0
..........................
A . (47)
b2 b1b0 ...........00
b b 0.............00
1 0
b 00..............00
0
For the above system of inequalities, we can use Lemma 1 with
xi n 1i n 1 , i 1, 2,..., n 1 . (48)
Than from this system of inequalities according to Lemma 1
bn 1
n n 1 q1 n 1 , (49)
1
n i b j 1bn j
where max .
1 i n 1
bn 1 j 1
According to Lemma 2 the inequality 1 j n is true for all integers
b j 1bn j Cnj11b0bn 1 (50)
and
14
n i b j 1bn j n i
Cnj11b0 bn 1
max max
1 i n 1 bn 1 1 i n 1 bn 1
j 1 j 1
(51)
n i n 1
b0 max C j 1
n 1 b0 C j 1
n 1 b0 2 n 1
1
1 i n 1
j 1 j 1
Having obtained the upper bound for the value ,
b0 2n 1 1 . (52)
we have also estimated the probability of system failure during the regeneration peri-
od of a random process in the redundancy model with recovery
bn 1
q q1 n 1 .
1 b0 2n 1 1
Let mk t k dG t – k is the moment of service time and m1 1 . At the initial
t 0
moment of time t 0 the system is in the state 0 (all elements are in order). We
denote by the time of the first failure of the ITMS from the moment when all of its
elements are in order. Theorem 1 of this paper implies the following theorem.
Theorem 2. Let there be a finite moment m2 . Then in the process
m2
q 0 probability
m1 1
0 1
P q x exp x . (53)
1 1
0 m
where the two-way estimate q is true for the failure probability
bn 1
bn 1 q q1 n 1 , n 1, 2,... , (54)
1 b0 2n 1 1
that is the time until the first failure of the ITMS which has asymptotically exponen-
tial distribution. This means a higher frequency of small periods of time before the
first fail and, given the large number of adaptive traffic lights in ITMSs, it indicates
the importance of finding new kinds of warm standby for them.
4 Results
The main result of the failure probability estimation presented above is the fairness of
two-sided estimate (54). For simple duplication, when n 1 , this estimate gives the
exact value of the probability
15
q 1 2 b0 . (55)
We can compare this result with result according to another method – with a similar
upper estimate obtained by A.D. Solovyev in [8], where there is no subtracted unit in
brackets in the denominator:
bn 1
bn 1 q q1 n 1 . (56)
1 2n 1 b0
So, the estimate of the failure probability (54) is more narrow than the estimate (56)
and can provide more exact predicted reliability of traffic lights with and without
warm standby. In case when the traffic lights fail with probability q1 , it is possible to
estimate the effect of warm standby use for adaptive traffic lights at two-way intersec-
tion. It can be done by means of determining the difference between vehicle service
time at the intersection with and without adaptive traffic control.
Comparing to the intersection with adaptive traffic control, the uncontrolled inter-
section with one major direction (street) has the next differences:
─ vehicles on the major road have no delays when passing the intersection;
─ vehicles on the minor road are obliged to slow down or even stop before entering
the major road and make sure of ability to move forward without traffic hindrances
on the main road.
It increases the time of passing the intersection and for vehicles on the minor road
it can be estimated as m II 2m I [9]. Hence, an average waiting time for the vehicles
on the minor (critical) movement direction can be estimated as the consequence of
adaptive traffic lights failure.
For example, let the traffic volume on one lane of the major road equals to
I 0.3 s 1 . Time of passing the intersection when traffic is permitted (for the major
road) equals to m I 2 s. Analogous time for the vehicles on minor road when there is
no adaptive traffic control at intersection equals to m II 2m I 4 s. Also let the traffic
volume on the minor road equals to II 0.075 s 1 . Then, when the traffic lights fail,
total intersection load equals to I II I m I II m II 0.6 0.3 0.9 . Then
average waiting time in a queue on the minor road [7] equals to
W II I m I (1 ) 1 . (57)
After example data substitution W II (0.6 2) / (1 0.9) 0.9 1 0.9 129 s. The
upper estimate of analogous time expenses on the minor road when there is an adap-
tive traffic control at intersection can be obtained regarding to restrictions from equa-
tion (31) and denoting that T TI TII 80 s. Then
TII TI (1 I )m II m I (1 m II ) TI 0.4 0.85 , (58)
16
wherefrom TII 25.6 s, TI 54.4 s.
The upper estimate of waiting time is obtained upon the Smith theorem [10]:
W II TI / 2 27.2 s. It means that waiting time is upper bounded by the value that is
almost 5 times less than in the case when there is no adaptive traffic control at inter-
section. At that, average waiting time for the vehicles on the major road does not ex-
ceed the value of W I TII / 2 12.8 s that allows to highlight advantages increasing
the reliability of adaptive traffic lights by warm standby use.
5 Conclusions
1. The Laplace transforms (7) and (18), which were obtained for the time interval of
passing over the intersection in a given direction when there are no restrictions on
the duration of the permissive signal, allow the determination of the ratio of values
of such restrictions from a practical point of view.
2. An upper estimate of the probability of failure of the system with one standby dur-
ing the regeneration period has been found. It leads to an exact value of the proba-
bility of failure, which coincides with a lower estimate.
3. This article defines the quantitative ratios that create the opportunity to assess the
feasibility of using warm standby in adaptive traffic lights, when quantitative char-
acteristics of traffic flows and the failure rates of TMS’ elements are known.
References
1. Cèlia, N,R.: Reliability, Availability and Maintainability Study of a Light Rail Transit Sys-
tem. Thesis, Polytechnic University of Catalonia (2014).
2. Tang, L.: Reliability assessments of railway signaling systems: A comparison and evalua-
tion of approaches. Thesis, Norwegian University of Science and Technology (2015).
3. Zhao, Mo: A methodology for reliability-based traffic signals alternative power capacity
design. Dissertation, University of Nebraska (2015).
4. Rössner, H.: Circuit configuration for signal transmitters with light-emitting diodes. US
Patent 6,069,452, 30 May 2000.
5. Oskina, M.A., Sergeev. B.S.: Резервированный светодиодный светофор (Redundant
LED Traffic Lights). Patent RU 2672314, 23 Nov 2018.
6. Wallace, W.A., Wojtowicz, J., Torrey, D., Renna, N., Tan J.: Guidelines for Traffic Signal
Energy Back-Up Systems. Final Report, SPR Research Project no. C-06-08. New York
(2009).
7. Gnedenko, B.V., Belyaev, Yu., Solovyev, A.D.: Mathematical Methods of Reliability
Theory. Academic Press, New York (1969).
8. Barzilovich, E., Beljaev, Ju., Kashtanov, V., Kovalenko, I., Solovev, A., Ushakov, I.: Во-
просы математической теории надежности (The Affairs of Mathematical Theory of Re-
liability). Radio i svyaz, Moscow (1983).
9. Highway capacity manual. Transportation research board, Washington (2000).
10. Smith, W.L.: Regenerative stochastic processes. Proc. Roy. Soc. A 232(1188), 6-31
(1955).