American Journal of Computational Mathematics, 2011, 1, 281-284
doi:10.4236/ajcm.2011.14034 Published Online December 2011 (http://www.SciRP.org/journal/ajcm)
The Wartime Bridge Reliability Evaluation Model Based
on Birth-and-Death Process
Duo-Dian Wang, Guo-Qing Qiu
Engineering Institute of Corps of Engineers, PLA University of Science & Technology, Nanjing, China
Received August 21, 2011; revised Septe mber 25, 2011; accepted October 8, 2011
Abstract
At first, the concept of bridge reliability is given, followed with its mathematic model. Then, based on the
analysis about the mechanism of the damage and repair of bridges, and the state diversion of bridge network,
the state diversion process is proved to be birth-and-death process. In the end, the state diversion balance
equation of bridge network is built, and the evaluation model of wartime bridge reliability is got. The model
is used in a certain example, and it is proved to be precise and credible.
Keywords: Wartime, Bridge Reliability, Evaluation, Birth-and-Death Process
1. Introduction
Mobile combat is the main type of the modern combat.
The battlefield conditions vary from minute to minute, so
maneuver rapidly or not can even decide the victory or be
defeated. During the war, maneuver path is the key factor
to obstruct each other’s mobility. Bridges included in the
maneuver paths are the main attack targets. In the paper,
how to evaluate the wartime bridge reliability is re-
searched.
The bridge reliability refers to its ability of the time in
the regulated function. Under the anti-war circumstances,
the bridge reliability mainly depends on its structure. On
wartime, Bridge is the key factor to both parties to ob-
struct each other’s mobility. The enemy destroys the bridge
to delay mobility while we rush to repair the bridge to
guarantee it.
Markov birth-and-death process method is widely ap-
ied to the research of bridge reliability. Huang Qi (see in
[1]) has studied on the application of the Markov birth-
and-death processes in the reliability theories. Furermore,
Wang Chao (see in [2]), Xia Hai-Bing (see in [3]) and Lu
Ying-Zhao (see in [4]) has studied on the application of
the prediction of bridge reliability. There isn’t any re-
search on the wartime bridge reliability now .
In the paper, birth-and-death process method is used to
evaluate the wartime bridge reliability, combined with
some related knowledge, including diagram theory, queu-
ing theory and reliabilit y theory. Brid g e reliabilit y is mea-
sured by the degree of reliability. Bridges are connected
with each other by roads, thus, constitutes bridge network.
Its scope depends on the maneuver mission. In Figure 1,
bridge D-16-DM/E-17-MU is marked bi. All the bridges
of the network are involved in a bridge gather. The bridge
gather is marked B. The reliability degree of bi is ri, and
the corresponding rel i a bili ty degree of B is R.
2. The Evaluation Model of Wartime Bridge
Reliability Based on the Birth-and-Death
Process
2.1. The Mechanism of the Damage and Repair
of Bridges
Attack from enemies has regulations. Air attack is the
main method. Various bridges can be sequenced accord-
ing to their importance. The importance degree can be
used to show how importance a bridge is, and it is
marked zk. The way to decide the importance of bridges
can be found in [6,7]. The bridges can be marked bk, and
k stands for its importance order. The enemy keeps stable
breakage strength . Each time they attacks one target only,
and just attacks the undamaged and the most important
one.
Our urgent repair also has regulations. The urgent repair
is carried on immediately after attack, and the allocation
of soldiers commonly follows the nearby principle. If the
strength of urgent repair is enough, we will rush to repair
all the attacked bridges. Most of time, we have enough
strength to repair the bridges after the enemies’ attack.

D.-D. WANG ET AL. 282
Figure 1. Bridge net of some place (see in [5]).
2.2. The State Diversion of a Bridge
The state of a bridge is divided into both kinds: damaged
and undamaged, and is marked state 0, 1. The mecha-
nism of state diversion of a bridge is in Figures 2 and 3.
The diversion rate from state 0 to state 1 is marked k

,
and the rate from state 1 to state 0 is marked k

. k is
the fought rate of a bridge ,and k is the broken
rate ,and k
d
P
x
is the repair rate. Therefore, there is for-
mulas as follows:
kk
kk
dp
x






k
(1)
The state diversion of a bridge is shown in Figure 4.
The fought rate of a bridge is decided by the power of
the enemy’s strike. It equals to the arrival rate of the
enemy. The broken rate of a bridge refers to the rate of
damage in the time of a unit (usually an hour) under av-
erage strike strength.
2.3. The State Diversion Analysis of Bridge
Network
The process of bridge network state diversion is a typical
undamaged
attacked
destroyed
undestroyed
damaged
undamaged
undamaged
unattacked
damaged
repaired
undamaged
damaged
unrepaired
Figure 2. The mechanism of state diversion of a bridge.
undamaged attackeddestroyeddamaged
damagedrepaired undamaged
Figure 3. Simplified mechanism of state diversion of a
brige.
0 1
k

k

Figure 4. State diver of a bridge.
birth and death process. The information desk in the ser-
vice system is the bridge network, and the customer is
the enemy. The process of our urgent repair can be seen
as making contributions to the customers.
The input process of birth and death process is a Pois-
son process; because it satisfies the following conditions
(see in [8]):
1) Steadiness. In a certain time zone, there are k
batches of enemy called to strike, which is re-
lated to the length of the time zone while it isn’t any re-
lation to the beginn ing of time. That is to say, the equals
of

k
Pt


k
Pt are the same during the time zone of [0, t] or
[a, a + t].
2) Non after-effect. The time of the enemy coming to
attack is independent during unrelated time zone. That is,
in the zone [a, a + t], the probability the en emy’s co ming
isn’t related to the time of that coming ahead of the time.
3) Commonness. There can be only a batch enemy in
enough small time zones to arrive, and two or more batch

D.-D. WANG ET AL. 283
to arrive in the meantime is impossible. If we use


t

stands for the probability of 2 or more batch to arrive
between


0,t, we can get


0ttt  
We try to get the analytic solutio n of the equ atio ns, the
stable state probability formulas are as follows:

k
P
.
0
12 0
11
kk
kk
kkk
PCP
C














(4)
4) Limitedness. The probability of a limited batch to
arrive in arbitrarily limited time is one.
In Figure 5, state 0 refers that all the bridges are un-
damaged. State 1 refers that only one bridge is damaged,
but others not. State k refers that k bridges are damaged,
but others not. The damaged bridge in state 1 is the one
with the importance1
b, the damaged bridges in state k
are the ones with the importance of first k ones,
12 . Therefore, the probability of the state k is
that of the bridge damaged. Th e reliability of bridge
is that
,,,
k
bb b
k
bk
b
Otherwise, there is
0
00
1
KK
kk
kk
PCP



 (5)
Therefore,
1
00
K
k
k
PC









 (6)
1
k
r (2)
The reliability of the bridge is: .
k
b1
kk
rP
The probability of input and output in each a state is
equal, therefore, the state balance equations are as follows: 3. Example




11 10
002211 1
113322 2
22111
11 11
kkkkkkk
kkkkk kk
PP
PP P
PP P
PP P
PP P

 
 

 

 












 (3)
The example is about the bridges in Figure 1 from ref-
erence 5. The intention of enemy’s attack is 1 time every
3 hours. The average repair time of the bridges and other
parameters are in Table 1:
With the method in reference 6, we can order the
bridges according to their importance.
0 1 2 3
0

2

k

1
k

k

k-2 k-1 kk+1
1

2

1

2

3

1

k

k

1
k


Figure 5. The state diversion of bridge network.
Table 1. Parameters of br idges in the bridge network.
Length Width
Number Bridge
name Bridge type Number of spans (m)(ft) (m)(ft)
Year
built ADTT (trucks/day) Average repair time(Hour)
1 E-16-MU Prestressed 1 34.1112.011.638.01994 810 1.5
2 E-16-LA Prestressed 2 77.9255.539.2128.51983 450 3
3 D-16-DM Prestressed 2 44.5146.014.246.51990 390 1.8
4 E-16-QI Prestressed 2 74.1243.230.7100.71995 1335 2.8
5 E-16-LY Prestressed 3 74.3243.734.1112.01985 1610 2.8
6 E-16-NM Prestressed 2 64.6212.028.092.01991 2955 1.5
7 E-16-MW Prestressed 2 72.7238.630.5100.01987 230 1.7
8 E-16-FK Steel-beam 4 69.2227.010.434.01951 1370 2.5
9 E-16-FL Steel-beam 4 54.0177.010.434.01951 765 2
10 E-16-Q Steel-beam 5 82.3270.012.240.01953 890 3.2
11 E-17-LE Steel plate girder 4 68.6225.019.764.51972 992 2.7
12 E-17-HS Steel plate girder 4 64.5211.710.434.01963 5 2.5
13 E-17-HR Steel plate girder 4 64.0209.810.434.01962 306 2.5
14 E-17-HE Steel plate girder 4 67.7222.210.434.01962 1290 2.8

D.-D. WANG ET AL. 284
Table 2. Repair parameters of bridges in the bridge network.
Bridge number 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Importance order
K
14 2 13 4 3 9 5 6 12 1 7 10 11 8
Repair time
k
t1.5 3 1.8 2.8 2.8 1.5 1.7 2.5 2 3.2 2.7 2.5 2.5 2.8
Repair probability k

0.67 0.33 0.56 0.360.360.670.590.4 0.5 0.31 0.37 0.4 0.4 0.36
Table 3. Wartime reliability of the bridges in the bridge network.
Order of bridge 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Importance order 14 2 13 4 3 9 5 6 12 1 7 10 11 8
Reliability
k
P0.9970 0.8516 0.9939 0.8753 0.86400.97680.93030.94250.98960.85160.9487 0.9809 0.98420.9530
10 2 5 4 7 811146
12 13931
bbbbbbbbb
bbbbb
 

The average probability o f enemy’s arrival is 0.33, the
probability with that the br idges are destroyed: 0.33
k


,
the other parameters are in Table 2.
The state diversion balance equations are as follows:




10
02 1
13 2
11 1312
12 1413
0.31 0.33
0.330.330.33 0.31
0.330.360.33 0.33
0.330.560.33 0.5
0.330.670.33 0.56
PP
PPP
PP P
PP P
PP













P
(7)
With matlab, we solve the equations, the solutions are
as follows:
012
345
678
91011
12 13 14
0.1394 =0.1484 =0.1484
=0.1360 =0.1247 =0.0697
=0.0575 =0.0513 =0.0470
=0.0232 =0.0191 =0.0158
=0.0104 =0.0061 =0.0030
PPP
PPP
PPP
PPP
PPP

According to the importance order, we can get the war-
time reliability of the bridges in Table 3.
Based on the result about bridge reliability in Table 3,
the decide maker can get his idea about maneuver pro-
ject.
4. Conclusions
The evaluation of wartime bridge reliability is an impor-
tant content of the research about wartime mobility
combat. At present, the bridge reliability research is gen-
erally limited by the peacetime appearance, while the
wartime research is seldom. On the battlefield, the bridge
eliability mainly depends on a series of destroy and re-
pair between us and the enemy due to the abruption and
quickness of the war.
r
In the paper, the definition of the wartime bridge reli-
ability is given on the foundation of general bridge reli-
ability. The state diversion of the bridge network is
proved to be a bir th-and- death pro cess, and an ev aluation
model of the bridge reliability is got. In the future, the
research will be carried on simulation exp erience and the
improvement of the model in the text.
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38
[4] Y.-Z. Lu and S.-H. He, “Prediction of Markov Fuzzy
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