“Markov chain for BMD”

author: “Alexander Levakov”

date: “Wednesday, November 19, 2014”

‘Would you tell me, please, which way I ought to go from here?’
‘That depends a good deal on where you want to get to,’ said the Cat.
‘I don’t much care where—’ said Alice.
‘Then it doesn’t matter which way you go,’ said the Cat.
Lewis Carroll, ALICE’S ADVENTURES IN WONDERLAND

BMD

A good deal of information about BMD program one can find here. (http://www.mda.mil/system/system.html).

library(expm)

## Loading required package: Matrix
## 
## Attaching package: 'expm'
## 
## The following object is masked from 'package:Matrix':
## 
##     expm

Main assumptions

We simplify this picture as 3 layers only: boost, midterm, terminal. Multilayer Ballistic Missile Defence We consider the probability of kill for each layer (stage of missile flight) Pkill to be the same. So the overall probability of kill at least for one layer i.e. stage of missile flight: \[P= 1 - (1-Pkill)^3\]
For example if Pkill=0.5 (worst case) we have

Pkill=0.5
P=1 - (1-Pkill)**3
print(P)

## [1] 0.875

Not so bad for one missile - 87.5%, but if we have a dozen of attacking missiles: \[Nkill=P*10\]

Nkill=P*10
print(Nkill)

## [1] 8.75

One missile will hit target. It’s not acceptable especially for missiles with 3 or 10 MIRVs.

Note

In order to make situation close to real even by some manipulations with probabilities we will consider next example where smart missile makes a hole in BMD system.

Research question with example

Our goal is to calculate the probability of hitting the target by missile carrying one RV or warhead P_S1,S6. We suppose that the flight of missile is composed by consequent events: \(S1\) - start, \(S2\) - boosting (boost stage), \(S3\) - orbiting (midterm stage), \(S4\) - reentry (terminal), \(S5\) - fail, \(S6\) - impact (hit the target). The events listed and described above (\(S1\), \(S2\), \(S3\), \(S4\), \(S5\), \(S6\)) can be considered as a Markov chain - a mathematical system that undergoes transitions from one state to another on a state space. It is a random process usually characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. (see http://en.wikipedia.org/wiki/Markov_chain).
markov chain

Note

MC is a transition matrix with 6 states. For instance, for \(S1\) we get three paths to leave: missile would not start due to technical problems P_s1,s1=0.02, missile would start (be launched) but go down due to technical problems P_s1,s5=0.02, missile would start and take a successful boost P_s1,s2=0.95. An absorbing state is a state from which it is impossible to leave: \(S5\) (missile technically failed or hit by BMD), \(S6\) (missile hit target). Here we consider RV has 100% reliability.

MC<-matrix(c(0.02,0.95,0,0,0.03,0,0,0,0.85,0,0.15,0,0,0,0,0.85,0.15,0,0,0,0,0,0.15,0.85,0,0,0,0,1,0,0,0,0,0,0,1),nrow=6,ncol=6,byrow=T)
print(MC)

##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0.02 0.95 0.00 0.00 0.03 0.00
## [2,] 0.00 0.00 0.85 0.00 0.15 0.00
## [3,] 0.00 0.00 0.00 0.85 0.15 0.00
## [4,] 0.00 0.00 0.00 0.00 0.15 0.85
## [5,] 0.00 0.00 0.00 0.00 1.00 0.00
## [6,] 0.00 0.00 0.00 0.00 0.00 1.00

solve(diag(nrow=4,ncol=4)-MC[1:4,1:4])%*%MC[1:4,5:6]

##           [,1]      [,2]
## [1,] 0.4046747 0.5953253
## [2,] 0.3858750 0.6141250
## [3,] 0.2775000 0.7225000
## [4,] 0.1500000 0.8500000

Alternative option

We can get the same result by multiplying our transition matrix MC by itself N times i.e. MC^N.

MC%^%16

##            [,1]        [,2]         [,3]         [,4]      [,5]      [,6]
## [1,] 6.5536e-28 3.11296e-26 1.323008e-24 5.622784e-23 0.4046747 0.5953253
## [2,] 0.0000e+00 0.00000e+00 0.000000e+00 0.000000e+00 0.3858750 0.6141250
## [3,] 0.0000e+00 0.00000e+00 0.000000e+00 0.000000e+00 0.2775000 0.7225000
## [4,] 0.0000e+00 0.00000e+00 0.000000e+00 0.000000e+00 0.1500000 0.8500000
## [5,] 0.0000e+00 0.00000e+00 0.000000e+00 0.000000e+00 1.0000000 0.0000000
## [6,] 0.0000e+00 0.00000e+00 0.000000e+00 0.000000e+00 0.0000000 1.0000000

It means that after launching smart missile 16 times i.e. launching 16 smart missiles we would get 59.5% probability of success to hit target in spite of 3 layers of BMD and some technical faults.

NS<-16*0.595
print(NS)

## [1] 9.52

So 9 missiles would hit targets.

Conclusion

Although this example of Markov Chain is for demonstration purpose only it can be used for real estimates of BMD effectiveness if one should expand state (events) space and would be able to get relevant probabilities proved by test flights (see http://rpubs.com/alex-lev/40511).