Dahlgren, A. C.

English

NASA Technical Reports Server

. , . - .  i 
Alfred C. Dahlgren 
https://ntrs.nasa.gov/search.jsp?R=19650002029 2020-03-17T00:21:32+00:00Z
Technical Report No. 32-655 
(Part /) 
Mathematical Models of Missile Launching 
Alfred C. Dahlgren 
1. Hamilton, Chief 
Systems Analysis Section 
J E T  P R O P U L S I O N  L A B O R A T O R Y  
PASADENA, CALIFORNIA 
C A L I F O R N I A  fNSTlTUTE OF TECHNOLOGY 
August 2 I ,  1964 
Copyright 0 1964 
Jet Propulsion Laboratory 
California Institute of Technology 
Prepared Under Contract No. NAS 7-100 
National Aeronautics & Space Administration 
JPL TECHNICAL REPORT NO. 32-655 (PART I )  
CONTENTS 
1. Introduction. . . . . . . . . . . . . . . . . . . . .  1 
11. The Mathematical launch Model . . . . . . . . . . . .  1 
111. Results. . . . . . . . . . . . . . . . . . . . . . .  2 
IV. Derivation of Results . . . . . . . . . . . . . . . . .  2 
References.. . . . . . . . . . . . . . . . . . . . . .  4 
. 
JPL TECHNICAL REPORT NO. 32-655 (PART I )  
ABSTRACT I /  6 ‘3 
By means of a mathematical model of the situation, including as- 
sumptions about the nature of the delays encountered, an estimate 
is presented of the expected number of days required to launch three 
missiles, each from its own pad, allowing no simultaneous (i.e., same- 
day) countdowns or launches. 
I. INTRODUCTION 
In space mission feasibility studies it is often necessary bility theory, for different launching configurations 
(see Refs. 1 and 2). The particular launching configura- 
tion examined here is that one of three missiles being 
launched, each from its own pad, allowing no simul- 
taneous (i.e., same-day) counting-down or launching. The 
following assumptions constitute the launch model. 
to have an estimate of the number of days required to 
launch a given number of missiles from a given (and 
possibly different) number of pads. This report (Part I 
of three parts) is one of a series, the purpose of which 
is to investigate this question with the aid of proba- 
II. THE MATHEMATICAL LAUNCH MODEL 
1. There are three identical pads and three identical 
missiles, each on its own pad. 
2. For each missile, a complete countdown and launch- 
ing in one day is possible, though not necessary. But 
once the counting has started on a given missile, 
the counting continues on that missile until it fires, 
no matter how many days are required. There is 
a probability p (a specified constant) that a missile 
will successfully count down and &e in one day. 
There is thus also a constant probability q = 1 - p 
that the missile will fail to complete a countdown 
3. Because p is a constant, failures to fire a given mis- 
sile do not influence the value of p in subsequent 
attempts to fire the same missile or others. Each 
missile is thus probabilistically independent of the 
other two. 
4. Prior to the start of the N-day period, the three 
missiles are erected, each on its own pad and ready 
to start counting. One of the three missiles is .then 
chosen arbitrarily, because all are identical. On the 
first day of the N-day period, the counting down is 
started on this missile, and continues until the mis- 
on that day, and will incur a one-day delay. In 
other words, if the countdown on a given missile 
stops on any day of counting, the counting must 
start over from the beginning the following morning. 
sile fires (the countdown started anew each clay), no 
matter how many days this requires. On the day 
after the firing, counting down is initiated on one 
of the two remaining missiles (again the choice is 
1 
. 
JPL TECHNICAL REPORT NO. 32-655 (PART I )  
arbitrary). This second missile also counts down for 
as many days as are required for it to fire, the count- 
down starting anew each day. On the day after the 
second missile is launched, the remaining missile 
starts counting, continuing until the missile fires on 
the Nth day, by assumption. The third and last mis- 
sile is always assumed to fire on the Nth day, what- 
ever the positions of the first and second missile 
launchings within the N-day period. In this launch- 
ing model, since no missiles are allowed to count 
down together on any day, no simultaneous (i.e., 
same-day) launches are possible. (The choosing at 
each firing of the next missile to be fired is logically 
equivalent to predefining the firing order before the 
start of the N-day period. The reader may take 
either point of view, depending on the practical 
application to which he intends to put the results 
of this report. See Section IV for further discus- 
sion of this point. 
5. Since each missile is on its own pad, it is not neces- 
sary in this launch model to take into account any 
turnaround time between missile firings, as has been 
done in other reports in this series. 
111. RESULTS 
1. The probability P ( N )  of launching (under the above 
conditions) three missiles from three pads in exactly 
N days from start of counting, is 
3. The variance u2 of the P(N) distribution (see Sec- 
tion IV) is given by 
P(N)=- p 3 q N - 3  (N - 1) (N - 2) 2 
39 
P2 
o2 = -
2. The mean day (counted from the first day of the 
N-day period), on which one can expect to launch 
the third and last missile, is found to be simply 
4. The moment-generating function M (e) is found 
to be 
3 
p = -days 
P 
p3e3e 
(1 - qee)3 M (e )  = 
IV. DERIVATION OF RESULTS 
In order to construct a sample space for the problem, 
we ask in how many different ways the three missiles 
can be launched. Since we have made it a condition of 
this launch model that there be no simultaneous (i.e., 
same-day) countdowns, it will be seen that only one 
specific type of launch sequence is possible for the three 
missiles, for the following reasons. As outlined in the 
introduction, the first missile to be fired must be con- 
sciously chosen from the three, with the second missile 
to be fired then chosen from the two remaining. This 
order is in reality a type of 1,2 ,3  launch sequence. Let 
US number the individual missiles arbitrarily but perma- 
nently as 1,2,3. It might seem at first glance that the 
order 1,2,3 and permutations of this order would be 
2 
. 
JPL TECHNICAL REPORT NO. 32-655 (PART I )  
distinct possible launch sequences, but this is not true. 
Granted, this actually would be the case if the condi- 
tions of the launch model did not require us to choose 
at each firing the missile to be launched next. But in so 
choosing, we remove at each firing the randomness other- 
wise associated with the question of which missile is 
next to be counted down. In other words, we constrain 
the actual launching order of the missiles to follow the 
single path that we choose for it. Because of this, only 
this one type of launch sequence is possible. 
It may further be noted that there is no difFerence 
logically between choosing the next missile at each 
launching as we go along, or equivalently predefining 
the same firing order before the N-day period even starts. 
(The more general case of choosing at each step is treated 
in this report for completeness.) If we did predefine the 
order, then it is obvious that this defined order would 
be the only sequence that could occur in the firing of 
the three missiles. Thus from another angle we have the 
same conclusion that if we choose at each firing the mis- 
sile to count down next, there are no other possible launch 
sequences than this one. Thus in the calculations below 
we need take account only of this one type of launch 
sequence in the sample space for the problem. 
Keeping the above in mind, let us assume that we 
choose at each firing such that the missiles do result in 
launching in the order 1,2,3 (with the numbering from 
above). The probability p l  that missile 1 will fire on the 
kth day (counted from the &st day of the N-day period), 
is equal to the probabilib that it will fail to fire for k - 1 
days, times the probability that it will fire in one day, 
or pl = pqk- ' .  Similarly, if missile 2 counts for m days 
(starting with the (k + 1)st day), p z  = p9"-'. Missile 3 
by assumption fires on the Nth day, and thus it counts 
for the number of days remaining in the N-day period 
after the (k + m)th day, or thus for N - k -m days. The 
probability p ,  of missile 3 firing on the Nth day is thus 
P3 = P q N - k - m - l -  
Out of a given N-day period, k and m can assume 
various different values, the limits on these values being 
determined by N. In order to express this situation mathe- 
matically, we will make m dependent on k, and the days 
for missile 3 dependent on m and k. We will then permit 
k to range over its maximum and minimum values within 
the N-day period. Doing this will automatically take care 
of the maximum and minimum values for m and the 
days for missile 3. The limits on k are as follows. The 
quantity k must be at least 1 day, but if k is almost as 
large as N, at the end of the N-day period we must leave 
at least one day each to launch missiles 2 and 3. Thus out 
of a given N days, k can range only over 1 4 k A N - 2. 
The m days for missile 2 starts on the (k + 1)st day, 
and m is at least one day, but again we must leave 
at least the Nth day in order to launch missile 3. Thus 
1 4  m 6 N - k - 1. From above, missile 3 counts for 
N - k - m days, and thus k and m ranging over their 
allowed values takes care of the maximum and minimum 
values for the days missile 3 counts. 
Further, the situation that missile 1 will take a par- 
ticular k, days, with missile 2 taking a particular m, days 
and missile 3 taking the remaining N - k, - m, days, is 
a compound event whose own compound probability p c  
therefore equals the product of the component event 
probabilities, or p ,  = p lp2p3 .  There can be different spe- 
cific k, and m,  combinations within the given Nday 
period. The compound event associated with a particu- 
lar k, and m, is an event mutually exclusive of those 
events associated with other k, and m, sets. Thus, since 
there are various values available to k, and m,, we must 
sum the compound probability p ,  = p lpzp3  over all pos- 
sible values of k and m within a given N-day period. By 
doing this we will arrive at the general probability ex- 
pression P(N) of launching 3 missiles in N days under 
the launch model conditions, or 
k = 1  m = i  
= N - z  N - k - i  2 p 9 k - i p 9 m - i p q N - k - m - 1  
k = 1  n r = i  
N - z  X - k - 1  
= p39"-3 Z Z (1) 
k = i  m=1 
or 
P(N) =- p39N-3 (N - 1) (N - 2) 2 
By assumption in this launch model, the possibility of 
failing to launch all three missiles was not included in 
the sample space for the problem. We would therefore 
keep counting until all three missiles were launched, 
which is equivalent to assuming an upper limit of infinity 
for N. At the other end of the spectrum, the smallest 
admissible value of N is three days, one day being the 
minimum necessary for counting down and launching 
each of the three missiles. 
We have a sample space for this launch model in which 
we have concluded that only one type of launch sequence 
is possible. If this sample space has actually taken into 
3 
JPL TECHNICAL REPORT NO. 32-655 (PART 1 )  
account all possible launch sequences, the total proba- 
bility over the whole sample space must equal 1. The 
total probability Pt  equals the sum of P ( N )  over all ad- 
missible avlues of N ,  or 
W W 
P t =  I: P ( N ) =  2 7 p3q”-3 ( N - 1) ( N  - 2) 
N = 3  N = 3  
Thus we conclude that in the sample space, we have 
accounted for all possible launch sequences, this model 
having only one type. Looked at from another angle, the 
result P ,  = 1 indicates that the expression for P ( N )  is 
correct, since on other grounds (as explained above), we 
had concluded that there can be only this one type of 
launch sequence in the sample space. 
Lastly, P (N) in this launch model plays the role of 
the probability function for a discrete random variable n, 
where n is the number of the day on which the third 
and last missile is launched. In other words, P ( N )  in 
reality is the probability that n = N .  Thus the mean day 
p (counted from the first day of the N-day period), on 
which one expects to launch the third and last missile, 
and the variance of the P ( N )  distribution may be cal- 
culated by the usual series: 
r- 
p =  E [ N ]  = N P ( N )  
N = 3  
and 
p and U* may also be calculated from the moment- 
generating function M (e ) ,  where 
M (e )  = E [eeN]  = I: W P ( N )  eeN 
s=s 
or 
p3e3@ 
(1 - qee)3 M ( e )  = 
This further implies that 
3 
=-days 
and also that 
or 
REFERENCES 
1 .  Solloway, C. B., A Simplified Statistical Model for Missile launching, Technical 
Report No. 32-43 1,  Jet Propulsion laboratory, Pasadena, California, May 1 ,  1963. 
2. loss, H., and C. B. Solloway, A Statistical Problem Related to the Launching of a 
Missile, Technical Report No. 32-1 24, Jet Propulsion Laboratory, Pasadena, Cali- 
fornia, July 20, 1961. 
4 


Mathematical models of missile launching, part i

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server