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A counter arrangement with constant resolving time
Physica X, no 9
November
1943
A COUNTER ARRANGEMENT W I T H CONSTANT RESOLVING TIME b y J. G I L T A Y Zusarnmenfassung E s w i r d eine d e r a r t i g e A b g n d e r u n g e i n e r y o n L e v e r t und S c h e e n x) b e n u t z t e n Z A h l v o r r i c h t u u g v o r g e s c h l a g e n , d a s s s i c h e i n e k o n s t a n t e r Aufl~Ssungszeit d e f i n i e r e n 1/isst. Die W a h r s c h e i n l i c h k e i t , class h i e r b e i eine b e s t i m m t e Z a h l y o n I m p u l s e n i n e i n e r g e g e b e n e n Z e i t gezAhlt w i r d , w i r d e x a k t b e s t i m m t . D a s P r o b l e m h g n g t z u s a m m e n m i t elnero bekannten yon E r 1 a n g gel6sten Problem aus der Fernsprecht e c h n i k 3).
Introduction. In a recent p a p e r 1) L e v e r t and S c h e e n deduce expi'essions for the n u m b e r of discharges mechanically c o u n t e d in a finite i n t e r v a l of time, when a G e i g e r-M fi 11 e r c o u n t e r is exposed to cosmic radiation. The v e r y interesting m e t h o d b y which the result is obtained can be applied to various o t h e r problems. This is especially the case for L e v e r t and S c h e e n's problem in a modified form, as will be explained below. The definition of the resolving time given b y L e v e r t and S c h e e n is not quite satisfactory as the supposed c o n s t a n c y of the i n t e r v a l of time between the m o m e n t at which the mechanical c o u n t e r has come to rest and the last pulse before this m o m e n t , is only an a p p r o x i m a t i o n if this pulse is not a counted one, the t r u e value of this i n t e r v a l being d e p e n d e n t on the state of the c o u n t e r when the pulse occurs. A really constant resolving time can be obtained b y p r o t e c t i n g the c o u n t e r from pulses t h a t occur before the c o u n t e r has come to rest again. A suitable a r r a n g e m e n t is schematically represented here (Fig. I). The electric pulses excited b y the G e i g e r-M fi 11 e r c o u n t e r device 1, ignite, after being amplified b y the amplifier 2, a t h y r a t r o n - - 725 Physica X
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-
-
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726
I. GILTAY
3, which energises the electro-mechanical counter 4, 5. So long as this counter is not at rest, contact 6 is open and amplifier 2 is biassed off b y a high negative tension. This tension is reduced to a nonblocking value as soon as contact 6 is closed again. Contact 7 is normally closed. J~
Fig. 1. This arrangement gives rise to a problem which may be stated as follows: A series of pulses is to be counted b y a counter. The pulses arriving less than a time h (resolving time) after the occurrence of a pulse which has been counted, are not counted and are supposed to disappear without leaving any trace. If the probability of the occurrence of a pulse in an arbitrary infinitesimal time dt is given by : ,
dqh,
(I)
the probability is required of counting x pulses in an interval of time t, the beginning of which is any arbitrary moment, in the following two cases: A) The counter device is put into action at the beginning of the interval t b y closing contact 7; B) the counter device has been at work for an infinite time (the contact 7 having been always closed). In our computations we have used a notation in agreement with that used in a recent publication dealing with similar problems 3). In this case the probability of counting x pulses in the time t will be denoted by: A,(t). T h e case A .
We choose the beginning of the interval t as our origin of the time axis. The probability of the occurrence of one pulse in each of the x + y infinitesimal intervals dup, beginning at the times up(p = 1 , 2 . . . . . x + y) and lying within the interval t, is given b y : (a . d u l / h ) (a . du2/h) . . . (a . d u , + y / h ) .
(2)
A COUNTER
ARRANGEMENT
WITH CONSTANT RESOLVING TIME
727
The probability of the absence of pulses in the remaining time of the interval is: e~""
(3)
so t h a t the compound probability is: e -~'tlj' • ( a / h ) *+y . dt~l . d u 2 . . . .
(4)
dztx+y.
We will now consider all the cases in which x of the x + y pulses are counted. Generally, each of the x pulses counted will precede some of the y pulses not counted. The first pulse, which in case A is always counted, m a y precede Yt pulses not counted, the second pulse counted Y2 etc., finally the x-th (last) pulse counted m a y precede y , pulses not counted. Pulses occurring after the end of the interval t are not considered. Of course: Yl + 3'2 + . . .
+ Y. = Y.
(5)
The moments of the occurrence of the x pulses counted m a y be: 1.#i, ~.t2, . . . ,
?:tx.
The moments of the occurrence of the yq pulses which occur between the q-th and the (q + 1)-th pulse counted m a y now be denoted : Uq:, %,2 . . . . , %,yq (q -- 1, 2, . . . , x). The condition for these pulses not to be counted is" g'(q ~'Ltq, l ~ U q , 2
~''"
~ [ q , yq--I ~glq, yq ~ ' z # q 2 F h '
(q~-. I , . . . , x ) ,
(6)
and moreover: u.,,. ~ t
(7)
W i t h the conditions (6) and (7) the following system of limits is consistent : %~u~,yq ~(%+h,t) 'l,l,q~ ~tq,yq__1 ~ Uq,yq ~q ~ ~tq,yq--2 ~ Uq,yq--1 .................... Ztq ~
Uq, I
~
(q =
I ..... x) '
(s~, .
..
Uq, 2
if (i, i) is used as a symbol for the smaller of the values i and 1". A.
necessary condition for counting the x pulses ul . . . . . u, is given b y :
O~ul, u l + h ~ u 2 , u 2 + h ~ u ~ ,
..., u..l,+h~u.,u.,~t.
(~),
728
j. GILTAY
With this condition the following system of limits is consistent: ( x - - 1) h ~ ux
~ t
(x---2) h < u x _ t
~u,--h
(x --- 3) h ~ u . _ 2
~ u._t -- h
. . . . . . . . .
(10)
. . . . . . . . . . . . . . . .
0 ~ ux
~ u2--h.
In the interval t it is impossible to count more than t/h pulses. The probability of x pulses to be counted in the interval t(x ~ t/h) and y not to be counted, from which y, after the q-th pulse counted (q = 1, . . . , x), can now be written" t
uz--h
e-'a'qh. (a/h)x+y. f dux f dux_t . . . (x--l)h (x--2)h ua--h
u,a--h
...fdu2fdux. h
0
x
(uq+h,t) ~
Uq,~
Uqfy
~q
II /du,," q Jfduq,_, ...fd'u,., " q
qflJ
Uq
Uq
(11)
Uq
or, performing the integrations, except the first: t
e--a.tlh
.
(a/h)'+'.f
du,. (u=--xh-+-h)X-' . ('-'h',~ii (~-- 1)!
(x--l)h
~=1 y ~ ! :
(h,t--u,)Vx
(12)
y,,!
By summing up all probabilities for which:
Yl + Y2 + . . . + Y, = Y we obtain : t
f,
:-,.,/h. (a/h)'+".j,~u,.
( u , - - x h + h ) *-t {h+h+...+h+(h,t-.u~)} y
~--
~
" i
~
. (13)
(x--1)h
In the last numerator is:
h+h+
... +h+(h,t---u,)=(x--1)h+(h,t--u,)= = (xh, xh
--
h + t -- u,).
To obtain Ax(t) the expression (13) must be summed up over y from 0 to co, so that t
A=(t) = e-°alh. talh~" ,,, . jf du, . (u=--xh - ~ - +h) ~ =-t .etaX,a(z_l)+a(t___ux)/h). (14~ {x--l)h
A COUNTER
ARRANGEMENT
WITH
CONSTANT RESOLVING
TIME
729
We now distinguish the cases: 1) x ~ t/h + I, for which: A~(t) = O, 2) x ~ t / h ,
but ~t/h+l,
for which:
t
A,(t)
=[{a(u,
- - x h + h)/h} *-1
( x - - 1)!
d (x--l)h
•
e--'~("x-'~+h)/h . a . d u . / h
(15.,
and 3) x ~ t/h, for which: t--h
A,(t)
= f{a(u,
J
- - x h + h)/h} ~-1
( x - - 1)!
• e -'(u'-'h)lh . a . d u . / h +
(x--l)h t
+ fia(u,--( x -xh - + 1)! h)}'-'
• a . du,,/h.
"
(16)
4--h
Now let a(u,, - - xh + h)/h = k
(17)
and [j] a symbol for the integer that fulfils the condition:
i -- i < [i] < y.
(18)
Then :
o
(x > [qh] + I)
~("~+~,_,_ ~ ,x--1) T " "-~ " d~ A:,(t)= •
(.~=[qh.]+ a)
a(tlk---x+l)
a(t/h--x)* x!
:
M-'
.
• e~('/h-*) + I ,__2__~)!. e - k . dk a tx--1
(19)
(x<[t/h])
a(tllv--x)
With the aid of a table of the incomplete F-function A . ( t ) can in any particular case easily be found from this expression. A . ( t ) can also be written in a completely integrated form, with the aid of the identity : k
fk" 0
. e -~ • dk -~ 1 - -
e -k "p=o ~ ~kp.
(m -- a positive integer). (20)
730
J. GILTAY
~ r e d e d u c e f r o m (19) a n d (20) :
"o 1
(x > It~hi + 1) e''a('/h--x+l) ,~x { a ( t / h - - x + 1)}p "p=o P!
(x = It~hi + 1)
-A,,(O = {a(t/h--x)y'. e-,,c,/~x~ + ~-,./,,-,,~ :~ {~(t/h.--x)}P x]
,=0
_ ~-,./h-,,+~ x~ {a(t/h--~ + 1)}p
(21)
P!
(x < It/hi)
F r o m (21) it is seen at once, t h a t , OO
E A,(t) = 1,
(22)
0
is satisfied, for, e x c e p t e d one t e r m which is e q u a l to 1, all t h e t e r m s a r e of t h e f o r m :
e_,~,l~__o { a ( t / h - - r ) } " •
s!
(s = 0, .
.
.. It~hi).
.. r; r ~ 1 .
.
"
(23)
'
As e v e r y c o m b i n a t i o n of r a n d s occurs twice, once w i t h a positive a n d once w i t h a n e g a t i v e sign, these t e r m s cancel out. N o w we define t h e e x p e c t a t i o n s • OO
M,(t) = ~ x' . Ax(t).
(24)
x~O
F r o m (21) it follows t h a t :
U,(t) = ~ x" . A.(t) + (It~hi + 1)'. Atqh]+l(t ) x~O
=E'~Jx' . A,(t) + (It~hi + 1 ) ' - x~0
(It/hi + 1)~ e--~*/~--t*/hJI . ./hl Z {a(t/h - - [t/h])}~ •
pffio
(25)
P!
E x c e p t t h e t e r m (It~hi + l) ~ all t e r m s are of t h e f o r m (23). E v e r y c o m b i n a t i o n of r a n d s occurs twice, once w i t h t h e coefficient
--(r + 1)' a n d once w i t h t h e coeffcient :
A COUNTER ARRANGEMENT WITH CONSTANT RESOLVING TIME
731
It is easily proved from (25) with the aid of (20) that : a(tlh--r)
Mdt ) =
k" e-k" dk. + 1 ) ' - - r'}. f ~.t"
[qh] Y~{(r r~0
(26)
0
The mean value of x is given by M 1(t), the mean square deviation of x from its mean value b y {Mr(t)} 2.
]ll2(t ) --
(27)
The formulae (19), (20) and (26) allow to compute all quantities in the case A that may be wanted. The case B. value of:
For computing this case we must first know the Lim
h. Ml(t)
v--~oo
(28,)
t
We will therefore approximate a(tlh--r)
-
,=oJ-i-f.
(29)
"
o il
The integration and summation of the function k' r!"
__
(30)
6 -k
must be extended over all values of k and all integer values of r within a triangle formed b y the k-axis, the r-axis and the line r
k
tlh + a . tl-------~-
I.
(31)
In a k-r plane (Fig. 2) the function (30) has its principal Values in the proximity of the line
r = k.
(32)
The intersection of the lines (31) and (32) is a point of the line a t/h. k --'-I+~-"
(33)
For our purpose, (29) will be sufficiently approximated if the line
732
j. GILTAY
(33) is used as a boundary instead of the line (31), so that, for large values of t: a--K-, tlh l+a
M~(t) '~,=o-- ~" e-k" dk
-- 1
+~"
o
and therefore: Lim h . M 1 ( t ) t->oo t I
a
1 +a
(34)
i ...........
-X::-::I_ /1\ I 0
/ ~
\ T~'t/h
a. t/b
Fig. 2. The probability of contact 6 being found open at any arbitrary moment of a very long time during which contact 7 is always closed equals the time during which contact 6 is open in that time, divided by that time itself and is thus given b y (28) or, according to (34), b y a
- -
(35)
l+a'
and that of contact 6 being found closed will be given by: 1
a l+a
1
=-- -l +-a
(36)
Let a finite interval of time t begin at any arbitrary m o m e n t then there is a probability (36) that the contact 6 will be found closed and a probability 1 A,(/), (37) l+a" that moreover x pulses will be counted in the interval t.
A COUNTER
ARRANGEMENT
WITH CONSTANT RESOI.VING TIME
733
There is a probability (35) for the contact~5 to be found open at the beginning of the interval t. Within a time h it certainly will be closed again, as the total time during which the contact 6 can be open without interruption equals the resolving time h. The a r b i t r a r y m o m e n t of the beginning of an interval t, has equal probability to hit a n y of the equal infinitesimal intervals d T w i t h i n a complete time h during which the contact 6 is open. Therefore, if the contact 6 is found open, the probability for t h a t contact to close after a time T, but before a time T + d T is:
ar/h 0
(1" < h) (T >
h).
(38)
The contact being closed after a time between T and T + dT, the probability of cornering x pulses in the remaining time t - - T can be expressed by:
A,(t - - T).
(39)
The probability Bx(t) of x pulses being counted in an interval g, which began at an a r b i t r a r y m o m e n t (the counter arrangement having been at work all the time), is now given by:
B,(t)-
1 1 +~"
(h,t)
A,(t) + 1 +------~" a -j'A,(t__ T) " dT/h.
(40)
0
If t > h, T can have all values between 0 and h, but if t < h, T can only v a r y from 0 to t. For t h a t reason the smaller of the values h and t appears as the upper limit of the integral in (40). If we define, analogous to (24): O0
N~(t) = Z x' . B.(t),
(41)
it follows from (40) and (24) that" (h,t)
Ni(t)-
+1 ~1"
M,(t) + 1-4---~'. a /'M,(t - - T) " dr/h.
(42)
0
"File mean value of x is found for i = 1, the mean square deviation is found b y an expression similar to (27). If t >> h and a ~ 1 there will be very little difference between
734
A C O U N T E R A R R A N G E M E N T W I T H CONSTANT R E S O L V I N G T I M E
values c o m p u t e d for analogous quantities in the cases A and B. As an example for the case, t h a t t is not large we mention :
Ao(h ) = e-2~,
Bo(h ) = e - ' / ( l + a).
(43)
Remark. The problem dealt with can be interpreted as a special case of a well-known congestion problem : A telephone traffic consisting of conversations of the duration h has one line available. The agreement is, t h a t calls which find the line engaged, disappear without leaving a n y trace. The probability for the line being engaged is given b y a formula due to A. K. E r I a n g . Our expression (35) agrees w i t h this formula. C. P a 1 m 2) has published a proof for the validity of the formula for conversations of different types, constant durations included. An i m p o r t a n t difference between our problem and the telephone problem is, t h a t in the latter it is of more ilnportance to sum up (13) over x t h a n over y, as the probability for y calls being lost in an interval t is then found. Received July 28th, 1943.
REFERENCES I) C. L e v e r t and W. L. S c h e e n , Probability fluctuations of discharges in a Geiger-Mfiller counter produced by cosmic radiation; Physica, Vol. X, no. 4, April 1943, page 225. 2) C. P a l m, Analysis of the Erlang traffic formulae for busy-signal arrangements; Ericsson Technics 1938, no. 4 page 39. 3) L. K o s t e n, Over blokkeerings- en wachtproblemen; diss. Delft 1942.
November
1943
A COUNTER ARRANGEMENT W I T H CONSTANT RESOLVING TIME b y J. G I L T A Y Zusarnmenfassung E s w i r d eine d e r a r t i g e A b g n d e r u n g e i n e r y o n L e v e r t und S c h e e n x) b e n u t z t e n Z A h l v o r r i c h t u u g v o r g e s c h l a g e n , d a s s s i c h e i n e k o n s t a n t e r Aufl~Ssungszeit d e f i n i e r e n 1/isst. Die W a h r s c h e i n l i c h k e i t , class h i e r b e i eine b e s t i m m t e Z a h l y o n I m p u l s e n i n e i n e r g e g e b e n e n Z e i t gezAhlt w i r d , w i r d e x a k t b e s t i m m t . D a s P r o b l e m h g n g t z u s a m m e n m i t elnero bekannten yon E r 1 a n g gel6sten Problem aus der Fernsprecht e c h n i k 3).
Introduction. In a recent p a p e r 1) L e v e r t and S c h e e n deduce expi'essions for the n u m b e r of discharges mechanically c o u n t e d in a finite i n t e r v a l of time, when a G e i g e r-M fi 11 e r c o u n t e r is exposed to cosmic radiation. The v e r y interesting m e t h o d b y which the result is obtained can be applied to various o t h e r problems. This is especially the case for L e v e r t and S c h e e n's problem in a modified form, as will be explained below. The definition of the resolving time given b y L e v e r t and S c h e e n is not quite satisfactory as the supposed c o n s t a n c y of the i n t e r v a l of time between the m o m e n t at which the mechanical c o u n t e r has come to rest and the last pulse before this m o m e n t , is only an a p p r o x i m a t i o n if this pulse is not a counted one, the t r u e value of this i n t e r v a l being d e p e n d e n t on the state of the c o u n t e r when the pulse occurs. A really constant resolving time can be obtained b y p r o t e c t i n g the c o u n t e r from pulses t h a t occur before the c o u n t e r has come to rest again. A suitable a r r a n g e m e n t is schematically represented here (Fig. I). The electric pulses excited b y the G e i g e r-M fi 11 e r c o u n t e r device 1, ignite, after being amplified b y the amplifier 2, a t h y r a t r o n - - 725 Physica X
-
-
-
46*
726
I. GILTAY
3, which energises the electro-mechanical counter 4, 5. So long as this counter is not at rest, contact 6 is open and amplifier 2 is biassed off b y a high negative tension. This tension is reduced to a nonblocking value as soon as contact 6 is closed again. Contact 7 is normally closed. J~
Fig. 1. This arrangement gives rise to a problem which may be stated as follows: A series of pulses is to be counted b y a counter. The pulses arriving less than a time h (resolving time) after the occurrence of a pulse which has been counted, are not counted and are supposed to disappear without leaving any trace. If the probability of the occurrence of a pulse in an arbitrary infinitesimal time dt is given by : ,
dqh,
(I)
the probability is required of counting x pulses in an interval of time t, the beginning of which is any arbitrary moment, in the following two cases: A) The counter device is put into action at the beginning of the interval t b y closing contact 7; B) the counter device has been at work for an infinite time (the contact 7 having been always closed). In our computations we have used a notation in agreement with that used in a recent publication dealing with similar problems 3). In this case the probability of counting x pulses in the time t will be denoted by: A,(t). T h e case A .
We choose the beginning of the interval t as our origin of the time axis. The probability of the occurrence of one pulse in each of the x + y infinitesimal intervals dup, beginning at the times up(p = 1 , 2 . . . . . x + y) and lying within the interval t, is given b y : (a . d u l / h ) (a . du2/h) . . . (a . d u , + y / h ) .
(2)
A COUNTER
ARRANGEMENT
WITH CONSTANT RESOLVING TIME
727
The probability of the absence of pulses in the remaining time of the interval is: e~""
(3)
so t h a t the compound probability is: e -~'tlj' • ( a / h ) *+y . dt~l . d u 2 . . . .
(4)
dztx+y.
We will now consider all the cases in which x of the x + y pulses are counted. Generally, each of the x pulses counted will precede some of the y pulses not counted. The first pulse, which in case A is always counted, m a y precede Yt pulses not counted, the second pulse counted Y2 etc., finally the x-th (last) pulse counted m a y precede y , pulses not counted. Pulses occurring after the end of the interval t are not considered. Of course: Yl + 3'2 + . . .
+ Y. = Y.
(5)
The moments of the occurrence of the x pulses counted m a y be: 1.#i, ~.t2, . . . ,
?:tx.
The moments of the occurrence of the yq pulses which occur between the q-th and the (q + 1)-th pulse counted m a y now be denoted : Uq:, %,2 . . . . , %,yq (q -- 1, 2, . . . , x). The condition for these pulses not to be counted is" g'(q ~'Ltq, l ~ U q , 2
~''"
~ [ q , yq--I ~glq, yq ~ ' z # q 2 F h '
(q~-. I , . . . , x ) ,
(6)
and moreover: u.,,. ~ t
(7)
W i t h the conditions (6) and (7) the following system of limits is consistent : %~u~,yq ~(%+h,t) 'l,l,q~ ~tq,yq__1 ~ Uq,yq ~q ~ ~tq,yq--2 ~ Uq,yq--1 .................... Ztq ~
Uq, I
~
(q =
I ..... x) '
(s~, .
..
Uq, 2
if (i, i) is used as a symbol for the smaller of the values i and 1". A.
necessary condition for counting the x pulses ul . . . . . u, is given b y :
O~ul, u l + h ~ u 2 , u 2 + h ~ u ~ ,
..., u..l,+h~u.,u.,~t.
(~),
728
j. GILTAY
With this condition the following system of limits is consistent: ( x - - 1) h ~ ux
~ t
(x---2) h < u x _ t
~u,--h
(x --- 3) h ~ u . _ 2
~ u._t -- h
. . . . . . . . .
(10)
. . . . . . . . . . . . . . . .
0 ~ ux
~ u2--h.
In the interval t it is impossible to count more than t/h pulses. The probability of x pulses to be counted in the interval t(x ~ t/h) and y not to be counted, from which y, after the q-th pulse counted (q = 1, . . . , x), can now be written" t
uz--h
e-'a'qh. (a/h)x+y. f dux f dux_t . . . (x--l)h (x--2)h ua--h
u,a--h
...fdu2fdux. h
0
x
(uq+h,t) ~
Uq,~
Uqfy
~q
II /du,," q Jfduq,_, ...fd'u,., " q
qflJ
Uq
Uq
(11)
Uq
or, performing the integrations, except the first: t
e--a.tlh
.
(a/h)'+'.f
du,. (u=--xh-+-h)X-' . ('-'h',~ii (~-- 1)!
(x--l)h
~=1 y ~ ! :
(h,t--u,)Vx
(12)
y,,!
By summing up all probabilities for which:
Yl + Y2 + . . . + Y, = Y we obtain : t
f,
:-,.,/h. (a/h)'+".j,~u,.
( u , - - x h + h ) *-t {h+h+...+h+(h,t-.u~)} y
~--
~
" i
~
. (13)
(x--1)h
In the last numerator is:
h+h+
... +h+(h,t---u,)=(x--1)h+(h,t--u,)= = (xh, xh
--
h + t -- u,).
To obtain Ax(t) the expression (13) must be summed up over y from 0 to co, so that t
A=(t) = e-°alh. talh~" ,,, . jf du, . (u=--xh - ~ - +h) ~ =-t .etaX,a(z_l)+a(t___ux)/h). (14~ {x--l)h
A COUNTER
ARRANGEMENT
WITH
CONSTANT RESOLVING
TIME
729
We now distinguish the cases: 1) x ~ t/h + I, for which: A~(t) = O, 2) x ~ t / h ,
but ~t/h+l,
for which:
t
A,(t)
=[{a(u,
- - x h + h)/h} *-1
( x - - 1)!
d (x--l)h
•
e--'~("x-'~+h)/h . a . d u . / h
(15.,
and 3) x ~ t/h, for which: t--h
A,(t)
= f{a(u,
J
- - x h + h)/h} ~-1
( x - - 1)!
• e -'(u'-'h)lh . a . d u . / h +
(x--l)h t
+ fia(u,--( x -xh - + 1)! h)}'-'
• a . du,,/h.
"
(16)
4--h
Now let a(u,, - - xh + h)/h = k
(17)
and [j] a symbol for the integer that fulfils the condition:
i -- i < [i] < y.
(18)
Then :
o
(x > [qh] + I)
~("~+~,_,_ ~ ,x--1) T " "-~ " d~ A:,(t)= •
(.~=[qh.]+ a)
a(tlk---x+l)
a(t/h--x)* x!
:
M-'
.
• e~('/h-*) + I ,__2__~)!. e - k . dk a tx--1
(19)
(x<[t/h])
a(tllv--x)
With the aid of a table of the incomplete F-function A . ( t ) can in any particular case easily be found from this expression. A . ( t ) can also be written in a completely integrated form, with the aid of the identity : k
fk" 0
. e -~ • dk -~ 1 - -
e -k "p=o ~ ~kp.
(m -- a positive integer). (20)
730
J. GILTAY
~ r e d e d u c e f r o m (19) a n d (20) :
"o 1
(x > It~hi + 1) e''a('/h--x+l) ,~x { a ( t / h - - x + 1)}p "p=o P!
(x = It~hi + 1)
-A,,(O = {a(t/h--x)y'. e-,,c,/~x~ + ~-,./,,-,,~ :~ {~(t/h.--x)}P x]
,=0
_ ~-,./h-,,+~ x~ {a(t/h--~ + 1)}p
(21)
P!
(x < It/hi)
F r o m (21) it is seen at once, t h a t , OO
E A,(t) = 1,
(22)
0
is satisfied, for, e x c e p t e d one t e r m which is e q u a l to 1, all t h e t e r m s a r e of t h e f o r m :
e_,~,l~__o { a ( t / h - - r ) } " •
s!
(s = 0, .
.
.. It~hi).
.. r; r ~ 1 .
.
"
(23)
'
As e v e r y c o m b i n a t i o n of r a n d s occurs twice, once w i t h a positive a n d once w i t h a n e g a t i v e sign, these t e r m s cancel out. N o w we define t h e e x p e c t a t i o n s • OO
M,(t) = ~ x' . Ax(t).
(24)
x~O
F r o m (21) it follows t h a t :
U,(t) = ~ x" . A.(t) + (It~hi + 1)'. Atqh]+l(t ) x~O
=E'~Jx' . A,(t) + (It~hi + 1 ) ' - x~0
(It/hi + 1)~ e--~*/~--t*/hJI . ./hl Z {a(t/h - - [t/h])}~ •
pffio
(25)
P!
E x c e p t t h e t e r m (It~hi + l) ~ all t e r m s are of t h e f o r m (23). E v e r y c o m b i n a t i o n of r a n d s occurs twice, once w i t h t h e coefficient
--(r + 1)' a n d once w i t h t h e coeffcient :
A COUNTER ARRANGEMENT WITH CONSTANT RESOLVING TIME
731
It is easily proved from (25) with the aid of (20) that : a(tlh--r)
Mdt ) =
k" e-k" dk. + 1 ) ' - - r'}. f ~.t"
[qh] Y~{(r r~0
(26)
0
The mean value of x is given by M 1(t), the mean square deviation of x from its mean value b y {Mr(t)} 2.
]ll2(t ) --
(27)
The formulae (19), (20) and (26) allow to compute all quantities in the case A that may be wanted. The case B. value of:
For computing this case we must first know the Lim
h. Ml(t)
v--~oo
(28,)
t
We will therefore approximate a(tlh--r)
-
,=oJ-i-f.
(29)
"
o il
The integration and summation of the function k' r!"
__
(30)
6 -k
must be extended over all values of k and all integer values of r within a triangle formed b y the k-axis, the r-axis and the line r
k
tlh + a . tl-------~-
I.
(31)
In a k-r plane (Fig. 2) the function (30) has its principal Values in the proximity of the line
r = k.
(32)
The intersection of the lines (31) and (32) is a point of the line a t/h. k --'-I+~-"
(33)
For our purpose, (29) will be sufficiently approximated if the line
732
j. GILTAY
(33) is used as a boundary instead of the line (31), so that, for large values of t: a--K-, tlh l+a
M~(t) '~,=o-- ~" e-k" dk
-- 1
+~"
o
and therefore: Lim h . M 1 ( t ) t->oo t I
a
1 +a
(34)
i ...........
-X::-::I_ /1\ I 0
/ ~
\ T~'t/h
a. t/b
Fig. 2. The probability of contact 6 being found open at any arbitrary moment of a very long time during which contact 7 is always closed equals the time during which contact 6 is open in that time, divided by that time itself and is thus given b y (28) or, according to (34), b y a
- -
(35)
l+a'
and that of contact 6 being found closed will be given by: 1
a l+a
1
=-- -l +-a
(36)
Let a finite interval of time t begin at any arbitrary m o m e n t then there is a probability (36) that the contact 6 will be found closed and a probability 1 A,(/), (37) l+a" that moreover x pulses will be counted in the interval t.
A COUNTER
ARRANGEMENT
WITH CONSTANT RESOI.VING TIME
733
There is a probability (35) for the contact~5 to be found open at the beginning of the interval t. Within a time h it certainly will be closed again, as the total time during which the contact 6 can be open without interruption equals the resolving time h. The a r b i t r a r y m o m e n t of the beginning of an interval t, has equal probability to hit a n y of the equal infinitesimal intervals d T w i t h i n a complete time h during which the contact 6 is open. Therefore, if the contact 6 is found open, the probability for t h a t contact to close after a time T, but before a time T + d T is:
ar/h 0
(1" < h) (T >
h).
(38)
The contact being closed after a time between T and T + dT, the probability of cornering x pulses in the remaining time t - - T can be expressed by:
A,(t - - T).
(39)
The probability Bx(t) of x pulses being counted in an interval g, which began at an a r b i t r a r y m o m e n t (the counter arrangement having been at work all the time), is now given by:
B,(t)-
1 1 +~"
(h,t)
A,(t) + 1 +------~" a -j'A,(t__ T) " dT/h.
(40)
0
If t > h, T can have all values between 0 and h, but if t < h, T can only v a r y from 0 to t. For t h a t reason the smaller of the values h and t appears as the upper limit of the integral in (40). If we define, analogous to (24): O0
N~(t) = Z x' . B.(t),
(41)
it follows from (40) and (24) that" (h,t)
Ni(t)-
+1 ~1"
M,(t) + 1-4---~'. a /'M,(t - - T) " dr/h.
(42)
0
"File mean value of x is found for i = 1, the mean square deviation is found b y an expression similar to (27). If t >> h and a ~ 1 there will be very little difference between
734
A C O U N T E R A R R A N G E M E N T W I T H CONSTANT R E S O L V I N G T I M E
values c o m p u t e d for analogous quantities in the cases A and B. As an example for the case, t h a t t is not large we mention :
Ao(h ) = e-2~,
Bo(h ) = e - ' / ( l + a).
(43)
Remark. The problem dealt with can be interpreted as a special case of a well-known congestion problem : A telephone traffic consisting of conversations of the duration h has one line available. The agreement is, t h a t calls which find the line engaged, disappear without leaving a n y trace. The probability for the line being engaged is given b y a formula due to A. K. E r I a n g . Our expression (35) agrees w i t h this formula. C. P a 1 m 2) has published a proof for the validity of the formula for conversations of different types, constant durations included. An i m p o r t a n t difference between our problem and the telephone problem is, t h a t in the latter it is of more ilnportance to sum up (13) over x t h a n over y, as the probability for y calls being lost in an interval t is then found. Received July 28th, 1943.
REFERENCES I) C. L e v e r t and W. L. S c h e e n , Probability fluctuations of discharges in a Geiger-Mfiller counter produced by cosmic radiation; Physica, Vol. X, no. 4, April 1943, page 225. 2) C. P a l m, Analysis of the Erlang traffic formulae for busy-signal arrangements; Ericsson Technics 1938, no. 4 page 39. 3) L. K o s t e n, Over blokkeerings- en wachtproblemen; diss. Delft 1942.