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Timeout thresholds for dialing machines
Volume 5, N/amber 2
OPERATIONS t~ESEARCH LETTERS
July 1986
TIMEOUT THRESHOLDS FOR DIALING MACHINES
David A. HOEFLIN and Yonalan LEVY AT&T Bell Laboratories, Holmdei. NJ 07733, USA Received October 1985 Revised March 1986
We consider the problem of determining an optimal fimeout threshold for calls to be usedby dialing machines. We derive the rate of completed calls as a simple functio,~ of the timeout threshold and show a condition for the exisier~c, of .~ unique optimal timcout threshold. The results can he used in determining the staff or equipment required. stochastic model applications • replacement * renewaJ processes
I. Introduction How long do you wait for someone to answer their phone - 5, 6, 10 rings? The answer will probably depend on such factors as whom you're calling, how sure you are that they are there to answer, and how desperate you are in reaching the other party. Now, changing the problem slightly, suppose you have a long list of phone numbers to call, and you wish to reach as many people on the list as quickly as possible; how long should you wait for an answer on each call? The use of dialing machines by industry and government makes the answer to the previous question of interest. We define a dialing machine as a machine that can store a list of phone numbers, call a phone number, detect ringing, time out and start a new call if there is no answer, or, if there is an answer, either play a tape or connect. the call to an operator. Clearly, by changing the timeout threshold, T, we can control ;~(T), the 'arrival ra~e' of completed calls. A~ ~pfimal choice of T can maximize the arrival rate generated by one dialing machine. Thus, given N madunes, each generating the maximum arrival rate, one can determine *hz. ~cq~,ired number of oFt,,,~t,~" ~r given a number of operators, one can find the minir~:,~r:, ~umber o~" machines (or the setting of ,~e .~.... _~ ~ate) needed to maintain a certain level of c,~¢upancy. Furthermore, maximizing the arrival rate not only leads to maximizing profit for the
individual but ~,I~o to more efficient use of the gk~bal network resources like ringing circuits and t ~ n k s held for ringing. This problem can be formulated as a variation of age replacement pelicies studied in renewal th.eory, cf., e.g., [1,2,4]. We differ from the usual age replacement problem in that we seek to maximize our "failure rate', ~,, instead of minimizing, and we allow a positive probability of "no failure', i.e., no answer. In Section 2 we derive an expression for A(T), the rate of answered calls, given the timmut threshold T, and show when there exists a unique,, finite, positive T* which maximizes X. Section 3 shows two examples for which we demonstrate our results and contains a few remarks on improper exponential distributions.
2. Analysis l e t H be a positive random variable representing the time from the start of dialing until the ~tart of the first fing~ The expected value of H is denotzd by H. Next, let P(t) be the probability that a call is answered with t seconds after the :tact ~f the firs~ rinse By .the dcf,.'nition of P, we have that P ( 0 ) = 0. We assume that P(t) is an improper c.d.L in the sense that P(0o)ffi a < I. Hence, there is a time, Tm~ (may be infinite), beyond which an answer is not expected, so there is positive probability, 1 - a , of no answer. Also,
0167-63"]7/86/$3.50 © 1986, Elsevier Science Publishers B.V. (No~-H~land)
57
OPERATIONS RESEARCH LETTERS
Volume 5, Number 2
we assume that P is independent of the time of day. For practical application, this need not be true, but one could solve the problem assuming dependence on the time of day, and hence find T which maximizes 2t as function of the time of day. For example, one could expect there to be a general 'day' P an a general 'evening' P which c o , ! a be used to determine two optimal values of T. To inc-.~:porate the possibility of a busy signal, let Pb denote the ~robability of a busy signal. Let K be a non-negative ~andom variable, which represents the time from the start of the busy signal to the time the machine starts a new call. Let ~ l = H + p b K , and P = ( l - p h ) P ; then all the following results could have H ~eplaced b y / t and P replaced by P. Let C(T) be the expected cycle time; that is, the expected time spent o n . ~ e ea)l attempt. Then, for a given P~ we have -
dPix)+(I-P(T))T
C(T)=H+ -
= H + T-
Plx)dx. i
As:~uming the dialing tria|s are independent, a dialing machine can be modeled as a renewal process. Therefore, by standard renewal arguments, cf. [1,2], the average rate of answered calls corresponding to a timeout T is X ( T ) = P ( T ) =_
C(T)
P(T) ~ + T .... (x)dx
Jttly 1986
Lemma I. Let f be a non-negative differentiable function, g be a positive di/ferentiable function, such that f ' > 0, g' > 0, and ( f ' / s ' ) is stricdy decreasing. Then, f / g is one of the following: (a) strictly increasing, (b) strictly decreasing, (c) strictly increasing and then strictly decreasing. The proof of Lemma 2 uses the equality
f'
g'
The proof shows the continuous function ( f / g ) " cannot go f r o ~ being zero to being positive, and it cannot be zero on an interval, if ( f ' / g ' ) is strictly decreasing (see the Apnendix). Theorem 1 follows by setting f - - P, g ffi C, so that ~, - - f / g and h -f ' / g ' , and noting that h(0)ffi ~,(oo)= 0, and that ~ ( T ) ~ 0 . Moreove,', Theorem 1 can be generalized to cover hazard functions that are first strictly increasing and then strictly decreasing, e.g., strictly concave hazard functions. Thus, such distributions as the improper exponential or the Weibuli have unique optimal timeout thresholds. If the condition of Theorem 1 is relaxed to h being non-increasing, the T* still exists, but it may not be unique.
3. Examples .
(I)
3.1. Improper exponential distribution for T_>0. Thus, !n the case P ' exists, it is stra;giltforward to determine a T* which maximizes h. There are two cases, either T* = Tm~ or T* satisfies ?,'(T*) =- 0. Interestingly, h'(T*) = 0 can be written as
P'(r*) )~(T*)= I _ P ( T , ) ,
As a first example, w.: present en exponentiallike distribution for P. Let
v(t)
:=
-
e- :"),
where !./ja is the mean tithe to answer given there is an answer. Then,
where the right-hand side is the hazard function of P. The following theorem gives sufficient conditions for the existence of a unique T*.
X(T)
Theocem 1. l f h(T), the hazard function of P, is strictly decreasing, then there exists a unique, finite, positive T* that maximizes h, and ~ is strictly increasing up to T* and strictly decreasing afterwards.
If H > O and a < 1, then the hypothesis of Theorem I is satisfied, and the optimal value T* has to satisfy the equation
The proof of Theorem 1 follows from the following known calculus result.
Note that the conditions H > 0 and a < 1 are also necessary for tke existence of a unique T* > 0. If
58
= -
H + T- a
/;,,- e1- "-)
-.
c -~'x)dx
pH e ~'r = 1 + I~T+ (1 - a ) "
Velume 5, Number 2
OPERATIONS RESL~.P.CH LEIq'ERS
H = 0 and a < 1, then h(0+)---c~# is the maxi~mm value, implying that the machine's timeout threshold is 0. If tl > 0 and a = 1, ~, is an increasing function of T Thug, the machine should always wait for art answer. Last, if H ffi 0 and a = 1, then .~ is a constant. Heace, it would not matter what T is, the arrival rate would ~emain the same. Often the exact value ot bt is unknown and on!y assume to be in some interval away from zero. Several intuitive ideas can be shown to be true. For each ~. Let T~* be the corresponding optimal threshold. Then, by simple manipulations, one can show the following results: (1) T~* is ~ strictly decreasing function of In. (2) h(/~, T~*) is f~ncreasing in p. (3) h(p, T) is Lipsehitz in T for fixed in. In fact, for 8 > 0, we have
I"
I-
,
~,
2
i7.
6 < ] 8. < 4 ( C ( , 4 , ) ) 2 [ 2P(T,*)(1 - P(T~*)) Brief derivations of these results are given in the Appendix. Thus, results one and two agree with our intuition that the shorter the mean time to answer, l/in, the shorter the optimal timeout and the higher the arrival rate. Thus~ given an interval where the true value of ~ lies, we can give upper and lower bounds on the true arrival rate and on T~*. The third result is true for a general differentiable distribution P, and it can be viewed as a measure of the insensitivity of h to T. The smaller the Llpschitz constant, the less sensitive h is to changes in T. The Lipschitz constant for 8 < 0, i.e., for T < T~*, is larger but is bounded by max r < r . P ' ( T ) / C ( 0 ) .
Jui~ 1986
for which ,~,,
> -
a
. + ~'-'(i
-
(3)
e,)
Using data from [5], we determined that the optimal value of T* to maximize h is T * - - 3 6 , i.e., approximately 6 rings. The data u ~ d was an~ average across various types of called parties and may not be representative of special targetted called popuiauons.
4. Conclusion We have derived the arrival rate of calls as a function of the :imeoL,t threshold of the dialing machine. Moreover, we have given conditions for the o~stence a~ld u,iqueness of a nonotrivial optimal timeout threshold, T*, wlfich maximizes 2L Although T* depends on the actual distribution of answer times, the arrival rate, 2~, is less sensitive to the choice of T. The examples given in Section 3 show that waiting until one is absoiutdy sure that there will be no answer may not be the optimal strategy for dialing machine use. The usefulness of our result can be seen in that by setting the timeout threshold, we can fix g at a given value and then determine the required number of operators, or amount of equipment, to handle the traffic.
Acknow~nt We would like to acknowlgdge J.C. Frauenthai L-, this problem. for his suggestiong and ;m,,,rest ~
3.2. Discrete distlqbutions
Appendix
In the discrete ca~e,
P"
x(.)
(2)
- ~ + n O - e . ) + Z'~jp,'
where n denotes the number of seconds after the start of the first ring, pj is the probability that the plz,)t~e is answered in the j t h second, P,~.'/Pi, and H is as before. Theorem 1 can be generalized to the discrete case by replacing derivatives with first order differences. Then, if the hazard function is strictly decreasing, the optimal n* is the first n
We provide a proof of Lemma 1 and a sketch of the proofs for the three statements in Section 3.1.
Proof of/.,emma I We need to only show that f / g cannot be strictly decreasing and then become increasing. First, observe that (f/g)" is co,afinuous and 59
Volume 5, Number 2
OPERATIONS RF~EARCH LETTERS
Derivations of results in Section 3.
can be w~tt~n as
(f'(t) f(t))g'(t)
(f) "(t)=
g,(t )
g(t)
g(:)"
(A.1)
Next, we show that (f/g)" cannot be zero and then become positive. Assume to the contrary that there exists tt, t: > 0, such that (g
), (t,)
(A.3)
By eq. (A.3), we have for t in (~, t 2)
f'(t) --> f(t)
(A.5)
g(t)"
By eqs. (A.2) and (A.3), we have for t in (h, t~.) f(t___~)> f(G___))
(A.6)
g(tl)"
By hypothesis, [ ' / g ' for t in (t 1, t 2)
/'it,______ 2) f q
,
is strictly decreasing, thus
t_____)
(A.7)
g'(t) "
Hence, we have the following inequal':ty for i in (ll' t2}:
f ( t ) > ~f ( t , ) = ~f ' ( t l ) > i f ( t ) > f(t~ gtt) ~(t,~ f(t,) ~ ~7)"
f(t)
s(t)
constant.
But this is a contradiction to f ' / g ' being strictly decreasing. This concludes the proof of Lemma 1.
60
oc,,
Thus, for p. fixed, f(t~, T) is strictly increasing, so there exists a unique T~* such that f(l~. T~*)--O. Further, for tt > .a,,/(tL, T~*) isstrictly increasing. Hence, for /x2:~,Z,, 0 = f ( ~ t : , T~*)
(A.S)
which is a contradiction. Therefore, f / g must be either a monotone function or a function which is increasing then decreasing. To prove strictness, we show that ( f / g ) ' cannot be zero on an interval. Again, assume to the contrary that ( f / g ) ' is zero on an interval 1. Then, using eq. (A.1), we have for l in I
/'(t) - f(t) =
lim f ( p , T ) = T-'*
(A.4)
g'(t,) g(tt)"
>
(I -,0"
/A~,, ~') > 0, f,,,(~, r)>0.
f'(t,)=f(h)
g'(t,)
a~
/r(~, r) >0, /(~,,o)
By eq. (A.I), we have
g(t)
m
f(y., T) = evr - 1 - # T -
(A.2)
>0.
g'(t)
To derive the three results of Section 3.1, first define
Then, for #, T > 0, =0
and for t in (t 1, t2) (f)'(t)
July 1986
References [I] R.E. Barlow and F. Proschan, "Comparison ot ,'..'.placement policies and renewal theory implications", Ann. Math.
Stat.35, 577-589 (1964). [2] E. Cinlar, Introduction to Stochastic Processes, PrenticeHall, Englewood Cliffs, NJ, 1975. [3] F.P. Duffy and R.A. Mercer, "A study of network dialing and customer behavior during direct-distance dialing call attempts in the U.S.A.", BSTJ 57, 1-33 (1978). [4] J. Medhi, StochasticProcesses,Wiley Eastern, New Delhi, 1982. [5] A. Myskja and O.O. Walman. "A statistical study of lelephone traffic data with emphasis on subscriber behavior", 7th International Teletraffic Congress, 1973.
OPERATIONS t~ESEARCH LETTERS
July 1986
TIMEOUT THRESHOLDS FOR DIALING MACHINES
David A. HOEFLIN and Yonalan LEVY AT&T Bell Laboratories, Holmdei. NJ 07733, USA Received October 1985 Revised March 1986
We consider the problem of determining an optimal fimeout threshold for calls to be usedby dialing machines. We derive the rate of completed calls as a simple functio,~ of the timeout threshold and show a condition for the exisier~c, of .~ unique optimal timcout threshold. The results can he used in determining the staff or equipment required. stochastic model applications • replacement * renewaJ processes
I. Introduction How long do you wait for someone to answer their phone - 5, 6, 10 rings? The answer will probably depend on such factors as whom you're calling, how sure you are that they are there to answer, and how desperate you are in reaching the other party. Now, changing the problem slightly, suppose you have a long list of phone numbers to call, and you wish to reach as many people on the list as quickly as possible; how long should you wait for an answer on each call? The use of dialing machines by industry and government makes the answer to the previous question of interest. We define a dialing machine as a machine that can store a list of phone numbers, call a phone number, detect ringing, time out and start a new call if there is no answer, or, if there is an answer, either play a tape or connect. the call to an operator. Clearly, by changing the timeout threshold, T, we can control ;~(T), the 'arrival ra~e' of completed calls. A~ ~pfimal choice of T can maximize the arrival rate generated by one dialing machine. Thus, given N madunes, each generating the maximum arrival rate, one can determine *hz. ~cq~,ired number of oFt,,,~t,~" ~r given a number of operators, one can find the minir~:,~r:, ~umber o~" machines (or the setting of ,~e .~.... _~ ~ate) needed to maintain a certain level of c,~¢upancy. Furthermore, maximizing the arrival rate not only leads to maximizing profit for the
individual but ~,I~o to more efficient use of the gk~bal network resources like ringing circuits and t ~ n k s held for ringing. This problem can be formulated as a variation of age replacement pelicies studied in renewal th.eory, cf., e.g., [1,2,4]. We differ from the usual age replacement problem in that we seek to maximize our "failure rate', ~,, instead of minimizing, and we allow a positive probability of "no failure', i.e., no answer. In Section 2 we derive an expression for A(T), the rate of answered calls, given the timmut threshold T, and show when there exists a unique,, finite, positive T* which maximizes X. Section 3 shows two examples for which we demonstrate our results and contains a few remarks on improper exponential distributions.
2. Analysis l e t H be a positive random variable representing the time from the start of dialing until the ~tart of the first fing~ The expected value of H is denotzd by H. Next, let P(t) be the probability that a call is answered with t seconds after the :tact ~f the firs~ rinse By .the dcf,.'nition of P, we have that P ( 0 ) = 0. We assume that P(t) is an improper c.d.L in the sense that P(0o)ffi a < I. Hence, there is a time, Tm~ (may be infinite), beyond which an answer is not expected, so there is positive probability, 1 - a , of no answer. Also,
0167-63"]7/86/$3.50 © 1986, Elsevier Science Publishers B.V. (No~-H~land)
57
OPERATIONS RESEARCH LETTERS
Volume 5, Number 2
we assume that P is independent of the time of day. For practical application, this need not be true, but one could solve the problem assuming dependence on the time of day, and hence find T which maximizes 2t as function of the time of day. For example, one could expect there to be a general 'day' P an a general 'evening' P which c o , ! a be used to determine two optimal values of T. To inc-.~:porate the possibility of a busy signal, let Pb denote the ~robability of a busy signal. Let K be a non-negative ~andom variable, which represents the time from the start of the busy signal to the time the machine starts a new call. Let ~ l = H + p b K , and P = ( l - p h ) P ; then all the following results could have H ~eplaced b y / t and P replaced by P. Let C(T) be the expected cycle time; that is, the expected time spent o n . ~ e ea)l attempt. Then, for a given P~ we have -
dPix)+(I-P(T))T
C(T)=H+ -
= H + T-
Plx)dx. i
As:~uming the dialing tria|s are independent, a dialing machine can be modeled as a renewal process. Therefore, by standard renewal arguments, cf. [1,2], the average rate of answered calls corresponding to a timeout T is X ( T ) = P ( T ) =_
C(T)
P(T) ~ + T .... (x)dx
Jttly 1986
Lemma I. Let f be a non-negative differentiable function, g be a positive di/ferentiable function, such that f ' > 0, g' > 0, and ( f ' / s ' ) is stricdy decreasing. Then, f / g is one of the following: (a) strictly increasing, (b) strictly decreasing, (c) strictly increasing and then strictly decreasing. The proof of Lemma 2 uses the equality
f'
g'
The proof shows the continuous function ( f / g ) " cannot go f r o ~ being zero to being positive, and it cannot be zero on an interval, if ( f ' / g ' ) is strictly decreasing (see the Apnendix). Theorem 1 follows by setting f - - P, g ffi C, so that ~, - - f / g and h -f ' / g ' , and noting that h(0)ffi ~,(oo)= 0, and that ~ ( T ) ~ 0 . Moreove,', Theorem 1 can be generalized to cover hazard functions that are first strictly increasing and then strictly decreasing, e.g., strictly concave hazard functions. Thus, such distributions as the improper exponential or the Weibuli have unique optimal timeout thresholds. If the condition of Theorem 1 is relaxed to h being non-increasing, the T* still exists, but it may not be unique.
3. Examples .
(I)
3.1. Improper exponential distribution for T_>0. Thus, !n the case P ' exists, it is stra;giltforward to determine a T* which maximizes h. There are two cases, either T* = Tm~ or T* satisfies ?,'(T*) =- 0. Interestingly, h'(T*) = 0 can be written as
P'(r*) )~(T*)= I _ P ( T , ) ,
As a first example, w.: present en exponentiallike distribution for P. Let
v(t)
:=
-
e- :"),
where !./ja is the mean tithe to answer given there is an answer. Then,
where the right-hand side is the hazard function of P. The following theorem gives sufficient conditions for the existence of a unique T*.
X(T)
Theocem 1. l f h(T), the hazard function of P, is strictly decreasing, then there exists a unique, finite, positive T* that maximizes h, and ~ is strictly increasing up to T* and strictly decreasing afterwards.
If H > O and a < 1, then the hypothesis of Theorem I is satisfied, and the optimal value T* has to satisfy the equation
The proof of Theorem 1 follows from the following known calculus result.
Note that the conditions H > 0 and a < 1 are also necessary for tke existence of a unique T* > 0. If
58
= -
H + T- a
/;,,- e1- "-)
-.
c -~'x)dx
pH e ~'r = 1 + I~T+ (1 - a ) "
Velume 5, Number 2
OPERATIONS RESL~.P.CH LEIq'ERS
H = 0 and a < 1, then h(0+)---c~# is the maxi~mm value, implying that the machine's timeout threshold is 0. If tl > 0 and a = 1, ~, is an increasing function of T Thug, the machine should always wait for art answer. Last, if H ffi 0 and a = 1, then .~ is a constant. Heace, it would not matter what T is, the arrival rate would ~emain the same. Often the exact value ot bt is unknown and on!y assume to be in some interval away from zero. Several intuitive ideas can be shown to be true. For each ~. Let T~* be the corresponding optimal threshold. Then, by simple manipulations, one can show the following results: (1) T~* is ~ strictly decreasing function of In. (2) h(/~, T~*) is f~ncreasing in p. (3) h(p, T) is Lipsehitz in T for fixed in. In fact, for 8 > 0, we have
I"
I-
,
~,
2
i7.
6 < ] 8. < 4 ( C ( , 4 , ) ) 2 [ 2P(T,*)(1 - P(T~*)) Brief derivations of these results are given in the Appendix. Thus, results one and two agree with our intuition that the shorter the mean time to answer, l/in, the shorter the optimal timeout and the higher the arrival rate. Thus~ given an interval where the true value of ~ lies, we can give upper and lower bounds on the true arrival rate and on T~*. The third result is true for a general differentiable distribution P, and it can be viewed as a measure of the insensitivity of h to T. The smaller the Llpschitz constant, the less sensitive h is to changes in T. The Lipschitz constant for 8 < 0, i.e., for T < T~*, is larger but is bounded by max r < r . P ' ( T ) / C ( 0 ) .
Jui~ 1986
for which ,~,,
> -
a
. + ~'-'(i
-
(3)
e,)
Using data from [5], we determined that the optimal value of T* to maximize h is T * - - 3 6 , i.e., approximately 6 rings. The data u ~ d was an~ average across various types of called parties and may not be representative of special targetted called popuiauons.
4. Conclusion We have derived the arrival rate of calls as a function of the :imeoL,t threshold of the dialing machine. Moreover, we have given conditions for the o~stence a~ld u,iqueness of a nonotrivial optimal timeout threshold, T*, wlfich maximizes 2L Although T* depends on the actual distribution of answer times, the arrival rate, 2~, is less sensitive to the choice of T. The examples given in Section 3 show that waiting until one is absoiutdy sure that there will be no answer may not be the optimal strategy for dialing machine use. The usefulness of our result can be seen in that by setting the timeout threshold, we can fix g at a given value and then determine the required number of operators, or amount of equipment, to handle the traffic.
Acknow~nt We would like to acknowlgdge J.C. Frauenthai L-, this problem. for his suggestiong and ;m,,,rest ~
3.2. Discrete distlqbutions
Appendix
In the discrete ca~e,
P"
x(.)
(2)
- ~ + n O - e . ) + Z'~jp,'
where n denotes the number of seconds after the start of the first ring, pj is the probability that the plz,)t~e is answered in the j t h second, P,~.'/Pi, and H is as before. Theorem 1 can be generalized to the discrete case by replacing derivatives with first order differences. Then, if the hazard function is strictly decreasing, the optimal n* is the first n
We provide a proof of Lemma 1 and a sketch of the proofs for the three statements in Section 3.1.
Proof of/.,emma I We need to only show that f / g cannot be strictly decreasing and then become increasing. First, observe that (f/g)" is co,afinuous and 59
Volume 5, Number 2
OPERATIONS RF~EARCH LETTERS
Derivations of results in Section 3.
can be w~tt~n as
(f'(t) f(t))g'(t)
(f) "(t)=
g,(t )
g(t)
g(:)"
(A.1)
Next, we show that (f/g)" cannot be zero and then become positive. Assume to the contrary that there exists tt, t: > 0, such that (g
), (t,)
(A.3)
By eq. (A.3), we have for t in (~, t 2)
f'(t) --> f(t)
(A.5)
g(t)"
By eqs. (A.2) and (A.3), we have for t in (h, t~.) f(t___~)> f(G___))
(A.6)
g(tl)"
By hypothesis, [ ' / g ' for t in (t 1, t 2)
/'it,______ 2) f q
,
is strictly decreasing, thus
t_____)
(A.7)
g'(t) "
Hence, we have the following inequal':ty for i in (ll' t2}:
f ( t ) > ~f ( t , ) = ~f ' ( t l ) > i f ( t ) > f(t~ gtt) ~(t,~ f(t,) ~ ~7)"
f(t)
s(t)
constant.
But this is a contradiction to f ' / g ' being strictly decreasing. This concludes the proof of Lemma 1.
60
oc,,
Thus, for p. fixed, f(t~, T) is strictly increasing, so there exists a unique T~* such that f(l~. T~*)--O. Further, for tt > .a,,/(tL, T~*) isstrictly increasing. Hence, for /x2:~,Z,, 0 = f ( ~ t : , T~*)
(A.S)
which is a contradiction. Therefore, f / g must be either a monotone function or a function which is increasing then decreasing. To prove strictness, we show that ( f / g ) ' cannot be zero on an interval. Again, assume to the contrary that ( f / g ) ' is zero on an interval 1. Then, using eq. (A.1), we have for l in I
/'(t) - f(t) =
lim f ( p , T ) = T-'*
(A.4)
g'(t,) g(tt)"
>
(I -,0"
/A~,, ~') > 0, f,,,(~, r)>0.
f'(t,)=f(h)
g'(t,)
a~
/r(~, r) >0, /(~,,o)
By eq. (A.I), we have
g(t)
m
f(y., T) = evr - 1 - # T -
(A.2)
>0.
g'(t)
To derive the three results of Section 3.1, first define
Then, for #, T > 0, =0
and for t in (t 1, t2) (f)'(t)
July 1986
References [I] R.E. Barlow and F. Proschan, "Comparison ot ,'..'.placement policies and renewal theory implications", Ann. Math.
Stat.35, 577-589 (1964). [2] E. Cinlar, Introduction to Stochastic Processes, PrenticeHall, Englewood Cliffs, NJ, 1975. [3] F.P. Duffy and R.A. Mercer, "A study of network dialing and customer behavior during direct-distance dialing call attempts in the U.S.A.", BSTJ 57, 1-33 (1978). [4] J. Medhi, StochasticProcesses,Wiley Eastern, New Delhi, 1982. [5] A. Myskja and O.O. Walman. "A statistical study of lelephone traffic data with emphasis on subscriber behavior", 7th International Teletraffic Congress, 1973.