Nobel, R.D.

Ommeren, J.C.W.

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VU Research PortalAn elementary proof of a basic result for the GI/G/1 queueOmmeren, J.C.W.; Nobel, R.D.1988document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Ommeren, J. C. W., & Nobel, R. D. (1988). An elementary proof of a basic result for the GI/G/1 queue. (SerieResearch Memoranda; No. 1988-20). 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May. 2021SERIE RESEARCH mEmORMIDn AN ELEMENTARY PROOF OF A BASIC RESULT FOR THE G I / G / 1 QUEUE J.C.W. van Ommeren R.D. Nobel Researchmemorandum 1988-20 March 1988 VRIJE UNIVERSITEIT Faculteit der Economische Wetenschappen en Econometrie A M S T E R D A M An elementary proof of a basic result for the GI/G/1 queue. By J. C. W. van Ommeren and R. D. Nobel, Operations Research Group, Dept. of Econometrics, Vrije Universiteit, Postbus 7161, 1007 MC Amsterdam. Abstract. The purpose of this note is to give an elementary proof of the result that the complementary waiting-time distribution function in the GI/G/1 queue is the sum of two exponential functions when the service time has a Coxian-2 distribution. Apart from its didactic interest the result is quite useful in the analysis of more complex queueing systems. 1. Introduction. An important model is the GI/G/1 queue in which the service time has a Coxian-2 distribution. That is, the service time S can be represented as S. with probability 1-b (1) S-+ S„ with probability b where S1 and S2 are independently exponentially distributed random variables with respective means l/^xand \/IM2, cf. Cox [1]. The Laplace transform of a Coxian-2 density is the ratio of a polynomial of degree at most one to a polynomial of degree two. The latter property defines the class of K2-distributions, cf. Cohen [2]. In fact, every K2-density is equivalent to a Coxian-2 density. Denoting by W (.) the probability distribution of the waiting time of a customer in the stationary situation and assuming service in order of arrival it follows from general results in Gohen [2], p 324, that for Coxian-2 service 2 t 1 - W (t) - £ a.e"^ for all t>0 (2) 3 i = 0 X where 0 < r\x < miri(jj,1 ,/z2) < rj2 a r e the r o o t s of the equa t i on O "> - x t x - ( ^ + /t2)x + plft2- (^2- ( l - b ^ x ) / e dA(t) - 0 (3) and the constants ai are given by a » ^ (*?i -/*a)(/*i -r/i ) a _. 1i (n?. -M?)(Mi -f]o) , 4 s Here A(t) denotes the probability distribution function of the interarrival time. Denoting by p the quotiënt of the mean service time and the mean interarrival time, the condition p<l is required in order to guarantee that •^mn-.DÏ){WI1<x} = W (x) exists and is a proper probability distribution where Wn is the delay in queue of the n-th arrival. For the special case of exponential service we actually have 1 - W (x) = ae""x where r) is the solution of the equation e dA(t) = 0 and a = The usefulness of the tractable analytical solution (2) is not restricted to C2-service. This solution can also be used to calculate simple and accurate - 2 -approximations for the general GI/G/1 queue by using extrapolation with respect to the squared coëfficiënt of variation of the service time, cf. Tijms [3]. The result (2) was obtained in Cohen [2] in a much more general framework by rather deep arguments using complex function analysis. In view of the practical importance of the result (2), it seems of some interest to give an elementary proof that can be used in introductory courses on queueing models. The elementary proof is based on the observation that a Coxian-2 distributed service time can be represented as a random sum of exponentially distributed service phases with the same means. This observation allows us to use a simple embedded Markov-chain analysis. In section 2 it is shown that the waiting-time distribution for the GI/G/1 model can be obtained by analyzing a special batch-arrival GIX/M/1 model. The embedded Markov-chain analysis is given in section 3. 2. The analysis for the waiting-time distribution. To give an elementary derivation of the waiting-time distribution we need the following well-known result. Suppose that Xa , X2 , .. are independent random variables with a common exponential distribution with mean 1/y. and that the random variable N has a geometrie distribution P{N-k} = p(l-p)k_1 for k -1, 2 If the random variable N is independent of the sequence [X^} then the random sum J?=1Xi is exponentially distributed with mean l/(p/i) . Consider now the C2-distributed service time S as in (1). Depending on whether Mi^2 o r lJ,2>tii w e represent Sx or S2 as a geometrically distributed sum of independent exponentials where the parameters p and JJ, of this representation are p=n2/fi1 , fi=fi1 respectively •p=p1/fi2 and M=M2 • For the case l^i^M2 •̂t c a n ^ e said that a customer represents a batch of independent service phases where the batch-size distribution {gk} is given by 1-b for k - 1 k-2 (5) bp(l-p) for k - 2, 3, .. with •p=*n2/n1 . Here the service time of each phase is exponentially distributed having mean \/\x with A*=PI • For the case of M2>A*i w e get the same representation (5) provided we replace b by l-(l-b)p and take p=/i1//x2 and y.=n2 . In other words the GI/C2/1 queue can be studied as a special GIX/M/1 queue with batch-size distribution (5). Irrespective of the special form of gk, we can define the following embedded Markov chain §k={ - 3 -XJJ — the number of uncompleted exponential service phases present in the system just prior to the arrival epoch of the n-th customer. Since p<l the Markov chain {2?̂ : n—1,2,..} is positive-recurrent. Moreover the chain is aperiodic. Let us denote its one-step transition probabilities by ptj and its n-step transition probabilities by pjjn) . Then from Markov-chain theory we know that the limiting distribution J n lim PfX^j} - lim p^"' , j - 0, 1, exists and is independent of the initial state i. Moreover, {w,} is the unique nonnegative solution of the system of linear equations co x4 = I X-P-H , j = 0, 1, .. (6) J i=o x XJ together with the normalization equation X°=oxj = •*-• The limiting distribution {TT,} leads directly to the waiting-time distribution of a customer arriving in the stationary situation. Since the phases are independent exponentials with a common mean l//j and because of the memoryless property of the exponential distribution the conditional delay in queue of a customer finding upon arrival j phases in front of him has an Erlang-j distribution. Hence W (t) -Ï*A1 -"ï e'^Oït^/kO for all t2:0 (7) " j=l J k = 0 In the next section it will be shown that for certain constants a, f3, rx and r2 7r.= ar,3+ /3r J for j - 0, 1, ... j 1 ^ 2 J > » Then, together with (7) we get 1 . W (t) - - ^ - e ^ 1 - ' ! ^ + - & * - e-(l-'a)/*t At the end of section 3 we will see that (8) and (2) coincide. (8) 3. Embedded Markov-chain analysis. In this section.we will first prove that there exist numbers TX , TZ, a and f3 such that Xj = ar1J+y0r2J for j=0,l,.. satisfies sirnultaneously the system of linear equations (6) and the normalization equation. Subsequently we will show - 4 -that this solution coincides with the steady-state distribution {TT̂  } . Put for abbreviation k = ƒ e"Mt ^ ^ ( t ) , n - 0 , 1, ... n o n* Then, we have obviously Pij ~i=J-i) + SA+i-j for j - 1, 2, ..; 1 - 0, lf .. (9) Here (j-i)+ - max(0, j-i) and g0=0. Note that the formulae (9) only apply for j**0 and that pi0= 1 - Ej^0 Pij • With (9) we can write the equations in (6) as x. - > x. Y g „k. „ ., for j - 1, 2, .. (10) J if-o J i=(j-i)+ ^ i+i-j Now define for any solution {x^} of (10) q, - t x g for j - 0, 1, .. (11) J i = 0 ^ J -*• If we take Xj=7Tj for all j>0, q̂  represents the probability that j uncompleted phases are present in the system just after the arrival of a customer. Using that gk — (l-p)gk_1 for k>3 (see (5)) we find qj= glXj-l+ §2Xj-2+ ( 1 - p ) J j 0 XiSj-l-i" (1-P)xj-2Sl • J = 2, 3, .. and so we obtain the recursion relation qj = (l-p)q..1 + (l-b)x..1 +(b+p-l)Xj_2f for j - 2, 3, ... (12) Now we rewrite (11) in a more convenient form by changing the order of summation. By summing over the diagonals n = i+i we get x. - Y y x.g .k ., for j - 1, 2, ... (13) 1 u . ,L,n i&n-i n-j J » > Next by inserting (11) in (13) and using the recursion (12) we find x. - (l-p)x. .+ (1-b) J x .k .+ (b+p-1) l x 0k . , j = 2, 3, .. (14) i r i-l . n-1 n-j u. n-2 n-i J -> J n=j J n=j J In fact the equations (14) are equivalent to the equations (13) in which the equation corresponding to j=l is omitted. Denote by A(s) = J"e"stdA(t) the Laplace-Stieltjes transform of the interarrival-time distribution function - 5 -A(t). Then, by substitution, we find that the system of equations (14) allows solutions of the form x̂  - rJ (j = 0, 1, . .) if r satisfies the equation Z " (1"P)Z A(/i(l-z)). (15) (l-b)z + b+p-1 In the sequel we assume that both b+p-1^0 and bs-O. If either condition is not satisfied the GI/C2/1 queue reduces to the well-studied GI/M/1 queue in which case (15) has a unique solution r>0 in the interval (-1, 1). Using the condition p<l it is a matter of routine analysis to show that (15) has two real roots r-i<Tz in the open interval (-1, 1) if b+p-1^0. In case b+p-l>0 we have T ^ O ,while rx>0 in case b+p-l<0. In both cases r2>l-p. Once we know that (14) allows solutions Xj=r1j and Xj=r2J it is immediate that any linear combination ^=ar1 j+/3r2d (j—0,1,..) is also a solution of the system (14). We now conjecture ni="aT1j+^r2J , j-0,1, . . , with a and /3 still to be specified. To prove this conjecture we must determine a and fi such that •x.i=ar1i+l3T2i , j-0,1,.., is a solution of system (6). Because the equations (6) with j—2,3., •. . are equivalent with (14) and xi=ar1i+^Tzi is a solution of (14), we have that the proposed solution satisfies the equations (6) for j-2,3,.. irrespective of the values of a and /3. So we still have to prove that there exist an a and a fi such that xJ=ar:L'j+j8r2'j is also a solution of both the remaining equation of system (6) 00 xi - 1 * 1 P u (16) X i-O ^ J"L and the normalization equation £™ = 0xj = ^" We do this by substituting the solutional form Xj=ar1j +/3r2J into these equations. With respect to (16) this leads after some algebra (using (13) with j=l) to the following equation otr' • + 0r, - bp{ ̂ 1 - r , ) ) + M(M(l-r?)) , ̂  f _ a + _ g _ ^ 11 1 2 F[ 'i+P-1 r2+p-l ^ L 'x+P-l ^2+P-l Ji Now we use the fact that rx and r2 satisfy (15), so we can replace A(/i(l-ri)), (i-1,2) by the left side of (15). After some simplifications this gives ar^+ fir2 = ar^+ @T^- bpA(p/i) J_ _ 7^+p-l r2+p-l ^ So we find, a/(r1+p-l) + y8/(r2+p-l) — 0. Further, substitution of the proposed solution in the normalizing equation £x^ = 1 gi-ves> a/(l~ri) + /3/(l-r2) = 1. From these two linear equations in a and /? we find - 6 -a _ (l-r^d-TaXl-p-r-,) $ _ (1-r, ) (1-r, ) (1-p-r, ) P('"2-Ti) P(ri-r2) Taking these values for a and /3 the solution Xj=ar1J+/3r2J , j=0,l,.., satisfies all the balance equations in (6) together with the normalization equation £XJ=1. So it remains to show that Xj=»7i\j for all j>0. For completeness we include the proof of this. The series JXj converges absolutely because both |rx| and |r2| are less than one. Thus, using that for the ergodic Markov chain iri - limn_>tapi jn> j w e find by iterating the balance equations (6) that xj = Zl = 0xiPijn) > f o r a ^ nal- Next- letting n-*», we get Xj = Jixi'r<3 > using the bounded convergence theorem. This gives Xj =CTTj for all j5:0 with C=2]XJ . But from the earlier analysis we know that £x.j=l, and so XJ=?TJ for all j. Finally, we show that formulae (2) and (8) coincide. First we remark that the transformation yii(l-z)=x in (15) leads to (3). This gives /i(l-r1)=r?2 and n(l-r2)=Vi • s ° , also using (18), we can express (8) in r/-^ and r\z and after some algebra this expression turns out to be equal to (2). Acknowledgement. We would like to thank prof. dr. H. C. Tijms for his valuable guidance during the course of this work, He suggested many improvements which made the text much more readable. References. [1] Cox, D. R. (1955). A use of complex probabilities in the theory of stochastic processes, Proc. Camb. Phil. Soc, 51, 313-319. [2] Cohen, J. W. (1982). The Single Server Queue, 2nd ed., North-Holland, Amsterdam. [3] Tijms, H. C. (1986). Stochastic Modelling and Analysis, Wiley, New York. 

English

An elementary proof of a basic result for the GI/G/1 queue

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SERIE RESEARCH mEmORMIDn 
AN ELEMENTARY PROOF OF A BASIC RESULT 
FOR THE G I / G / 1 QUEUE 
J.C.W. van Ommeren 
R.D. Nobel 
Researchmemorandum 1988-20 
March 1988 
VRIJE UNIVERSITEIT 
Faculteit der Economische Wetenschappen en Econometrie 
A M S T E R D A M 
An elementary proof of a basic result for the GI/G/1 queue. 
By 
J. C. W. van Ommeren and R. D. Nobel, 
Operations Research Group, 
Dept. of Econometrics, 
Vrije Universiteit, 
Postbus 7161, 
1007 MC Amsterdam. 
Abstract. 
The purpose of this note is to give an elementary proof of the result that the 
complementary waiting-time distribution function in the GI/G/1 queue is the sum 
of two exponential functions when the service time has a Coxian-2 distribution. 
Apart from its didactic interest the result is quite useful in the analysis of 
more complex queueing systems. 
1. Introduction. 
An important model is the GI/G/1 queue in which the service time has a 
Coxian-2 distribution. That is, the service time S can be represented as 
S. with probability 1-b 
(1) 
S-+ S„ with probability b 
where S1 and S2 are independently exponentially distributed random variables 
with respective means l/^xand \/IM2, cf. Cox [1]. The Laplace transform of a 
Coxian-2 density is the ratio of a polynomial of degree at most one to a 
polynomial of degree two. The latter property defines the class of K2-
distributions, cf. Cohen [2]. In fact, every K2-density is equivalent to a 
Coxian-2 density. Denoting by W (.) the probability distribution of the 
waiting time of a customer in the stationary situation and assuming service in 
order of arrival it follows from general results in Gohen [2], p 324, that for 
Coxian-2 service 
2
 t 
1 - W (t) - £ a.e"^ for all t>0 (2) 
3 i = 0 X 
where 0 < r\x < miri(jj,1 ,/z2) < rj2 a r e the r o o t s of the equa t i on 
O "> - x t 
x - ( ^ + /t2)x + plft2- (^2- ( l - b ^ x ) / e dA(t) - 0 (3) 
and the constants ai are given by 
a » ^ (*?i -/*a)(/*i -r/i ) a _. 1i (n?. -M?)(Mi -f]o) , 4 s 
Here A(t) denotes the probability distribution function of the interarrival 
time. Denoting by p the quotiënt of the mean service time and the mean 
interarrival time, the condition p<l is required in order to guarantee that 
•^
mn-.DÏ){WI1<x} = W (x) exists and is a proper probability distribution where 
Wn is the delay in queue of the n-th arrival. For the special case of exponential 
service we actually have 1 - W (x) = ae""x where r) is the solution of the 
equation 
e dA(t) = 0 and a = 
The usefulness of the tractable analytical solution (2) is not restricted 
to C2-service. This solution can also be used to calculate simple and accurate 
- 2 -
approximations for the general GI/G/1 queue by using extrapolation with respect 
to the squared coëfficiënt of variation of the service time, cf. Tijms [3]. 
The result (2) was obtained in Cohen [2] in a much more general framework by 
rather deep arguments using complex function analysis. In view of the practical 
importance of the result (2), it seems of some interest to give an elementary 
proof that can be used in introductory courses on queueing models. The elementary 
proof is based on the observation that a Coxian-2 distributed service time can 
be represented as a random sum of exponentially distributed service phases with 
the same means. This observation allows us to use a simple embedded Markov-
chain analysis. In section 2 it is shown that the waiting-time distribution for 
the GI/G/1 model can be obtained by analyzing a special batch-arrival GIX/M/1 
model. The embedded Markov-chain analysis is given in section 3. 
2. The analysis for the waiting-time distribution. 
To give an elementary derivation of the waiting-time distribution we need the 
following well-known result. Suppose that Xa , X2 , .. are independent random 
variables with a common exponential distribution with mean 1/y. and that the 
random variable N has a geometrie distribution P{N-k} = p(l-p)k_1 for k -
1, 2 If the random variable N is independent of the sequence [X^} then 
the random sum J?=1Xi is exponentially distributed with mean l/(p/i) . 
Consider now the C2-distributed service time S as in (1). Depending on 
whether Mi^2 o r lJ,2>tii w e represent Sx or S2 as a geometrically distributed 
sum of independent exponentials where the parameters p and JJ, of this 
representation are p=n2/fi1 , fi=fi1 respectively •p=p1/fi2 and M=M2 • For the case 
l^i^M2 •^t c a n ^ e said that a customer represents a batch of independent service 
phases where the batch-size distribution {gk} is given by 
1-b for k - 1 
k-2 (5) 
bp(l-p) for k - 2, 3, .. 
with •p=*n2/n1 . Here the service time of each phase is exponentially distributed 
having mean \/\x with A*=PI • For the case of M2>A*i w e get the same representation 
(5) provided we replace b by l-(l-b)p and take p=/i1//x2 and y.=n2 . In other words 
the GI/C2/1 queue can be studied as a special GIX/M/1 queue with batch-size 
distribution (5). Irrespective of the special form of gk, we can define the 
following embedded Markov chain 
§k={ 
- 3 -
XJJ — the number of uncompleted exponential service phases present in 
the system just prior to the arrival epoch of the n-th customer. 
Since p<l the Markov chain {2?^ : n—1,2,..} is positive-recurrent. Moreover the 
chain is aperiodic. Let us denote its one-step transition probabilities by ptj 
and its n-step transition probabilities by pjjn) . Then from Markov-chain 
theory we know that the limiting distribution 
J 
n 
lim PfX^j} - lim p^"' , j - 0, 1, 
exists and is independent of the initial state i. Moreover, {w,} is the unique 
nonnegative solution of the system of linear equations 
co 
x4 = I X-P-H , j = 0, 1, .. (6) 
J i=o x XJ 
together with the normalization equation X°=oxj = •*-• 
The limiting distribution {TT,} leads directly to the waiting-time distribution 
of a customer arriving in the stationary situation. Since the phases are 
independent exponentials with a common mean l//j and because of the memoryless 
property of the exponential distribution the conditional delay in queue of a 
customer finding upon arrival j phases in front of him has an Erlang-j 
distribution. Hence 
W (t) -Ï*A1 -"ï e'^Oït^/kO for all t2:0 (7) 
" j=l J k = 0 
In the next section it will be shown that for certain constants a, f3, rx and r2 
7r.= ar,3+ /3r J for j - 0, 1, ... j 1 ^ 2 J > » 
Then, together with (7) we get 
1 . W (t) - - ^ - e ^ 1 - ' ! ^ + - & * -
 e-(l-'a)/*t 
At the end of section 3 we will see that (8) and (2) coincide. 
(8) 
3. Embedded Markov-chain analysis. 
In this section.we will first prove that there exist numbers TX , TZ, a and f3 
such that Xj = ar1J+y0r2J for j=0,l,.. satisfies sirnultaneously the system of 
linear equations (6) and the normalization equation. Subsequently we will show 
- 4 -
that this solution coincides with the steady-state distribution {TT^  } . 
Put for abbreviation 
k = ƒ e"Mt ^ ^ ( t ) , n - 0 , 1, ... 
n
 o n* 
Then, we have obviously 
Pij ~i=J-i) + SA+i-j for j - 1, 2, ..; 1 - 0, lf .. (9) 
Here (j-i)+ - max(0, j-i) and g0=0. Note that the formulae (9) only apply 
for j**0 and that pi0= 1 - Ej^0 Pij • With (9) we can write the equations in 
(6) as 
x. - > x. Y g „k. „ ., for j - 1, 2, .. (10) 
J if-o J i=(j-i)+ ^ i+i-j 
Now define for any solution {x^} of (10) 
q, - t x g for j - 0, 1, .. (11) 
J i =
 0 ^ J -*• 
If we take Xj=7Tj for all j>0, q^  represents the probability that j uncompleted 
phases are present in the system just after the arrival of a customer. Using 
that gk — (l-p)gk_1 for k>3 (see (5)) we find 
qj= glXj-l+ §2Xj-2+ ( 1 - p ) J j 0 XiSj-l-i" (1-P)xj-2Sl • J = 2, 3, .. 
and so we obtain the recursion relation 
qj = (l-p)q..1 + (l-b)x..1 +(b+p-l)Xj_2f for j - 2, 3, ... (12) 
Now we rewrite (11) in a more convenient form by changing the order of 
summation. By summing over the diagonals n = i+i we get 
x. - Y y x.g .k ., for j - 1, 2, ... (13) 
1 u . ,L,n i&n-i n-j J » > 
Next by inserting (11) in (13) and using the recursion (12) we find 
x. - (l-p)x. .+ (1-b) J x .k .+ (b+p-1) l x
 0k . , j = 2, 3, .. (14) i r i-l . n-1 n-j u. n-2 n-i J 
-> J n=j J n=j J 
In fact the equations (14) are equivalent to the equations (13) in which the 
equation corresponding to j=l is omitted. Denote by A(s) = J"e"stdA(t) 
the Laplace-Stieltjes transform of the interarrival-time distribution function 
- 5 -
A(t). Then, by substitution, we find that the system of equations (14) allows 
solutions of the form x^  - rJ (j = 0, 1, . .) if r satisfies the equation 
Z
 "
 (1
"
P)Z
 A(/i(l-z)). (15) (l-b)z + b+p-1 
In the sequel we assume that both b+p-1^0 and bs-O. If either condition is not 
satisfied the GI/C2/1 queue reduces to the well-studied GI/M/1 queue in which 
case (15) has a unique solution r>0 in the interval (-1, 1). Using the condition 
p<l it is a matter of routine analysis to show that (15) has two real roots r-i<Tz 
in the open interval (-1, 1) if b+p-1^0. In case b+p-l>0 we have T ^ O ,while rx>0 
in case b+p-l<0. In both cases r2>l-p. Once we know that (14) allows solutions 
Xj=r1j and Xj=r2J it is immediate that any linear combination ^=ar1 j+/3r2d 
(j—0,1,..) is also a solution of the system (14). 
We now conjecture ni="aT1j+^r2J , j-0,1, . . , with a and /3 still to be 
specified. To prove this conjecture we must determine a and fi such that 
•x.i=ar1
i+l3T2i , j-0,1,.., is a solution of system (6). Because the equations 
(6) with j—2,3., •. . are equivalent with (14) and xi=ar1i+^Tzi is a solution 
of (14), we have that the proposed solution satisfies the equations (6) for 
j-2,3,.. irrespective of the values of a and /3. So we still have to prove that 
there exist an a and a fi such that xJ=ar:L'j+j8r2'j is also a solution of both 
the remaining equation of system (6) 
00 
xi - 1 * 1 P u (16) 
X
 i-O ^ J"L 
and the normalization equation £™
 = 0
xj = ^" 
We do this by substituting the solutional form Xj=ar1j +/3r2J into these 
equations. With respect to (16) this leads after some algebra (using (13) with 
j=l) to the following equation 
otr' 
• + 0r, - bp{ ^ 1 - r , ) ) + M(M(l-r?)) , ^  f _ a + _ g _ ^ 11 
1 2 F[ 'i+P-1 r2+p-l ^ L 'x+P-l ^2+P-l Ji 
Now we use the fact that rx and r2 satisfy (15), so we can replace A(/i(l-ri)), 
(i-1,2) by the left side of (15). After some simplifications this gives 
ar^+ fir2 = ar^+ @T^- bpA(p/i) J_ 
_ 7^+p-l r2+p-l ^ 
So we find, a/(r1+p-l) + y8/(r2+p-l) — 0. Further, substitution of the proposed 
solution in the normalizing equation £x^ = 1 gi-ves> a/(l~ri) + /3/(l-r2) = 1. 
From these two linear equations in a and /? we find 
- 6 -
a _ (l-r^d-TaXl-p-r-,) $ _ (1-r, ) (1-r, ) (1-p-r, ) 
P('"2-Ti) P(ri-r2) 
Taking these values for a and /3 the solution Xj=ar1J+/3r2J , j=0,l,.., 
satisfies all the balance equations in (6) together with the normalization 
equation £XJ=1. So it remains to show that Xj=»7i\j for all j>0. For completeness 
we include the proof of this. The series JXj converges absolutely because 
both |rx| and |r2| are less than one. Thus, using that for the ergodic Markov 
chain iri - limn_>tapi jn> j w e find by iterating the balance equations (6) that 
xj = Zl =
 0
xiPijn) > f o r a ^ nal- Next- letting n-*», we get Xj = Jixi'r<3 > using 
the bounded convergence theorem. This gives Xj =CTTj for all j5:0 with C=2]XJ . 
But from the earlier analysis we know that £x.j=l, and so XJ=?TJ for all j. 
Finally, we show that formulae (2) and (8) coincide. First we remark that the 
transformation yii(l-z)=x in (15) leads to (3). This gives /i(l-r1)=r?2 and 
n(l-r2)=Vi • s ° , also using (18), we can express (8) in r/-^ and r\z and after 
some algebra this expression turns out to be equal to (2). 
Acknowledgement. 
We would like to thank prof. dr. H. C. Tijms for his valuable guidance during 
the course of this work, He suggested many improvements which made the text much 
more readable. 
References. 
[1] Cox, D. R. (1955). A use of complex probabilities in the theory of 
stochastic processes, Proc. Camb. Phil. Soc, 51, 313-319. 
[2] Cohen, J. W. (1982). The Single Server Queue, 2nd ed., North-Holland, 
Amsterdam. 
[3] Tijms, H. C. (1986). Stochastic Modelling and Analysis, Wiley, New York. 


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An elementary proof of a basic result for the GI/G/1 queue

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