185. CONTROLLED STOCHASTIC MODEL OF A COMMUNICATION SYSTEM. E. G. COFFMAN, Jr. AT&T Bell Laboratories, Murray Hill, New Jersey 07974.
Queueing. Performance and Control in ATM (ITC-13) J.W. Cohen and C.D. Pack (eds.) Elsevier Science Publishers B.V. (North-Holland) e lAC. 1991
185
CONTROLLED STOCHASTIC MODEL OF A COMMUNICATION SYSTEM
E. G. COFFMAN, Jr. AT&T Bell Laboratories, Murray Hill, New Jersey 07974 B. M. IGELNIK Institute of Communication Engineering, Moscow, USSR Y. A. KOOAN
Faculty of Industrial Engineering and Management, The Technion, Haifa, Israel
We consider a stochastic model of buffering in a data communication system, with source and sink transmission parameters depending on the number of active sources. For models in this control setting we analyze an effective numerical method for evaluating the equilibrium distribution of buffer content The theoretical basis of the method is established first. Then, it is shown that the method has the same complexity, in terms of the total number of sources, as known analytical methods for the model with constant parameters. Asymptotics for tail probabilities at high buffer levels and under heavy load are also derived, and the complexity of their computation is compared with that of evaluating explicit formulas. In comparison to earlier results, our approach reduces the complexity of computing the probability of overflow and its asymptotic estimates. The speed-up stems from the application of interpolation schemes.
1• . INTRODUCTION
We analyze a generalization of the buffer model studied by Anick, Mitra and Sondhi [1]. (The analysis in [1] is also covered in a recent text on computer storage problems [2].) The model consists of an infinite capacity storage device that receives messages from a finite number, N, of information sources; these sources independently and asynchronously alternate between the transmitting (active) and idle (passive) states. Transitions between these two states are described by a continuous Markov chain whose rates depend on the number of active sources. More precisely, let AO' ... , AN-1 and Ill' ... , IlN be given positive numbers. If at time t there are i active sources, then in the interval [t, t + At] each passive source transits to the active state with probability Ai llt + o(llt), OS i S N -1, and each active source transits to the passive state with probability Il; At + o(llt), 1 Si S N. Hence, the number of active sources is described by a birth-and-death process. A routine analysis shows that the equilibrium distribution of this process is given by
(N)
P;=Po· ,
Aj_1 II I -,
j=l
.
IS1SN,
Ilj N
where P 0 is determined from .:E P;
,=0
= 1.
(1.1)
As a convenient normalization, we assume that sources transmit at a uniform rate of 1 unit of information per unit of time. Thus, when i sources are active simultaneously, the instantaneous receiving rate at the buffer is i. The buffer accumulates information when the receiving rate exceeds the maximum transmission rate of the output channel. This rate, C i' is also allowed to depend on the number of active sources i, 0 S i S N. (For a different model of buffer behavior with variable input and output rates, see also the work of Halfin [5].) It is helpful to assume that the C ; 's are not integers. We also assume that c; < N for at least one i, since otherwise the buffer is always empty. Such a model can be used for the analysis of different flow control schemes, as shown by Weiss [14]. In [14] two particular control strategies were studied by asymptotic techniques based on large deviation theory. In both cases the maximum output rate was fixed. For the first strategy A;_l = m(i) Aand Ili = m(i) Il, 1 Si S N, where m(i) is a nondecreasing function and m(1) = 1. The multiplier on Il makes sources more liable to turn off as the number of active sources increases; the multiplier on A. then ensures that short-term average traffic is equal to longterm average traffic. Equivalently, the multiplier on A. can be regarded as a sort of memory - even though sources are forced to turn off, they still wish to transmit, so they are
186
more liable to turn on. The second strategy may be used in certain voice coding schemes where bits are dropped from packets in overload conditions. Here A; = A is fixed but J.L; increases with i for i ~ i l' where i 1 is a given overload parameter. Only approximate expressions were obtained in [14] for the probability of overflowing a prescribed level. To evaluate the accuracy of these approximations exact values are needed. To this end. we develop an effective numerical method for computing the equilibrium probability distribution of buffer content. The method contrasts with the techniques in [1] in that it is not based on evaluating explicit formulas. This contrast applies also to other. more recent papers [8.10] devoted to different generalizations of the basic model. The remainder of the paper is organized as follows. Section 2 begins with the differential equations governing the stationary probability that i sources are active and the buffer content is no greater than x. Section 2 then gives the main results concerning the properties of eigenvalues and how they can be used in a numerical solution of the equations. Section 3 describes procedures for calculating eigenvectors and eigenvalues. and then evaluates the complexity of the numerical method. Section 4 presents asymptotic approximations for the tail probabilities at high buffer levels and under heavy load. lllustrative numerical results on the evaluation of simple control strategies are also given. Section 4 concludes with a brief study of asymptotics for the model with constant parameters. 2. THE MAIN RESULTS Denote by b(t) the buffer content at time t. and let G (x. t) = Pr ( b (t) > x}. The existence of the stationary probability G(x) = lim G(x. t) is assured by the t~
..
stability condition r < C. where r and c are equilibrium expectations of the number of active sources and the maximum output rate. respectively. G(x) is also called the probability of overflow atx. Taking into account (1.1). we can express the stability condition in the explicit form
[N] ; Aj_l N l: i . IT - - < Co + l: N
;=1
I
j=l
J.Lj
;=1
[N]; Aj_l c; . IT - - . I
j=1
J.Lj
(2.1)
> O. 0 S i Si. - 1 and (2.2)
Ci
- i
< O. i. Si S N.
Thus, the function C i increases from 0 to N.
-
dF ;(x)
dx
(i-c;)
F; (x) == 0
Setting
for
=
+ (i+l)J.L;+IF;+l(x). OSiSN. X~O. (2.3)
Hereafter. boldface denotes matrices. arrows denote column vectors, and primes denote transposition. In matrix notation. with the column ~ vector ~
= (F o(x) • ...• F N(x»
F(x) ~
A F(x).x
~
,
we
•
have
dFi:)
=
O. where A = (aU)' with
a;j
= o. Ii - j I > 1.
0 Si, j S N ,
(N-i)A;+iJ.L; C; -
i (2.4)
a;,;+1
=-
ai,;-1
=-
(i+1)J.L;+1 c; -
i
(N -i + 1) A;_l c; -
i
In scalar-product notation. the unknown overflow probability is given by G(x) = 1 - (I, F(x». where lis a column vector of N + 1 l·s. To find F(x) it is necessary to find the eigenvalues and eigenvectors of the matrix A, and to sPeQfy the initial conditions for the differential ~
.
equation
dF(x)
~
~ = A F(x).x ~ O.
Let Z i' N - n + 1 S i S N. be the n negative eigenvalues of ~ the matrix A and let Cl> i be the associated right eigenvectors. Then assuming that the matrix A has a simple spectrum we have ~
~
F(x) = F(oo)
+
N
1:
e
,,X
~
ai ~i •
(2.5)
i=N-n+l
Henceforth we assume that the sequence Co • ...• CN satisfies the following additional condition. There exists a unique integeri. e {O. 1. ...• N} such that c; - i
is no greater than x. i ~ {O. 1. ...• N} we obtain
.
i changes sign only once as i
Denote. by F i (x), 0 S i S N, x ~ 0, the stationary probability that i sources are active and the buffer content
where the coefficients a; are determined by the initial conditions. To make formula (2.5) usable for computation, we need to (i) investigate the properties of the eigenvalues of A; (ii) prove that conditions (2.1) and (2.2) enable one to express the coefficients ai' N -n + 1 Si S N. in terms of the negative eigenvectors in an explicit form; (ill) define a numerical method for computing the z;. N - n + 1 S i S N; and (iv) determine a procedure that finds the associated eigenvectors ~; . Items (i) and (ii) are treated in the remainder of this section, and items (iii) and (iv) are dealt with in the next section.
187
Theorem 1. Suppose condition (2.2) holds and consider the matrix A defined by (2.4). Then (a) z =0 is an eigenvalue of A, i.e. A is singular; (b) all eigenvalues of A are real,' (c) if condition (2.1) holds then the negative eigenvalues of A are separated by the negative eigenvalues of the submatrix A 2 , defined as the intersection of the last N - i. + 1 rows and colwnns of A; (d) A has a simple spectrwn,' and (e) if condition (2.1) holds then n =N -i. + 1. Proof. Statement (a) is quite easy to verify and is left to the reader. Statements (b). (c). and (d) are proved by the three lemmas below. Proofs of these lemmas are omitted. owing to space constraints. (The proofs appear in an expanded version of this paper which is available from the authors.) Lemma 1. Let A be represented in the following form:
A
=
[AI C2
The following result applies Theorem 1 and gives the basis for finding the coefficients a i in (2.5). Theorem 2. lfthe stability condition (2.1) holds, then the coefficients ai' i. ~ i ~ N, are uniquely determined by the initial conditions. Proof. If (2.1) holds then by Theorem 1. n =N - i. + 1 and the eigenvalues Z i' i. ~ i ~ N. are negative and pairwise distinct. It is evident that F i (0) = 0 for i. ~ i ~ N. By supplementing these equations with the tridiagonal structure of A. we see that at least N - i. last -+ components of the vector A F(O) are zer~~ Repeating this argument we conclude that (AJ F(O»N = o. -+
o ~ j ~ N -i.. But dFt) = A F(x), x ~ 0, implies Aj F(O) = F(j) (0). and therefore FW) (0) = o. o ~ j ~ N - i.. Substituting (2.5) into this last equation and setting $ N = 1 for each ~ = ($ o. ...• $ N ) '. we find N . N ~. .z{ a i = O. 1 ~ j ~ N - i •• and .~. a i = - F N (00 ).
Cl] A2
1=,.
'='.
where AI' A 2 , C 2 and Cl are respectively i. x i., (N-i.+l) x(N-i.+l), (N-i.+l) xi. and (i. x (N - i. + 1» submatrices of A. Then under the conditions of Theorem 1 all eigenvalues of A 1 are positive while all eigenvalues of A2 are negative. Each of Al and A2 has a simple spectrwn.
Recognizing the determinant V of this linear system in ai' i. ~ i ~ N. as a Vandermonde determinant. we have V = n (zi - Zj)' Note that V ~ O. because i.Si
Z
N'
1
i.
2
(1)
Zi
(1)
> Zi > zi+l'
ze) > zeL
F N (00 )
> ... > zJJ)
Z· n __
i. ~ i ~ N •
J -,
(2.7)
i. S j S N zj - z i j~i
•..• ZN of
Let ZI > Z2 > ... > Zi.-l > 0> Zi.+l > let z(l) > z(l) > ... > il) > 0 be the
eigenvalues of AI' and let 0 > be the eigenvalues of A 2 . Then
=-
where F N (00)
= PN'
This completes the proof.
From Theorem 2 and G(x) G(x) = -
_
-+ -+
= 1-(1. F(x)} we obtain
N
1:
a ie
',;( -+ -+
(I, cfJ i) •
(2.8)
i=i.
1 ~ i ~ i. - 1.
(2.6a)
,-"here the a i's are found from (2.7).
(2.6b) 3. CALCULATIONS Lemma 3. Under the conditions of Theorem 1 there exists an eigenvalue zi. of A such that z~l) > z.
'.
'.
> z~2)
'.
.
(2.6c)
Lemmas 1 and 2 and statement (a) of Theorem 1 imply the existence of N distinct real eigenvalues (including z0 = 0) of the matrix A. The characteristic polynomial of A has real coefficients. Therefore. the remaining eigenvalue z.
'.
This section describes first the calculations of the eigenvectors and eigenvalues, and then the complexity of the Qverall method. Numerics are illustrated in two examples. -+ ' E1genvectors - Let z be some eigenvalue of A, and let cfJ
be the associated right eigenvector. Then if M = {mij} = DA. D = diag{ -co' 1-c 1 • .... N-cN}' we have Z D ~ = M~, which represents the system
z(i -ci)$i = mi,i-l $i-l + mu $i +
of A is real. By Lemma 3 either zP> > z· ~ 0 or (2)
'.
mi,i+l $i+l' 0 ~ i ~ N -1 (3.1)
'.
zi. < Z i. < O. Thus statements (b) and (c) of the theorem are implied by Lemmas 1-3. and the inequalities (2.6a)-(2.6c) separate the eigenvalues of A by the eigenvalues of A 1 and A 2'
= mN,N-l $N-l + mNN $N' (3.2) = 1 and mN,N-l = AN-l ~ O. we find a unique
z(N-cN)$N
Since $N value of $ N -1 from (3.2). Substituting the values of $ N
188
and 'N-l into (3.1) for i = N -1. we find a unique value of 2. Repeating this elimination procedure yields all components of~. (The last equation for i =0 is implied by the equations for i =1•...• N. since the rank of M is N.)
'N _
recursive relation is implied by (2.4): Si
=-
=
(z(i -c i )+Ai(N -0+ illi) ' i - ' i + 1 (i + 1) Ili + 1 A. (N + 1 - i) &-1
1
~
i
~
N- 1•
(3.3)
with the initial conditions 'N = 1. , N -1 = (NIlN + z(N-CN» I AN-I. Thus the computation of one eigenvector has a complexity of order /eN while the -+ -+ calculation of (I. ~) has a complexity of order (k+ I)N. where k is a constant independent of N. Eigenvalues - It is sufficient to calculate the eigenvalues 2 of A 2. To verify this. let k} and )} be the decreasing sequences of eigenvalues of A and A 2' respectively. From the proof of statement (c) of the 2 2 theorem it follows that ) > k+ > i. ~ k ~ N - 1. 0 > z j > z~2). and that in the interval (2) (2) • • • th th [z k • Z k + .1. I. ~ k ~ N - 1. ere are no 0 er 2 2 eigenvalues of the matrix A. Then given ) and one can find zk by standard numerical methods. Accordingly. for a given accuracy. the calculation of zk requires O(N k) operations where N k is the number of operations needed to calculate z~2).
{z
(z1
z1
z 1 z1 21'
z1
z1 21'
z12
z
To calculate ) we use the bisection method [13]. Let be any negative number and let E be the unit matrix. Then the number of sign changes in the principal minors of the matrix A2 - zE is equal to the number n_ (z) of eigenvalues of A2 that are no greater than z. Let n+ (z) = N -i. + I-n_ (z) be the number of eigenvalues of A2 that are greater than z. For! k and i k such that n+(! k) ~ k-i.,
z~2)
E (: k' ik]'
E
Now in terms of d
we
have
= (: k+ik)/2.
if
z1
n~(d) ~ k-i..
z1 )
n+(zk) < k-i.. 2
) E (d. zk]; then otherwise. (! k' d]. Thus. one step of the algorithm increases
zi
the accuracy of 2 ) by a factor of two. It is clear that the calculation of all (N - i. + 1) negative eigenvalues of the matrix A2 requires O(M 1 (N -i.» operations where M 1 is the number of operations required for the calculation of the principal minors of A 2 - zE. Complexity ~ tbe Metbod - The computation of G (x) requires (I, ~i)' ai' and zi' i. S; i ~ N. The principal minors of the matrix A2 - zE are computed in O(N -i.) operations. This is shown by the fact that. if Si is the minor fonned by elements of the last i + 1 rows and columns of the matrix A 2 - z E. then the following
+
(N-OIlN-i+iAN-i}
.
N - 1 - CN-i
Si-l -
i(N-i+l)AN_iIlN_i+l i~2 (N· -1-cN_i )(N· - I + 1 -cN-i+l ) Si-2 •
With the above in mind. it is convenient to represent the system (3.1) as , i -1
{Z
with So
=-
{NIlN N- cN
+
z}. and
SI = [(NIlN+ z(N- cN»(A N _ l +(N-l)IlN_l + z(N -1-cN_l »-NAN_l IlN]/[(N -cN)(N -1- cN_l)]
Hence we see that all negative eigenvalues of A can be computed in O(N - i.)2 operations. The complexity of computing the coefficients a i' i. ~ i ~ N. and the -+ -+ quantities (I. ~ i)' 1 ~ i ~ N has the same order. The complexity of the numerical method for determining G (x) is then O«N-i.)2). so the times to compute the equilibrium distribution in the generalized and constantparameters models differ by at most a constant factor. Numerical Results - We present below illustrative results evaluating two simple control strategies with constant maximum output rates. For the first strategy Ai -1 = A and Il i = Il if i < i l' but Ai -1 = mA and Ili = mll if i ~ i 1 • where il E {O. 1 •...• N} is a fixed parameter. The following values of the other parameters were used: A = 0.5. Il = 1. m = 2. C = 15.5. N = 45. The threshold i 1 was chosen close to zero and close to N (i 1 = 3 and i 1 = 42). The results are given in Table 1 and in Fig. 1. As one might expect. both graphs lie between the corresponding graphs for the model with the constant parameters A =' 0.5. Il= 1 and A= 1. 1l=2. Table 2 and Fig. 2 depict results evaluating a second strategy: A = const.• Ili = Il if i < i 1 and Ili = kll if i ~ i 1. The values of the parameters were taken to be A=0.5. 1l=0.5. c= 15.5. N=45. k=2. il =42. The results show that the asymptotics of G (x) for large x are the same as for the model with constant parameters Ai == A and Ili == kll. Even when i 1 is close to N and hence the buffer is rarely controlled. such a result can be explained by the fact that the buffer content reaches high levels when the number of active sources is sufficiently large. 4. ASYMPTOTIC BEHAVIOR OF THE PROBABILITY OF OVERFLOW Asymptotics for Large x - Equation (2.8) implies that ' / JC
-+ -+
G(x) - - a i • e • (I, ~i'>'
as
x ~
00 •
(4.1)
Formula (4.1) does not materially decrease the complexity of computing G(x), since the calculation of ai. reduces to that of z i for i. S i ~ N. However. we derive below an
189 log10 G(X)
c = 15.5
""'
....
....
Table 1
,,
....
'11
log10 G(X)
,,
'Ii 'Ii
,
'
...
\
il = 3
il = 42
-0.7
0
-.327045
-.328119
-0.8
4
-.552951
-.48467
-0.9
8
-.809409
-.612951
-1.0
X
'11
-0.6
12
-1.06585
-.737184
16
-1.32234
-.867409
20
-1.57874
-.994632
\ \
,,
,,
""'",10.5,11= 1 ,, ,, ,, , ,, '" , ,, ,, " ,,
,, ,,
,, ,
,, ,,
,, ,
,, ,
,, ,, ,
N = 45. c = 15.5
/", A. = 1. J.1 = 2
" ,,
,,
-1.7~----~----~------~----~----~---
4
12
8
16
log10 G(x) N = 45
Figure 1
c = 15.5
Table 2 log 10 G(x)
x
A. = 0.5
il = 42
J.1=1
A. = 0.5 k=2
-l.O~----~----~----~------~----~--
4
8
12
Figure 2
16
20
0
-.326137
-.220241
4
-.454671
-.381925
8
-.582951
-.523732
12
-.711841
-.660127
16
-.839409
-.800524
20
-.967632
-.940318
00
N
= 45. c = 15.5
20
00
190
asymptotic formula for G (x) that is more appropriate in the case of large N and i. «N. It decreases the complexity by a factor of O«N -i. )Ii.) relative to (4.1).
Substituting (4.6) into (4.5) and applying (1.1), we conclude that as x ~ 00
N
G(x)
= F N(oo) l: i=i.
G(x) -
(-z.)
IT
Qx(zi)
J
-+-+.
(J, Cl> i'>' ... , (J, Cl> N ), respectively. That is,
=
f
O. Denote by Lx; i •• ...• N(Z) the interpolating polynomial Lx(z) constructed with the abscissas (zi.' ... , zN). Then by Aitken's formula [4] we have (4.7)
Now by applying the triangle calculation scheme shown below,
N '
FN(OO) (n + 1) !
1
'..Il Zi
,=0 (ci-Ol ,=1
i.SjSN zi - Zj
where Qx(z) = e'DC 'P(z), and 'P(z) is the Lagrange interpolating polynomial with the points of interpolation or abscissas 0 > Z i > ... > Z N taking the values 0,
'P(z)
.
N (n+l)!Po I.Il
j~i
-+-+
N ~. } Il Ie " .xx n+l { i=1 '
I
N!PN 'P(Zi ) •
Substituting (2.7) into (2.8) we can write
(-zi) ,
i=i.
(4.3)
Lx; i •• i.+l (0) L X ;i.+l(O)
'.
FN(OO) G(x)
=
'.
Lx· i + 1 i + 2 (0)
Differentiations and substitution into (4.3) gives
. ••
Lx;i ••...• N(O)
(4.8)
Lx;N-l (0)
(n+l)!
Lx; N-l.N(O)
Comparinf ~4.1) and (4.4) and taking into account that Z and 'P(n+ - ) (1), 0 ~ k ~ n + 1, are bounded, we find that asx ~ 00.
we reduce the number of operations by a factor of least two. If £ > 0 is a given desired relative accuracy of the computation of Lx; i •• ...• N (0), then as an approximation we can replace Lx; i •• .... N(O) by Lx; i •• ...• j(O) for somej, ~ j ~ N; i.e., with
i.
Lx;
I •• ...•
j. j+ 1 (0) - Lx;
j ••..••
j(O)
Lx;i ••...• j(O)
But by (2.10) the continued product can be put in the form N
IT
(-zi)
=
(4.6)
i=i. N
N! .Il J.li N ,=1 Il (c .-i) 1=0 '
{Nl :A: ......A~.
I_ 1
i=O
1
,
[
)
~
(Ci-i)
}/i,tr_l-1
zi .
we can set Lx; i ••...• N(O) = Lx; i •• ...• j(O), if £j < £. By the scheme (4.8) the complexity of computing G(x) is reduced by a factor of two if j=N and by more than a factor of two if j < N. We now show that the interpolation technique is especially effective under heavy load. Define the traffic intensity as p = TIc and consider the case of beavy load when p is close to 1. To simplify the discussion, consider
191
the model with constant parameters c, A and Il= 1. Then P -- (1 AN +A) c· In th'IS case th e Iarges t negatt've
=-
eigenvalue is (see [1]) z. I.
consequently
Z i.
-
0
as
(1-p)(1+A) and 1 - c/N P ~ 1. Then
[7]
Kosten, L., "Stochastic Theory of a Multi-Entry Buffer (I)," Delft Progress Report, 1 (1974), 1018.
[8]
Kosten, L., "Liquid Models for a Type of Information Storage Problem," Delft Progress Report, 11 (1986), 71-86.
[9]
Marcus, M. and Minc H., A Survey of Matrix Theory and Matrix Inequalities, A1lyn and Bacon Inc., Boston, 1964.
[10]
Mitra, D., "Stochastic Theory of a Fluid Model of Producers and Consumers Coupled by a Buffer," Adv. Appl. Prob., 20 (1988), 646-676.
[11]
Mitra, D. and Anick, A., "Asymptotic Queue Behavior in a Data Handling System with a Large Number of Sources," typescript, (1979), AT&T Bell Laboratories, Murray Hill, NJ 07974 (available from first author).
[12]
Morrison, J. A., "Asymptotic Analysis of a DataHandling System with Many Sources," SIAM J. Appl. Math., 49 (1989), 617-637.
[13]
Voevodin, V. V., Computing Principles of Linear Algebra, Nauka, Moscow, 1977 [in Russian].
[14]
Weiss, A., "A New Technique for Analyzing Large Traffic Systems," Adv. Appl. Prob., 18 (1986),506-532.
Lx; i •• ...• N (0) - Lx. i. (0) and r x I G(x) - PNe' 'P(zi) I as p 1
wherePN
~=
{-
= [1
:A
r.
'I'(z.)
~
1,
(4.9)
= (1_~)N. and
(z'. +l-A) - " (z •• + l-A)2+ 4A } I lA.
Formula (4.9) requires the calculation of only one negative eigenvalue Z i. and reduces the complexity of computing G (x) by a 0 (N) factor. It is obvious that formula (4.9) is effective when Z i is close to zero. However, the approach extends to
l~ger values
of
IZ i. l
As a specific example, it is interesting to note that for = -0.5, N in the range 33-99, and c = 16.666,
Z i.
calculations of G(x) by the exact formula [1] and the approximate formula with only three negative eigenvalues zi.' zi.+l and zi.+2 produce results that are practically the same. It is natural to conjecture for the general model that Z i. - 0 (and hence (4.9» remains valid as p ~ 1.
REFERENCES [1]
Anick, D., Mitra, D., Sondhi, M. M., "Stochastic Theory of a Data-Handling System with Multiple Sources," Bell Syst. Tech. Journal, 61 (1982), 1871-1894.
[2]
Aven, 0.1., Coffman, E. G., Jr., and Kogan, Ya. A., Stochastic Analysis of Computer Storage, Riedel Publ. Co., Amsterdam, 1987.
[3]
Bellman, R., Introduction to Matrix Analysis, McGraw-Hill, New York, 1960.
[4]
Berezin, I. S. and Zhidkov, N. P., Computing Methods, Vols. 1,2, Pergamon Press, Oxford, 1965.
[5]
Halfin, S., "The Backlog of Data in Buffers with Variable Input and Output Rates," in Performance of Computer/Communication Systems, H. Rudin and W. Bux (eds.), Elsevier (North-Holland), IFIPS (1984),307-319.
[6]
Knessl, C., Matkowsky, B. J., Schuss, Z., and Tier, C., "Asymptotic Theory of Large Deviations for Markov Jump Processes," SIAM J. Appl. Math., 4S (1985), 1006-1028.
