Queueing Systems 43, 5–28, 2003 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
On Matrix-Geometric Solution of Nested QBD Chains ∗ SUNG HO CHOI
[email protected] Global Standards and Strategy Telecommunication R&D Center, Information & Communication Business, Samsung Electronics Co. LTD Suwon, P.O. Box 105 416, Maetan-3dong, Paldal-gu Suwon-si, Gyeonggi-do 442-742, Korea BARA KIM
[email protected] School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA KHOSROW SOHRABY
[email protected] School of Interdisciplinary Computing and Engineering, University of Missouri-Kansas City, 5100 Rockhill Road, Kansas City, MO 64110-2499, USA BONG DAE CHOI
[email protected] Telecommunication Mathematics Research Center and Department of Mathematics, Korea University, 1, Anam-dong, Sungbuk-ku, Seoul 136-701, Korea
Received 8 March 2001; Revised 10 July 2002 Abstract. In this paper, a generalization of the level dependent Quasi-Birth-and-Death (QBD) chains is presented. We analyze nested level dependent QBD chains and provide the complete characterization of their fundamental matrices in terms of minimal non-negative solutions of a number of matrix quadratic equations. Our results provide mixed matrix-geometric solution for the stationary distribution of nested QBD chains. Applications in overload control in communication networks are also discussed. Keywords: nested QBD chains, level dependent QBD chains, overload control, fundamental matrix, matrix-geometric solution
1.
Introduction
Consider a finite Quasi-Birth-and-Death (QBD) continuous-time Markov chain on the state space {(i, j ): 0 i M + 1, 1 j m} with an irreducible generator matrix 0 0 A0 U 0 0 0 0 1 D A U 0 D A U 0 2 , Q= .. .. .. .. .. . . . . . M A U M +1
0
D
A1
∗ A subset of this paper was presented at IEEE INFOCOM’2000 under the title of “General QBD processes
with applications to overload control”.
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S.H. CHOI ET AL. ∗
where D, A0 , A, A1 and U are m × m matrices. Let Q be a submatrix of Q and X be ∗ the inverse matrix of −Q , i.e., 1 A U 0 0 0 2 D A U ∗ Q ≡ .. . . . . . . . . . ... and . M A U X(1, 1) X(1, 2) . . . X(1, M) X(2, 1) X(2, 2) X(2, M) ∗ −1 , =X ≡ −Q . . .. .. .. . X(M, 1)
X(M, 2)
...
X(M, M)
where X(i, k) (1 i M, 1 k M) are m × m matrices. The fundamental ∗ matrix X of a transient generator matrix Q is defined as the minimal nonnegative left ∗ ∗ (and right) inverse matrix of −Q , say X = (−Q )−1 [8]. The (j, l)th element of the matrix X(i, k) (say, X(i, k)j,l ) is the expected amount of time the Markov chain spends in state (k, l) before it leaves the set of levels {1, 2, . . . , M} starting from state (i, j ). Hajek [6] obtained the first and last columns and rows of the fundamental matrix X (say, X(i, 1), X(i, M), 1 i M, and X(1, k), X(M, k), 1 k M) in the mixed matrix-geometric form. This was used for finding the stationary probability distribution of the generator Q in the mixed matrix-geometric form. For more references about mixed matrix-geometric representation of the stationary probability distribution of the QBD generator Q, see [15,16]. In this paper, we provide a complete characterization of the fundamental matrix in the mixed matrix-geometric form for the generator Q∗ 1 2 .. ∗ Q = . M −1 M
B0 D
U A
0 U .. .
0
..
. A D
0 0 .. . , U B1
∗
which is an extended version of the generator Q . The results are applicable to the analysis of level-dependent QBD chains with generators of the block form (0, 1) (0, 2) .. . Q=
(0, M1 ) (1, 1) .. . (1, M2 − 1) (1, M2 )
A
00
D0 .. . 0 0 .. . 0 0
U0 A0 .. .
0 U0 .. .
..
D0 0
A0 D1
0 0 .. .
. U0 A1 .. .
U1 .. .
..
D1 0
A1 D1
.
0 0 U1 A11
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
(0, 1) .. Q . 0 (0, M1 ) = (1, 1) 0 .. .
|
... 0
(1, M2 )
D1 .. . 0
0 .. .
| | ||
U0
| |
7
0 ... Q1
0 .
(1)
The above form arises naturally in applications to overload control with a single threshold [21]. A more general generator of the form 0 0 ... 0 Q0 Q0,1 Q1,0 Q1 Q1,2 0 ... 0 Q2,3 0 Q2,1 Q2 0 , (2) Q= .. .. .. .. .. . . . . . 0 Q Q ... Q N−1,N−2
0 where for 0 n N Bn0 Dn . Qn = ..
Qn,n−1
Un An .. .
0
... 0 Un .. .
... ... .. .
Dn An 0 . . . 0 Dn
. . . Dn . . . = , 0
0 0 .. .
N−1
N−1,N
QN,N−1
QN
, Un Bn1
0
Qn,n+1 = ...
Un
, ...
will also be considered. We call the Markov chain whose generator is of the form (2) a nested QBD chain. Note that the name of nested QBD chain is natural from the structure of Q (see figure 1). Detail definition will be given in section 2. This form arises in applications to overload controls with multiple thresholds [4,10–12]. The analysis will involve the determination of the fundamental matrices of the generators Qn , 0 n N. For the level-dependent QBD chains of the form (1), Ye and Li [21] introduced the folding algorithm to obtain the stationary probability distribution. Their work was extended to more general chain Q as in (2) with N = 1 [11–13] by using fundamental matrices of the generators Q0 and Q1 . These matrices were found by a generalization of the folding algorithm. Choi et al. [4] extended Li’s work to the general case of N 2. However, the approach using the folding algorithm does not give any information of the structure of the rows of the fundamental matrices. On the other hand, in this paper, a complete characterization of the fundamental matrix of a transient QBD generator is provided in the mixed matrix-geometric form. This yields a mixed matrix-geometric solution for the stationary distribution of the nested QBD chains. Further, our approach allows the size of matrix Q to be infinity in the sense that QN in (2) may have infinite
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S.H. CHOI ET AL.
Figure 1. Structure of the nested QBD chain.
dimension. This case cannot be treated by the generalized folding algorithm reported in [4,10–13,21]. Ibe and Keilson [7] gave the solution in detail with multiple thresholds (say, N 2) for the scalar case where the size m of each block is 1 using Green’s function method. Our model generalizes their work to general m. Further, the restriction of utilization (defined in section 2) of QBD blocks Qn is removed. The work in [7] assumes that all the utilizations are strictly less than 1.
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
9
Main results in this paper were introduced in [5] with some numerical examples but without proofs. This paper gives detail proofs to complete the results and removes some restrictions which were in [5]. The remainder of this paper is organized as follows. In section 2, we define nested QBD chains and provide an example in overload control. We present relationships between the stationary distribution of the chain and its associated fundamental matrices. In section 3, we give a complete characterization of the fundamental matrix of a transient QBD generator in the mixed matrix-geometric form. In section 4, we give the summary and conclusion.
2.
A nested QBD chain
A matrix Q is called a generator if its off-diagonal elements are non-negative and its diagonal elements are nonpositive and row sums are all less than or equal to zero. If the row sums of a generator Q are all zero, i.e., Qe = 0, where e is a column vector of appropriate size with all 1, then Q is called a conservative generator. Consider a nested QBD chain with three-dimensional state space {(n, i, j ): 0 n N, 1 i < Mn + 1, 1 j mn } with a (conservative) generator Q as in (2), where n represents the period, i the level and j the phase of the chain (see figure 1). We assume that 1 N < ∞, 1 Mn < ∞ for 0 n < N, and that 1 mn < ∞ for 0 n N. 1 Mn ∞ for n = N, Here Qn (0 n N) is a transient QBD generator such as 0 ... 0 Bn0 Un 0 1 An Un 0 . . . 0 D 2 n 0 0 Dn An Un 3 , Qn = .. . .. .. .. .. . . . . . .. Dn An Un Mn 0 ... 0 Dn Bn1
(3)
where Bn0 , Dn , An , Un and Bn1 are all mn × mn matrices. Qn,n−1 (1 n N) and Qn,n+1 (0 n N − 1) are the matrices related to transitions of the Markov chain with change of the period (say n → n − 1 or n → n + 1), each of which has only one nonzero block, whose size is mn × mn−1 for Qn,n−1 and mn × mn+1 for Qn,n+1 , such as
Qn,n−1
1 = ... Mn
1 0
... ...
+ ln+1 Dn
0
... ...
Mn−1 0
and
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S.H. CHOI ET AL.
Qn,n+1
1
1 . = .. Mn 0
...
...
− ln+1 0
Un
...
Mn+1
...
0
Note that if ln+ = Mn and ln− = 1 for all n, then the generator Q in (2) is the so-called level-dependent QBD chain [1–3,19,21]. Example. Overload control based on thresholds with multiclass messages [7] (see figure 2). Consider a single sever queue with 3 classes of traffics numbered 0, 1, 2 and the system capacity K ∞. Class k messages are generated according to a Poisson process with rate λk (0 k 2). The service times of all messages are exponentially distributed with mean 1/µ. The overload control in the system with multiple sets of thresholds ((an , bn ), 1 n 2) such that 1 a1 < min{b1 , a2 } max{b1 , a2 } < b2 < K + 1 is exercised in the following manner. We assume that the system is empty initially. Until the queue length (which is defined as the number of messages in the system) reaches b1 , the congestion status of the queue is assigned as 0 (i.e., n = 0). Once the queue length reaches b1 from below, we assign the congestion status of the queue as 1 during the period until the queue length reaches a1 from above (which changes the congestion status of the queue from 1 to 0) or reaches b2 from below (which changes the congestion status of the queue from 1 to 2). The congestion status of 2 will change into 1 when the queue length reaches a2 from above. During the congestion status of the queue is n (0 n 2), the arriving messages whose class is less than n are blocked and lost. Then we have a Markov chain whose state space is {(n, i , j ): 0 n 2, an < i < bn+1 , j = 1} where a0 ≡ −1 and b3 ≡ K + 1. Then the period n represents the congestion status of the queue and the level i represents the queue length. This Markov chain is an example of nested QBD chains with state space {(n, i, j ): 0 n N, 1 i < Mn , 1 j mn }, where (n, i, j ) = (n, i − an , j ), Mn = bn+1 − an − 1, mn = 1, ln+ = an+1 − an and ln− = bn − an . 2 Let λ+ n = k=n λk for n = 0, 1, 2, then we can identify the 1 × 1 matrices as follows: B00 = −λ+ 0,
B01 = −λ+ 0 − µ,
B20 = −λ+ 2 − µ,
B21 = −µ,
Dn =
Dn
= µ,
An = −λ+ n − µ,
B10 = B11 = −λ+ 1 − µ, Un = Un = λ+ n
for n = 0, 1, 2.
Figure 2. A special model of overload control based on thresholds when N = 2.
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
11
The above model can be easily extended to the system where the arrival process when the congestion status is n is an MAP (Markovian Arrival Process) represented by (C n , D n ) [4,14], where C n and D n are m × m matrices. In that case, the phase j (1 j m) represents the phase of the arrival process and the block matrices are as follows B00 = C 0 ,
B01 = C 0 − µI,
B20 = C 2 − µI,
B21 = −µI,
Dn =
Dn
= µI,
B10 = B11 = C 1 − µI, Un = Un = D n
An = C n − µI,
for n = 0, 1, 2.
We assume that the nested QBD chain Q is irreducible. If MN < ∞, then Q is ergodic since it is an irreducible finite Markov chain. For the case that MN = ∞, assuming that DN + AN + UN is irreducible it is easily shown that Q is ergodic if and only if the “utilization” ρN < 1 (see [17, theorem 1.7.1]), where ρN is defined as follows: Let κN be the stationary probability vector of DN + AN + UN , i.e., κN (DN + AN + UN ) = 0 and κN e = 1, where e is the column vector of appropriate size with all 1. Then ρN is defined as ρN ≡ κN UN e/(κN DN e). Assume that Q is ergodic. Then there exists unique stationary distribution. Let p(n, i, j ) denote the stationary probability that the nested QBD chain Q in (2) is in state (n, i, j ). Define p(n, i) ≡ (p(n, i, 1), . . . , p(n, i, mn )) for 1 i < Mn + 1 and pn ≡ (p(n, 1), p(n, 2), . . . , p(n, Mn )) for 0 n N, and p ≡ (p0 , p1 , . . . , pN ). In order to obtain the probability vector p, we need to solve the equation pQ = 0 and pe = 1. In the following theorem we express the stationary probability vector p in terms of the fundamental matrices Xn defined as
Xn (1, 1) Xn (2, 1) Xn ≡ .. . Xn (Mn , 1)
Xn (1, 2) Xn (2, 2) .. .
... ... .. .
Xn (Mn , 2)
...
Xn (1, Mn ) Xn (2, Mn ) ≡ (−Qn )−1 , .. . Xn (Mn , Mn )
where Xn (i, k) (1 i, k Mn ) are all mn × mn matrices. Theorem 1. Suppose that the nested QBD chain Q in (2) is ergodic. Then the stationary probability vector p of Q is expressed in terms of fundamental matrices as follows: + + cπ0 X0 (l1 , k), p(n, k) = cπn− Xn (ln− , k) + cπn+ Xn (ln+ , k), − cπN XN (lN− , k),
n = 0, 1 k M0 , n = 1, . . . , N − 1, 1 k Mn , n = N, 1 k < MN + 1. (4)
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S.H. CHOI ET AL.
Here πn− , 1 n N, and πn+ , 0 n N − 1 are mn -dimensional probability vectors − + , πN−1 , πN− ) are the invariant probability vector for so that π ≡ (π0+ , π1− , π1+ , . . . , πN−1 the following stochastic matrix 0 0 U0+ 0 0 0 0 0 . . . 0 0 0 0 D1− 0 0 U1− 0 0 0 . . . 0 0 0 0 0 D+ 0 0 U + 0 0 0 . . . 0 0 0 0 0 1 1 0 0 D− 0 0 U − 0 . . . 0 0 0 0 0 2 2 + + 0 0 D2 0 0 U 2 0 . . . 0 0 0 0 0 , (5) P = .. .. .. .. .. .. .. .. . . .. .. .. .. . . . . . . . .. . . . . − − 0 0 0 0 0 0 0 . . . 0 DN−1 0 0 UN−1 0 0 0 0 0 0 0 . . . 0 D+ 0 0 U + N−1
0
0
0
0 0 0 0...0
0
0 DN−
N−1
0
where Un+ = Xn (ln+ , Mn )Un , Un− = Xn (ln− , Mn )Un and Dn− = Xn (ln− , 1)Dn ,
Dn+
=
Xn (ln+ , 1)Dn ,
0 n N − 1, 1 n N − 1, 1 n N.
Here c is the normalizing constant. Proof. It is readily checked that p given by (4) satisfies pQ = 0 by using π P = π and Xn Qn = −I for all n. Remark. 1. In fact, the matrix P above is the one-step transition probability matrix for the Markov chain embedded at the epochs immediately after period-transitions of the nested QBD chain Q, and (4) is derived from the theory of Markov regenerative processes (see [9, theorem 9.30]). 2. Since the matrices Xn (i, k) can be obtained in a mixed matrix-geometric form (see next section), the stationary probability distribution (p(n); n = 0, . . . , N) has also a mixed matrix-geometric form. 3. In this paper we assume that transitions to outside of a period occur only when the nested QBD chain is in the boundary levels 1 or Mn (see figure 1). The nested QBD chains in this paper can be easily extended to ones in which the levels where transitions to outside of the period are possible is not necessarily the boundary levels. For the extended chains, all theory in this paper hold with slight modification. The reasons why we assume that the levels from which period-transitions are possible are the boundary levels are that (i) this assumption simplifies notation and that (ii) the above extension is not necessary for almost applications (especially, in overload controls in communication networks).
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
3.
13
Fundamental matrix of a transient QBD generator
In this section, we provide the complete characterization of for a transient QBD generator Q∗ . We denote 1 B0 U 0 0 2 D A U 0 3 0 D A U . Q∗ = .. .. . . .. . .. .. . M −1 0 D A M 0 0 D X(1, 1) X(1, 2) . . . X(2, 1) X(2, 2) . . . −1 =X ≡ −Q∗ . .. .. . X(M, 1)
X(M, 2)
...
the fundamental matrix X , U B1 X(1, M) X(2, M) .. .
(6)
,
X(M, M)
where all submatrices are m×m. We consider both cases for which M is finite or infinite. Throughout this section, we assume that Q∗ is irreducible and transient, and D + A + U is irreducible. When D + A + U is conservative, let κ be its invariant distribution vector, i.e., κ(D + A + U ) = 0, κe = 1, and define the utilization ρ ≡ κU e/(κDe). Define G1 , G2 , R1 and R2 as the minimal nonnegative solutions of the following equations [17,18], respectively, D + AG + U G2 = 0, R 2 D + RA + U = 0,
DG2 + AG + U = 0, D + RA + R 2 U = 0.
We deal with the cases for which M is finite or infinite separately. 3.1. The case of 2 M < ∞ For the case the M < ∞, we will use the following assumption separately. • Assumption (a): D + A + U is transient, or conservative and ρ = 1. • Assumption (b): D + A + U is conservative and ρ = 1. First, we need the following lemma. Lemma 2. Under the assumption (a), sp(G1 G2 ) < 1 and sp(G2 G1 ) < 1, and B0 + U G1 (B0 G2 + U )GM−2 2 B[G1 , G2 ] ≡ (D + B1 G1 )GM−2 DG2 + B1 1
(7)
is invertible. Proof. See appendix A for the proof of the first part. To prove the second part, we first note that if sp(G1 G2 ) < 1 and sp(G2 G1 ) < 1, then both I − G1 G2 and I − G2 G1 are invertible. Now suppose that B[G1 , G2 ] is not invertible. Then there
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S.H. CHOI ET AL.
exists a nonzero 2m × 1 column vector [v1T , v2T ]T such that B[G1 , G2 ][v1T , v2T ]T = 0. Let M−i T T v2 for 1 i M. Then we have Q∗ [x1T , x2T , . . . , xM ] = 0 xi = Gi−1 1 v1 + G2 ∗ by direct substitution. Since Q is invertible, xi = 0 for i = 1, . . . , M, and so x1 − G2 x2 = (I − G2 G1 )v1 = 0 and xM − G1 xM−1 = (I − G1 G2 )v2 = 0. Since I − G2 G1 and I − G1 G2 are invertible from the first part of this lemma, we have both v1 = 0 and v2 = 0. Therefore [v1T , v2T ]T = 0, which is a contradiction. The following theorem is an extended version of [6, theorem 3, part (i)]. Theorem 3. Under the assumption (a), the first and last columns of the fundamental matrix for the generator Q∗ in (6) are given by M−i V (2, 1) X(i, 1) = Gi−1 1 V (1, 1) + G2
X(i, M) =
Gi−1 1 V (1, M)
+
GM−i V (2, M) 2
for 1 i M, for 1 i M,
where V (i, j ) (i = 1, 2, j = 1, M) are m × m matrices defined by −1 V (1, 1) V (1, M) ≡ − B[G1 , G2 ] . V (2, 1) V (2, M) Proof.
The theorem is proved by checking Q∗ X = −I directly.
Lemma 4. Under the assumption (a), for 2 k M − 2, B0 + U G1 (B0 G2 + U )Gk−2 0 0 2 M−k−1 DG2 + A U U G2 −U Gk1 B (k) [G1 , G2 ] ≡ M−k DGk−1 D A + U G −DG 1 1 2 0 0 (D + B1 G1 )GM−k−2 DG2 + B1 1 is invertible. Proof. Suppose that B (k) [G1 , G2 ] is not invertible. Then there exists a nonzero 4m × 1 column vector [v1T , v2T , v3T , v4T ]T such that B (k) [G1 , G2 ][v1T , v2T , v3T , v4T ]T = 0. Let k−i Gi−1 1 i k, 1 v1 + G2 v2 , xi = i−(k+1) M−i v3 + G2 v4 , k + 1 i M. G1 T T ] = 0 by direct substitution. Since Q∗ is invertible, Then we have Q∗ [x1T , x2T , . . . , xM xi = 0 for all i (1 i M) and so
x1 − G2 x2 = (I − G2 G1 )v1 = 0, xk+1 − G2 xk+2 = (I − G2 G1 )v3 = 0,
xk − G1 xk−1 = (I − G1 G2 )v2 = 0, xM − G1 xM−1 = (I − G1 G2 )v4 = 0.
Since both I − G2 G1 and I − G1 G2 are invertible, v1 = 0, v2 = 0, v3 = 0 and v4 = 0. These are contradictory to [v1T , v2T , v3T , v4T ]T = 0.
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
15
The following theorem together with theorem 3 gives the complete characterization of the fundamental matrix for the generator Q∗ in the mixed matrix-geometric form. Theorem 5. Under the assumption (a), the kth columns of the fundamental matrix for the generator Q∗ in (6) are given as follows: k−i 1 i k, Gi−1 1 V (1, k) + G2 V (2, k), X(i, k) = i−(k+1) M−i G1 V (3, k) + G2 V (4, k), k + 1 i M, for 2 k M − 2, and i−1 W (2, k), G1 W (1, k) + Gk−i−1 2 X(i, k) = i−k M−i G1 W (3, k) + G2 W (4, k),
1 i k − 1, k i M,
for 3 k M − 1, where 0 V (1, k) −1 −I V (2, k) (k) , = B [G1 , G2 ] 0 V (3, k) 0 V (4, k) 0 W (1, k) −1 0 W (2, k) (k−1) [G1 , G2 ] −I . W (3, k) = B 0 W (4, k) Proof.
The theorem is proved by checking Q∗ X = −I directly.
The following theorem gives the characterization of the fundamental matrix X in terms of the matrices R1 and R2 . Theorem 6. Under the assumption (a), the rows of the fundamental matrix for the generator Q∗ in (6) are given as follows. (i) The first and last rows are given by
where
(1, 2)R2M−k , (1, 1)R1k−1 + V X(1, k) = V
1 k M,
(M, 2)R2M−k , (M, 1)R1k−1 + V X(M, k) = V
1 k M,
(1, 1) V V (M, 1)
(1, 2) V V (M, 2)
B0 + R 1 D ≡ R2M−2 (R2 B0 + D)
R1M−2 (U + R1 B1 ) R 2 U + B1
here the inverse exists. (ii) The ith row is given by (i, 2)R2i−k , (i, 1)R1k−1 + V 1 k i, V X(i, k) = k−(i+1) M−k (i, 4)R2 , i + 1 k M, (i, 3)R1 +V V
−1 ,
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S.H. CHOI ET AL.
for 2 i M − 2, and (i, 2)R2i−k−1 , 1 k i − 1, (i, 1)R1k−1 + W W X(i, k) = (i, 4)R2M−k , (i, 3)R1k−i + W i k M, W for 3 i M − 1, where (i) (i, 1) V (i, 2) V (i, 3) V (i, 4) ≡ [0 −I 0 0] B [R1 , R2 ] −1 , V 2 i M − 2, −1 (i−1) (i, 1) W (i, 2) W (i, 3) W (i, 4) ≡ [0 0 −I 0] B [R1 , R2 ] , W 3 i M − 1, here (i) [R1 , R2 ] B −R1i D B0 + R 1 D i−2 R (R2 B0 + D) R2 U + A ≡ 2 0 D 0 R2M−i−1 D 2 i M − 2,
R1i−1 U U A + R1 D −R2M−i U
0 0 , R1M−i−2 (U + R1 B1 ) R 2 U + B1
and it is invertible. Proof. The theorem is proved by modifying the proofs of lemma 2, theorem 3, lemma 4 and theorem 5 (see appendix D). In the remaining of this section, we consider the case for which D + A + U is conservative and ρ = 1. A subset E of the state space of a Markov chain is called a closed set if for all i and j such that i ∈ E and j ∈ E c , j cannot be reached from i (refer to [20, pp. 154, 205]). Let . .. ... ... . Q≡ (8) D A U .. .. .. . . . We start with the following lemma. Lemma 7. If Q is irreducible, then each of G1 G2 and G2 G1 has only one irreducible closed set. Proof.
See appendix B.
Note that a stochastic matrix with only one irreducible closed set has the eigenvalue 1 and its algebraic multiplicity is one. The following theorem gives mixed matrixgeometric expression for every column of the fundamental matrix of the generator Q∗ when D + A + U is conservative and ρ = 1.
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
17
Theorem 8. Under the assumption (b), there exists an m-dimensional column vector y satisfying (U − D)e = (D + A + U )y. Further if Q is irreducible then the fundamental matrix for the generator Q∗ in (6) is given as follows. (i) The first and last columns are given by M−i V (2, 1) for 1 i M, X(i, 1) = (i − 1)e − y v1,1 + Gi−1 1 V (1, 1) + G2 i−1 M−i X(i, M) = (i − 1)e − y v1,M + G1 V (1, M) + G2 V (2, M) for 1 i M, (9) where v1,1 and v1,M are 1 × m vectors and V (i, j ) (i = 1, 2, j = 1, M) are all m × m matrices, and they are given by a solution of v1,1 v1,M −I 0 , (10) B[y, G1 , G2 ] V (1, 1) V (1, M) = 0 −I V (2, 1) V (2, M) where
B[y, G1 , G2 ] ≡
U e − (B0 + U )y (M − 2)D + (M − 1)B1 e − (D + B1 )y (B0 G2 + U )GM−2 B0 + U G1 2 (D + B1 G1 )GM−2 DG2 + B1 1
and (10) has a solution. (ii) The kth column is given by k−i (i − 1)e − y v1,k + Gi−1 1 V (1, k) + G2 V (2, k), 1 i k, X(i, k) = V (3, k) + GM−i V (4, k), (i − k − 1)e − y v3,k + Gi−(k+1) 2 1 k + 1 i M, for 2 k M − 2, and k−i−1 W (2, k), (i − 1)e − y w1,k + Gi−1 1 W (1, k) + G2 1 i k − 1, X(i, k) = M−i W (4, k), (i − k)e − y w3,k + Gi−k 1 W (3, k) + G2 k i M,
(11)
(12)
for 3 k M − 1, where v1,k , v3,k , w1,k and w3,k are 1 × m vectors and V (j, k) and W (j, k) (1 j 4) are all m × m matrices, and they are given by solutions of v1,k w1,k 0 0 V (1, k) W (1, k) V (2, k) −I W (2, k) 0 (k−1) [y, G1 , G2 ] B (k) [y, G1 , G2 ] = = , B , 0 −I v3,k w3,k V (3, k) 0 W (3, k) 0 V (4, k)
W (4, k) (13)
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S.H. CHOI ET AL.
where
U e − (B0 + U )y B0 + U G1 (B0 G2 + U )Gk−2 2 k U (−ke + y) −U G DG + A 2 1 (k) B [y, G1 , G2 ] = D[(k − 1)e − y] DGk−1 D 1 0 0 0 0 0 0 −Uy U U GM−k−1 2 M−k D(y + e) A + U G1 −DG2 (∗) (D + B1 G1 )GM−k−2 DG2 + B1 1 with (∗) = [(M − k − 2)D + (M − k − 1)B1 ]e − (D + B1 )y and each equation in (13) has a solution. Proof. Refer to the proof of theorem 3 in [6] for the existence of vector y satisfying (U − D)e = (D + A + U )y. (i) (9) is proved by checking Q∗ X = −I directly, if (10) has a solution. So, it remains to show that 2m × (2m + 1) matrix B[y, G1 , G2 ] has a pseudo-inverse, i.e., B[y, G1 , G2 ] has rank of 2m. Since the number of columns of the matrix B[y, G1 , G2 ] (say, 2m + 1) is equal to the rank of B[y, G1 , G2 ] plus the dimension of the kernel of B[y, G1 , G2 ], it suffices to show that the dimension of the kernel is less than or equal to 1. Let v ≡ [a, v1T , v2T ]T be a nonzero element of the kernel of B[y, G1 , G2 ], i.e., B[y, G1 , G2 ]v = 0. Let M−i v2 . xi ≡ a (i − 1)e − y + Gi−1 1 v1 + G2 T T ] = 0 by direct substitution. Since Q∗ is invertible, Then we have Q∗ [x1T , x2T , . . . , xM we have xi = 0 for all i (1 i M), and therefore (14) x1 − G2 x2 = a (G2 − I )y − e + (I − G2 G1 )v1 = 0.
Let κ (1) and κ (2) be the invariant probability vectors of G1 G2 and G2 G1 , respectively, i.e., κ (1) G1 G2 = κ (1) κ (2) G2 G1 = κ (2)
and and
κ (1) e = 1, κ (2) e = 1.
(15) (16)
Note that from lemma 7, κ (1) and κ (2) are uniquely determined by (15) and (16), and that κ (1) = κ (2)G2 By (14),
and
κ (2) = κ (1)G1 .
κ (2)(x1 − G2 x2 ) = a κ (1) − κ (2) y − 1 = 0.
By appendix C, (κ (1) − κ (2) )y < 1. Hence a = 0, and so (I − G2 G1 )v1 = 0. Thus v1 = c1 e for some c1 since G2 G1 is a stochastic matrix with only one irreducible closed set.
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
19
On the other hand, xM − G1 xM−1 = (I − G1 G2 )v2 = 0. Hence v2 = c2 e for some c2 . Since x1 = (c1 + c2 )e = 0, we have c2 = −c1 . Thus v = (0, c1 eT , −c1 eT )T for some c1 . Hence the dimension of the kernel is less than or equal to 1. (ii) (11) and (12) are proved by checking Q∗ X = −I directly, if each equation in (13) has a solution. So, it remains to show that each equation in (13) has a solution. This is proved by showing that for 2 k M −2, 4m×(4m+2) matrix B (k) [y, G1 , G2 ] has a pseudo-inverse, i.e., B (k) [y, G1 , G2 ] has rank of 4m. Since the number of columns of the matrix B (k) [y, G1 , G2 ] is 4m + 2, it suffices to show that the dimension of kernel of B (k) [y, G1 , G2 ] is less than or equal to 2. Let v = [a, v1T , v22 , b, v3T , v4T ]T be a nonzero element of the kernel of B (k) [y, G1 , G2 ], i.e., B (k) [y, G1 , G2 ]v = 0. Let k−i a (i − 1)e − y + Gi−1 1 i k, 1 v1 + G2 v2 , xi ≡ v3 + GM−i v4 , k + 1 i M. b (i − k − 1)e − y + Gi−k−1 1 2 T T ] = 0 by direct substitution and so xi = 0 for all i Then we have Q∗ [x1T , x2T , . . . , xM (1 i M). Since x1 − G2 x2 = a (G2 − I )y − e + [I − G2 G1 ]v1 = 0,
it is proved that a = 0 and v1 = c1 e for some c1 , as in the proof of (i). From xk − G1 xk−1 = [I − G1 G2 ]v2 = 0, we have v2 = c2 e for some c2 . Since x1 = (c1 + c2 )e = 0, we have c2 = −c1 . Similarly, from xk+1 − G2 xk+2 = 0
and
xM − G1 xM−1 = 0,
we have b = 0, v3 = c3 e and v4 = −c3 e for some c3 . Hence v = (0, c1 eT , −c1 eT , 0, c3 eT , −c3 eT )T for some c1 and c3 . Hence the dimension of the kernel is less than or equal to 2. The following theorem gives an expression for the fundamental matrix in terms of R1 and R2 under the assumption (b). Theorem 9. Under the assumption (b), there exists m-dimensional row vector z satisfying κ(D − U ) = z(D + A + U ). Further, if Q is irreducible then the fundamental matrix for the generator Q∗ in (6) is given as follows. (i) The first and last rows are given by (1, 1)R1k−1 + V (1, 2)R2M−k X(1, k) = v˜ 1,1 [(k − 1)κ − z] + V (M, 1)R1k−1 + V (M, 2)R2M−k X(M, k) = v˜M,1 [(k − 1)κ − z] + V
for 1 k M, for 1 k M,
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S.H. CHOI ET AL.
(i, j ) (i = 1, M, j = 1, 2) are all m × m where v˜1,1 and v˜M,1 are m × 1 vectors and V vectors and they are given by a solution of (1, 1) V (1, 2) v˜1,1 V (M, 1) V (M, 2) v˜M,1 V κD − z(B + D) κ (M − 2)U + (M − 1)B − z(U + B ) 0 1 1 M−2 × R (U + R1 B1 ) B0 + R 1 D 1
R2M−2 (R2 B0 + D) −I 0 = , 0 −I
R 2 U + B1 (17)
which has a solution. (ii) The ith row is given by (i, 2)R2i−k , (i, 1)R1k−1 + V v˜i,1 (k − 1)κ − z + V 1 k i, X(i, k) = (i, 4)R2M−k , (i, 3)R1k−(i+1) + V (k − i − 1)κ − z +V v ˜ i,3 i + 1 k M, for 2 i M − 2, and (i, 2)R2i−k−1 , (i, 1)R1k−1 + W w i,1 (k − 1)κ − z + W 1 k i − 1, X(i, k) = (i, 4)R2M−k , (i, 3)R1k−i + W w i,3 (k − i)κ − z + W i k M, (i, j ) and i,j (j = 1, 3) are m × 1 vectors and V for 3 i M − 1, where v˜i,j and w (i, j ) (1 j 4) are all m × m matrices, and they are given by solutions of W (i) (i, 1) V (i, 2) v˜i,3 V (i, 3) V (i, 4) B [z, R1 , R2 ] v˜i,1 V = [0 −I 0 0], (i−1) (18) (i, 1) W (i, 2) w (i, 3) W (i, 4) B [z, R1 , R2 ] i,3 W w i,1 W = [0 0 −I 0], where (i) [z, R1 , R2 ] B κD − z(B0 + D) (−iκ + z)D R1i D B0 + R 1 D i−2 R (R2 B0 + D) R2 U + A ≡ 2 0 −zD 0 D M−i−1 D 0 R2
(i − 1)κ − z U R1i−1 U U (κ + z)U A + R1 D −R2M−i U
0
0 0 (∗∗) M−i−2 R1 (U + R1 B1 ) R 2 U + B1
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
21
with (∗∗) = κ[(M − i − 2)U + (M − i − 1)B1 ] − z(U + B1 ) and each equation in (18) has a solution. Proof. The theorem is proved by modifying the proofs of lemma 7 and theorem 8 (see appendix D). Remark. If Q∗ with B0 = B1 = A in (6) is irreducible, then Q is irreducible. Hence theorem 3 together with 8(i) is an extension of [6, theorem 3], and theorem 6(i) together with theorem 9(i) is an extension of [6, theorem 4]. Further, we have no assumption on the size of M for the existence of the solutions of (10), (13), (17) and (18). Note that Hajek [6] showed the existence of the solutions of (10) and (17) for the case of sufficiently large M. In the next subsection, we consider the case of infinite levels. 3.2. The case that M = ∞ Now we consider the case that M = +∞. 1 B0 2D Q∗ (∞) ≡ 3 0 .. .. . .
Let U A D
0 U A .. .
0 0 U .. .
... ..
.
(19)
.
We assume that Q∗ (∞) is irreducible and transient, and D + A + U is irreducible. Note that if D + A + U is conservative and ρ 1, then B0 + U must be not conservative. It is possible that D + A + U is conservative and B0 + U is conservative if ρ > 1. When D + A + U is conservative and ρ = 1, we assume that Q in (8) is irreducible. Theorem 10. The columns and rows of the fundamental matrix for Q∗ (∞) in (19) are given as follows. (i) The first column is given by −1 X(i, 1) = −Gi−1 1 (B0 + U G1 ) ,
i 1.
and the kth (k 2) column is given as follows: When D + A + U is transient, or D + A + U is conservative and ρ = 1, i−1
G1 V (1, k) + Gk−i 2 V (2, k), 1 i k, X(i, k) = i k + 1, Gi−k 1 X(k, k), where k−2 −1 −1 V (1, k) = Gi−1 1 (B0 + U G1 ) (B0 G2 + U )G2 (DG2 + A + U G1 ) , V (2, k) = (DG2 + A + U G1 )−1 .
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S.H. CHOI ET AL.
When D + A + U is conservative and ρ = 1,
k−i
+ Gi−1 (i − 1)e − y v1,k 1 V (1, k) + G2 V (2, k), X(i, k) = Gi−k 1 X(k, k),
1 i k, i k + 1,
where y is an m-dimensional column vector satisfying (U − D)e = (D + A + U )y and
, V
(1, k) and V
(2, k) are given by a solution of v1,k k−2 + U )y B + U G (B G + U )G U e − (B 0 0 1 0 2 2 DG2 + B1 (k − 2)D + (k − 1)B1 e − (D + B1 )y (D + B1 G1 )Gk−2 1
v1,k 0 (20) × V
(1, k) = −I V
(2, k) with B1 = A + U G1 , which has a solution. (ii) The first row is given by X(1, k) = −(B0 + R1 D)−1 R1k−1 ,
k 1,
and the ith (i 2) row is given as follows: When D + A + U is transient, or D + A + U is conservative and ρ = 1,
(i, 2)R2i−k , 1 k i, (i, 1)R1k−1 + V V X(i, k) = k i + 1, X(i, i)R1k−i , where (i, 1) = (B0 + R1 D)−1 R2i−2 (R2 B0 + D)(R2 U + A + R1 D)−1 , V (i, 2) = −(R2 U + A + R1 D)−1 . V When D + A + U is conservative and ρ = 1,
(i, 2)R2i−k , 1 k i,
(i, 1)R1k−1 + V (k − 1)κ − z + V v˜i,1 X(i, k) = k i + 1, X(i, i)R1k−i ,
, where z is an m-dimensional row vector satisfying κ(D − U ) = z(D + A + U ) and v˜i,1
V (i, 1) and V (i, 2) are given by a solution of
(i, 2)
(i, 1) V v˜i,1 V κD − z(B0 + D) κ (i − 2)U + (i − 1)B1 − z(U + B1 ) = [0 −I ] × R1i−2 (U + R1 B1 ) B0 + R 1 D
R2i−2 (R2 B0 + D)
R 2 U + B1
with B1 = A + R1 D, which has a solution. Proof. (i) By the probabilistic interpretation of the matrix G1 and the fundamental matrix, it is easily seen that X(1, 1) = −(B0 +U G1 )−1 (refer to proof of [6, theorem 1]).
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
23
By simple probabilistic arguments, it is also seen that X(i, 1) = Gi−1 1 X(1, 1) for i 2. For a fixed k (k 2), consider 1 B0 U 0 0 ... 2 0 D A U 3 0 D A U ∗ . Q (k) = .. .. .. .. .. . . . . . 0 D A U k 0 0 D A + U G1 Observe that [X(k)](i, j ) = X(i, j ) (1 i, j k), where X(k) is the fundamental matrix for Q∗ (k). When D + A + U is transient, or D + A + U is conservative and k−i
ρ = 1, by theorem 3, we have [X(k)](i, k) = Gi−1 1 V1 + G2 V2 for 1 i k, where −1 B0 + U G1 (B0 G2 + U )Gk−2 0 V1 2 = −I V2
0 DG2 + A + U G1 (B0 + U G1 )−1 (B0 G2 + U )Gk−2 (DG2 + A + U G1 )−1 2 = . −(DG2 + A + U G1 )−1 When D + A + U is conservative and ρ = 1, by theorem 8, we have [X(k)](i, k) = k−i
[(i − 1)e − y]v1
+ Gi−1 1 V1 + G1 V2 for 1 i k, where v1 , V1 and V2 are given by a solution of equation (20), which has a solution. By the probabilistic interpretation of G1 and the fundamental matrix, it is seen that X(i, k) = Gi−k 1 X(k, k) for i k + 1. (ii) is proved by modifying the proof of (i) (see appendix D). 4.
Summary and conclusion
We now summarize all the steps for finding the steady-state distribution (p(n, k)) for the nested QBD chain with infinitesimal generator Q given as in (2). Algorithm. 1. Solve the rate matrices Rn,1 and Rn,2 (0 n N) from the quadratic matrix equations 2 Dn + Rn,1 An + Un = 0, Rn,1 2 Un = 0 Dn + Rn,2 An + Rn,2
to obtain the minimal nonnegative solutions Rn,1 and Rn,2 , respectively. 2. Obtain rows of fundamental matrices; Xn (ln+ , k) (0 n N − 1, 1 k Mn ) and Xn (ln− , k) (1 n N, 1 k Mn ) in terms of Rn,1 and Rn,2 using theorem 6 or theorem 9 for finite case, or theorem 10 for infinite case. 3. Obtain P matrix in the form (5) using Xn (ln− , Mn ) (1 n N−1), Xn (ln+ , Mn ) (0 n N − 1), Xn (ln− , 1) (1 n N), and Xn (ln+ , 1) (1 n N − 1).
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S.H. CHOI ET AL.
4. Solve the simultaneous equations π P = π and π e = 1. 5. Write p(n, k) using π , Xn (ln+ , k) and Xn (ln− , k) as in (4). The algorithm is mainly based on simple block matrix operations including addition, multiplication and solving systems of linear equations except the first step. Most of the computation is devoted to the first step where rate matrices Rn,1 and Rn,2 (0 n N) are obtained. We can use efficient algorithm proposed recently in [1,2] to obtain the rate matrices. We considered a general class of nested QBD chains and its applications to overload control. A general solution methodology to obtain the steady-state probability for the nested QBD chains was given using fundamental matrices of transient QBD matrices. We have also given complete characterization of the fundamental matrices for both the finite case and the infinite case in mixed matrix-geometric forms. Appendix A. The proof of the first part of lemma 2 Let ν ≡ maxi (−Ai,i ), and A ≡ I +(1/ν)A, D ≡ (1/ν)D, and U ≡ (1/ν)U . Consider a Markov chain {(Nn , Jn ): n = 0, 1, 2, . . .} with the one step transition probability matrix P of the form: .. . . . . . . . . . . D A U −1 . P ≡ 0 D A U D A U 1 .. .. .. .. . . . . Note that if D +A+ U is not conservative, then P is substochastic i.e., P e = e. Observe that [G1 ]i,j = P t < ∞, Jt = j | (N0 , J0 ) = (1, i) , [G2 ]i,j = P τ < ∞, Jτ = j | (N0 , J0 ) = (0, i) where t = inf{n > 0: Nn = 0} and τ = inf{n > 0: Nn = 1} (inf(∅) = ∞, as usual). Define stopping times tk (k 1) and τk (k 0) of Nn as follows: τ0 = 0, tk = inf{n > τk−1 : Nn = 0} and τk = inf{n > tk : Nn = 1} for k 1. Then it can be seen that [G1 G2 ]i,j = P τ1 < ∞, Jτ1 = j | (N0 , J0 ) = (1, i) . More generally, we have (G1 G2 )n i,j = P τn < ∞, Jτn = j | (N0 , J0 ) = (1, i) . If D + A + U is conservative and ρ < 1, then Nn → −∞ as n → ∞ with probability 1. If D + A + U is conservative and ρ > 1, then Nn → +∞ as n → ∞ with probability 1.
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
25
If D + A + U is not conservative (equivalently, if it is transient), then with probabil/ {. . . , −1, 0, 1, . . .} except for finitely many n. Therefore, for all cases, with ity 1, Nn ∈ propability 1, τk = ∞ except for finitely many k. Hence (G1 G2 )n → 0 as n → ∞. So we have sp(G1 G2 ) < 1. Similarly we can prove that sp(G2 G1 ) < 1. B. The proof of lemma 7 Let E ≡ {j : Ui,j > 0 for some i}. We will show that (i) for all i, j ∈ E, [(G1 G2 )n ]i,j > 0 for some n > 0, and that (ii) for j ∈ E c , j th column of (G1 G2 ) is the zero column, which imply that G1 G2 has the unique irreducible closed set and the set is E. (i) Suppose that i, j ∈ E. Let P , Nn , Jn , tk and τk be defined as in appendix A. Since P is irreducible, P {(Nn1 , Jn1 ) = (0, k) | (N0 , J0 ) = (1, i)} > 0 for some n1 > 0, where k is a state such that U k,j > 0. Hence ∞ ! (G1 G2 )n i,j P τn < ∞, Jτn = j for some n | (N0 , J0 ) = (1, i) n=1
P (Nn1 , Jn1 ) = (0, k), (Nn1 +1 , Jn1 +1 ) = (1, j ) | (N0 , J0 ) = (1, i) = P (Nn1 , Jn1 ) = (0, k) | (N0 , J0 ) = (1, i) U k,j > 0.
Hence [(G1 G2 )n ]i,j > 0 for some n > 0. (ii) Suppose that j ∈ E c . Since the j th column of U is the zero column, the j th column of G2 is the zero column. Hence the j th column of G1 G2 is the zero column. By (i) and (ii), G1 G2 has the unique irreducible closed set. Similarly it is proved that G2 G1 has the unique irreducible closed set. C. The proof of (κ (1) − κ (2))y < 1 in the proof of theorem 8 Let P , Nn and Jn be defined as in appendix A. Let f (n, i) ≡ n − yi . Then it is easily checked that {f (Nn , Jn ): n = 0, 1, 2, . . .} is an {Fn : n = 0, 1, 2, . . .}martingale, where Fn is the σ -field generated by {(Nk , Jk ): k = 0, . . . , n}. Let t ≡ inf{n > 0: Nn = 0}. Then, since t is a stopping time with respect to {Fn }, we have E f (Nt ∧n , Jt ∧n ) | (X0 , J0 ) = (1, i) = 1 − yi , where t ∧ n ≡ min{t, n}. Since f (Nt ∧n , Jt ∧n ) is bounded below, by Fatou’s lemma, we have −(G1 y)i 1 − yi . Hence, (C.1) (I − G1 )y i 1, for all i = 1, . . . , m. We claim that
(I − G1 )y
k
< 1,
for some k ∈ {1, . . . , m}.
(C.2)
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S.H. CHOI ET AL.
If (C.2) does not hold, (I − G1 )y = e. Hence y = e + G1 y = · · · = ne + Gn1 y, which is impossible since every entry of the right-hand side goes to infinity. Thus (C.2) holds. Let k be the phase such that [(I − G1 )y]k < 1. It is easily checked that there exists a phase i0 ∈ E ≡ the irreducible closed set of G1 G2 , such that P (Nt ∧u , Jt ∧u ) = (1, k) | (N0 , J0 ) = (1, i0 ) > 0 for some u 0. Conditioning on (Nt ∧u , Jt ∧u ), −(G1 y)i0 = E[f (Nt , Jt ) | (N0 , J0 ) = (1, i0 )] is written as −(G1 y)i0 = P (Nt ∧u , Jt ∧u ) = (1, k) | (N0 , J0 ) = (1, i0 ) × E f (Nt , Jt ) | (Nt ∧u , Jt ∧u ) = (1, k) ! P (Nt ∧u , Jt ∧u ) = (n, j ) | (N0 , J0 ) = (1, i0 ) + (n,j )=(1,k)
× E f (Nt , Jt ) | (Nt ∧u , Jt ∧u ) = (n, j ) . By (C.2), we have E f (Nt , Jt ) | (Nt ∧u , Jt ∧u ) = (1, k) = −(G1 y)k < 1 − yk = f (1, k), and by the martingale property and Fatou’s lemma, we have E f (Nt , Jt ) | (Nt ∧u , Jt ∧u ) = (n, j ) f (n, j ). Therefore, we have −(G1 y)i0 < P (Nt ∧u , Jt ∧u ) = (1, k) | (N0 , J0 ) = (1, i0 ) f (1, k) ! P (Nt ∧u , Jt ∧u ) = (n, j ) | (N0 , J0 ) = (1, i0 ) f (n, j ) + (n,j )=(1,k)
= E f (Nt ∧u , Jt ∧u ) | (N0 , J0 ) = (1, i0 ) = f (1, i0 ) (by martingale property) = 1 − yi0 . Hence,
(I − G1 )y
i0
< 1.
(C.3)
> 0, and so, by (C.1) Since i0 ∈ E ≡ the irreducible closed set of G1 G2 , we have κi(1) 0 and (C.3), we have
κ
(1)
−κ
(2)
y = κ (I − G1 )y = (1)
m ! i=1
κi(1) (I − G1 )y i < 1.
MATRIX-GEOMETRIC SOLUTION OF NESTED QBD CHAINS
27
D. A remark for proofs of theorems 6, 9 and 10(ii) If κ(B0 + D) 0
and
κ(U + B1 ) 0,
(D.1)
then theorems 6, 9 and 10(ii) are proved by applying theorems 3, 5, 8 and 10(i) to the dual generator −1 ∗ T Q diag(κ, . . . , κ) Q ≡ diag(κ, . . . , κ) of Q∗ . For example, (D.1) holds if B0 = B1 = A as in [6]. However (D.1) does not hold in general. So, we have to prove theorems 6, 9 and 10(ii) directly. In fact, theorems 6, 9 and 10(ii) are proved by modifying the proofs of lemmas 2, 4 and 7 and theorems 3, 5, 8 and 10(i). For example, we give a proof of theorem 6(i) by modifying the proofs of lemma 2 and theorem 3. Proof of theorem 6(i). Since sp(G1 G2 ) < 1 and sp(G2 G1 ) < 1, we have sp(R1 R2 ) < 1 and sp(R2 R1 ) < 1 by duality (see [8]). We claim that B0 + R 1 D R1M−2 (U + R1 B1 ) B [R1 , R2 ] ≡ R 2 U + B1 R2M−2 (R2 B0 + D) 1 , R2 ] is not invertible. Then there exists a nonzero is invertible. Suppose that B[R 1 , G2 ] = 0. Let x˜i = v˜1 R1i−1 + v˜2 R2M−i 1 × 2m row vector [v˜1 , v˜2 ] such that [v˜1 , v˜2 ]B[G for 1 i M. Then we have [x˜1 , x˜2 , . . . , x˜M ]Q∗ = 0 by direct substitution. Since Q∗ is invertible, x˜i = 0 for i = 1, . . . , M, and so x˜1 − x˜2 R2 = v˜1 (I − R1 R2 ) = 0 and x˜M − x˜M−1 R1 = v˜2 (I − R2 R1 ) = 0. Since sp(R1 R2 ) < 1 and sp(R2 R1 ) < 1, I − R1 R2 and I − R2 R1 are invertible. Hence v˜1 = 0 and v˜2 = 0. Therefore [v˜1 , v˜2 ] = 0, which is 1 , R2 ] is invertible. a contradiction. Therefore B[R Now, the theorem is completed by checking XQ∗ = −I directly. References [1] N. Akar, N.C. Oguz and K. Sohraby, A novel computational method for solving finite QBD processes, Comm. Statist. Stochastic Models 16(2) (2000) 273–311. [2] N. Akar, N.C. Oguz and K. Sohraby, An invariant subspace approach in M/G/1 and G/M/1 type Markov chains, Comm. Statist. Stochastic Models 13(3) (1997) 381–416. [3] L. Bright and P.G. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-anddeath processes, Comm. Statist. Stochastic Models 11(3) (1995) 497–525. [4] B.D. Choi, S. Choi, B. Kim and D.K. Sung, Analysis of priority queueing system based on thresholds and its application to signaling system No.7 with congestion control, Comput. Networks 32 (2000) 149–170. [5] S. Choi, K. Sohraby and B. Kim, General QBD processes with applications to overload control, in: Proc. of IEEE INFOCOM ’2000, March 2000, pp. 165–172.
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[6] B. Hajek, Birth-and-death processes on the integers with phases and general boundaries, J. Appl. Probab. 19 (1982) 488–499. [7] O.C. Ibe and J. Keilson, Overload control in finite-buffer multiclass message systems, Telecommunication Systems 2 (1994) 121–140. [8] J.G. Kemeny, J.L. Snell and A.W. Knapp, Denumerable Markov Chains (Springer, New York, 1976). [9] V.G. Kulkarni, Modeling and Analysis of Stochastic Systems (Chapman & Hall, London, 1995). [10] L.A. Kulkarni and S.Q. Li, Performance analysis of rate based feedback control for ATM networks, in: Proc. of IEEE INFOCOM ’97, April 1997. [11] W.C. Lau and S.Q. Li, Sojourn-time analysis on nodal congestion in broadband networks and its impact on QoS specifications, in: Proc. of IEEE INFOCOM ’96, March 1996, pp. 1327–1337. [12] S.Q. Li, Overload control in a finite message storage buffer, IEEE Trans. Commun. 37(12) (1989) 1330–1338. [13] S.Q. Li and H.D. Sheng, Generalized Folding-algorithm for sojourn time analysis of finite QBD processes and its queueing applications, Comm. Statist. Stochastic Models 13(3) (1996) 507–522. [14] D.M. Lucantoni, K.S. Meier-Hellstern and M.F. Neuts, A single-server queue with server vacations and a class of non-renewal arrival processes, Adv. Appl. Probab. 22 (1990) 676–705. [15] V. Naumov, Matrix-multiplicative approach to quasi-birth-and-death processes analysis, in: MatrixAnalytic Methods in Stochastic Models, eds. A.S. Alfa and S.R. Chakravarthy, Lecture Notes in Pure and Applied Mathematics, Vol. 183 (Marcel Dekker, New York, 1996). [16] V. Naumov, Modified matrix-geometric solution for finite QBD processes, in: Advances in Algorithmic Methods for Stochastic Models, eds. Latouche and Taylor (Notable Publications, 2000). [17] M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models (Johns Hopkins Univ. Press, Baltimore, MD, 1981). [18] M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications (Marcel Dekker, New York, 1989). [19] V. Ramaswami and P.G. Taylor, Some properties of the rate operators in level-dependent quasi-birthand-death processes with a countable number of phases, Comm. Statist. Stochastic Models 12(1) (1996) 143–164. [20] R.W. Wolff, Stochastic Modeling and the Theory of Queues (Prentice-Hall, Englewood Cliffs, NJ, 1989). [21] J. Ye and S.Q. Li, Folding algorithm: A computational method for finite QBD processes with leveldependent transitions, IEEE Trans. Commun. 42(2–4) (1994) 625–639.
