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The Invariant Measure of Homogeneous Markov Processes in The Quarter-Plane: Representation in Geometric Terms

We consider the invariant measure of a homogeneous continuous- time Markov process in the quarter-plane. The basic solutions of the global balance equation are the geometric distributions. We first show that the invariant measure can not be a finite linear combination of basic geometric distributions, unless it consists of a single basic geo- metric distribution. Second, we show that a countable linear combina- tion of geometric terms can be an invariant measure only if it consists of pairwise-coupled terms. As a consequence, we obtain a complete characterization of all countable linear combinations of geometric dis- tributions that may yield an invariant measure for a homogeneous continuous-time Markov process in the quarter-plane.


Introduction
Homogeneous continuous-time Markov processes in the quarter-plane have translation invariant transition rates, except for the rates along the horizontal and vertical boundaries. In literature, many examples exist of such processes with geometric invariant measure, including Jackson networks and queuing networks with negative customers, see [4] for an overview. The compensation approach of Adan [2,1] has revealed that Markov processes without transitions to the east, north and northeast can have an invariant measure that is a countable linear combination of geometric terms.
The present paper provides a complete characterization of all countable linear combinations of geometric distributions that may yield an invariant measure for homogeneous Markov processes in the quarter-plane. In particular, our contributions are as follows. First we show that the invariant measure cannot be a finite linear combination of geometric distributions, unless it consists of a single geometric distribution. Second, we show that a countable linear combination of geometric terms can be an invariant measure only if it consists of pairwise-coupled terms, i.e., only if each 2-dimensional geometric distribution in the sequence shares a parameter with the previous term in the sequence.
Alternative approaches to analyzing the invariant measure of Markov processes in the quarter-plane are available in the literature. Most notably, generating functions have been used for the analysis of a variety of such problems, see, e.g., [7,9]. A general theory is provided in [6,8], in which the invariant measure is characterized via transforms. However, an explicit expression for the invariant measure is usually hard to obtain. The present paper characterizes invariant measures of a tractable form, i.e., linear combinations of geometric terms.
The remainder of this paper is structured as follows. In Section 2 we present the model and some definitions. The main results of this paper are given in Section 3.

Model
Consider a two-dimensional continuous-time Markov process on the pairs (i, j) of non-negative integers. We refer to {(i, j)|i > 0, j > 0}, {(i, j)|i > 0, j = 0}, {(i, j)|i = 0, j > 0} and (0, 0) as the interior, the horizontal axis, the vertical axis and the origin of the state space, respectively. The transition rate from (i, j) to (i + s, j + t) is denoted by q s,t (i, j). The process is homogeneous in the sense that for any (i, j) and (k, l) in the interior of the state space for all s and t. Moreover, (1) holds for all pairs (i, j), (k, l) on the horizontal and vertical axis respectively. Note that the first equality of (1) implies that the rates of transitions leaving from each part of the state space are translation invariant. The second equality ensures that also rates entering the same part of the state space are translation invariant. 1 Transitions are restricted to adjoining points (horizontally, vertically and diagonally), i.e., q s,t (k, l) = 0 if |s| > 1 or |t| > 1. We introduce, for i > 0, j > 0, the notation q s,t (i, j) = q s,t , q s,0 (i, 0) = h s and q 0,t (0, j) = v t . Finally, let −q 0,0 , −h 0 and −v 0 denote the outgoing rates in the interior, at the horizontal axis and at the vertical axis respectively. The model and notation are illustrated in Figure 1. We will refer to this type of process as a homogeneous Markov process. In the remainder of this paper, if not explicitly stated, we assume that a Markov process is ergodic.

Candidate geometric measures
Our interest is homogeneous Markov processes with an finite invariant measure that can be expressed as a linear combination of geometric measures. In particular, we will consider geometric measures that satisfy the balance equations in the interior of the space, i.e., measures of the form ρ i σ j , with (ρ, σ) ∈ C, If m is a linear combination of terms Γ ⊂ C we say that m is induced by Γ, see Figure 2(a).

Uncoupled partitions
Different ways of partitioning set Γ will be introduced here. These partitions play an essential role in the analysis later on.

implies that σ =σ; and is 3) uncoupled if it is both horizontally and vertically uncoupled.
We call the partition with the largest number of components a maximal partition.
Lemma 1. Among all the horizontally uncoupled partitions, there exists a unique maximal horizontally uncoupled partition.
Existence of maximal unique (vertically) uncoupled partitions follows similarly. Examples of maximal uncoupled partition, horizontally, vertically uncoupled partitions can be found in Figures 2(b), 2(c) and 2(d) respectively. We denote the number of components in the maximal horizontally uncoupled partition by H and the components themselves as Γ h The elements of this partition can be obtained by taking the union of elements from Using the maximal uncoupled partition, we can introduce measures m k , defined as This allows us to write m(i, j) = U k=1 m k (i, j). Remark : Note that H, V and U are not necessarily finite. With a slight abuse of notation we will write expressions involving for instance H k=1 , or k = 1, . . . , H, also when H is infinite, i.e., the number of components is countably infinite. The same assumption holds for V .

Analysis
We consider the structure and cardinality of Γ. In Section 3.1 we consider the number of uncoupled components in Γ. In Section 3.2 we study the structure of Γ in more detail. Finally, in Section 3.3 we consider the cardinality of Γ.

Number of uncoupled components in Γ
The following theorem is the first main result. It states that an invariant measure can not be induced by a set Γ of which the uncoupled partition contains multiple components.

Theorem 1. Consider a homogeneous Markov process P and its invariant
The proof of the theorem is deferred to the end of this section. We first introduce some additional notation. For any set Γ i ⊂ Γ let Note that B h (Γ) and B v (Γ) are the balance equations for the measure induced by Γ at the horizontal and vertical boundary respectively.

Lemma 2. Consider a homogeneous Markov process P and any finite mea-
where the last equality follows from the fact that m is a finite measure, see also Definition 1.
Note that Lemma 2 does not require m to be the invariant measure of P , it can be any finite measure. The following lemma is a key element for the proof of Theorem 1.
Proof. Since m is the invariant measure of P , it satisfies the balance equations at state (i, 0). Therefore, The exchange of summations is justified by Lemma 2.
Suppose that H is finite. From (7) , 0), i > 0 is satisfied. Using the same reasoning balance at the vertical axis is satisfied. Balance in the interior is satisfied by the assumption that m is induced by Γ ⊂ C. Finally, balance in the origin is implied by balance in other parts of the state space.
Proof of Theorem 1. We will show that the measures m k , k = 1, . . . , U, satisfy all balance equations. Let measure m k be induced by Γ k . By definition of C this implies that all m k , k = 1, . . . , U satisfy the balance equations in the interior of the state space. For the balance equation for m k at state (i, 0) at the horizontal boundary we obtain By Lemma 2, the interchange of the summations leading to the second equality is valid. The last equality follows from Lemma 3.
In similar fashion it follows that the balance equations at the vertical boundary are satisfied. As a consequence, we have shown that m 1 , · · · , m U are invariant measures of P . When U > 1, this contradicts to the fact that there is a unique invariant measure for ergodic Markov process P .

Structure of Γ
In this section the structure of set Γ will be discussed. From Theorem 1 it follows if the number of components in the maximal uncoupled partition is greater than one, then a measure induced by Γ cannot be the invariant measure of homogeneous Markov process P . In this section we investigate the measure induced by a set with one uncoupled component. To this end, we introduce the notion of a pairwise-coupled set.
Pairwise coupling allows the explicit characterization of the structure of set Γ that is required if the measures induced by Γ are the invariant measure of homogeneous Markov process P . The following corollary is a simple application of Theorem 1.  Proof. Suppose that where (ρ, σ) ∈ C and (ρ,σ) ∈ C. It follows from the definition of C that σ andσ are the roots of the following quadratic equation in x, Note that the maximal vertically uncoupled partition of {(ρ, σ), (ρ,σ)} consists of the two singleton components {(ρ, σ)} and {(ρ,σ)}. It follows from Lemma 3 that B v ({(ρ, σ)}) = B v ({(ρ,σ)}) = 0. Therefore, σ andσ are the roots of From a comparison of the coefficients of (9) and (10) it follows that either a) one of the roots will be 1, contradicting the definition of C, or b) the transition rates of P are such that P is not irreducible and hence not ergodic. Hence, m as defined in (8) can not be the invariant measure of P . Using the same argument it follows that a form α(ρ, σ)ρ i σ j + α(ρ,σ)ρ iσj cannot be the invariant measure of P .
Before proving the final lemma, i.e., that Γ satisfying 2 < |Γ| < ∞ can not induce an invariant measure, we introduce a final piece of notation and some technical results that will help in the presentation of the remaining proofs.
The following technical result will greatly simplify the presentation of our final proofs of the cases when the measure is induced by Γ satisfies 2 < |Γ| < ∞.
Proof. From the definition in (11) it follows that From the above the first row of inequalities follow directly. The remaining inequalities follow directly from (12).
The second technical lemma that we will need is readily verified and stated without proof.
The final result, together with Lemma 4 it provides the proof of Theorem 2 is the following lemma.
Lemma 7. Consider a homogeneous Markov process P , if 2 < |Γ| < ∞, then no measure induced by Γ can be in the invariant measure of P .
For the first case we consider the relations which by Lemma 3 are required to hold if m is the invariant measure of P . We will construct s 1 , s 2 , t 1 and t 2 that satisfy By Farkas' Lemma this leads to a contradiction to (13). The s 1 , s 2 , t 1 and t 2 are constructed by considering the auxiliary measurē m = α(ρ 1 , σ 1 )ρ i 1 σ j 1 + α(ρ n , σ n )ρ i n σ j n and the homogeneous Markov processP , that has the same transition rates as P in the interior of the state space and ratesh 1 ,h −1 ,v 1 andv −1 along the boundaries. We now consider the . Also, from Lemma 6 it follows that s 1 ≥ 0, s 2 ≤ 0, t 1 ≥ 0, t 2 ≤ 0. Therefore, s 1 , s 2 , t 1 and t 2 satisfy (15). This concludes the proof of the first case.
To summarize the contributions of the present paper, we combine Theorems 1 and 2 into the following corollary. Note that if Γ has one element, then m as geometric product form and many examples exist in literature, see also [3]. The existence of invariant measures with countably many terms has been demonstrated in [2].