A compensation approach for two-dimensional Markov processes

Abstract

Several queueing processes may be modelled as random walks on a multi-dimensional grid. In this paper the equilibrium distribution for the case of a two-dimensional grid is considered. In previous research it has been shown that for some two-dimensional random walks the equilibrium distribution has the form of an infinite series of products of powers which can be constructed with a compensation procedure. The object of the present paper is to investigate under which conditions such an elegant solution exists and may be found with a compensation approach. The conditions can be easily formulated in terms of the random behaviour in the inner area and the drift on the boundaries.

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“…In the present case we could only prove absolute convergence on the set A, which was determined by N. If N = r, then A represents the whole first quadrant of the grid, but for N > r, there are also states for which xm,n(OO) was formally defined, but may not converge absolutely. A similar feature was already encountered in the case of the asymmetic shortest queue problem (d. [2]) and also in the case of general random walks on the first quadrant with transitions to neighbours only (d. [4]). The remaining question is of course: how large can N be?…”

“…In [4] it has been investigated for which random walks on the first quadrant of the plane a compensation approach could be developed. In order to keep the analysis tractable, a restriction was made to random walks with transitions to neighbouring states only.…”

“…The model to be treated in the present paper satisfies the latter condition for n~r (compare fig. 1), but it does not belong to the restricted class considered in [4], since for inner states the transitions are not restricted to neighbouring states. At first sight the difference is not essential, although it will make the analysis more cumbersome.…”

“…In fact, it becomes more and more apparent that finding feasible starting solutions is the essential difficulty of the compensation approach. In the first exploration of the compensation approach for the symmetric shortest queue problems (d. [I]), this aspect of finding a starting solution still looked like a simple question, since there was a natural candidate, but later explorations placed this aspect more and more in the center (d. [3,4,6,7]). So, Section 5 may well be called the key part of this paper.…”

“…To create sufficient freedom we introduce extra coefficients for the solution in states (m, n) with n = 0,1, ... , r -1 instead of using the other r -1 roots of equation The analogous procedure would not have worked for the vertical boundary, since we would have been forced by the inner conditions (2.1) for the states on the lines m = r, r + 1, ... , m+ r -1 to set the coefficients of (3n equal to am + cam, yielding insufficient freedom to satisfy the vertical boundary coriditions (2.2)-(2.3). In the case of the horizontal boundary, however, the lines n = 0,1, ... ,r -1 only allow transitions to the line n = r and not to any state with n > r. Because of this feature, the equations (2.1) for inner states do not restrict the freedom of choice for the en' Note that the way of compensating of (3.4) was also used in [4] Q.…”