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On partially homogeneous nearest-neighbour random walks in the quarter plane and their application in the analysis of two-dimensional queues with limited state-dependency

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Abstract

This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such random walks are characterized by the fact that the one-step transition probabilities are functions of the state-space. We show that stationary behaviour is investigated by solving a finite system of linear equations, two matrix functional equations, and a functional equation with the aid of the theory of Riemann (–Hilbert) boundary value problems. This work is strongly motivated by emerging applications in flow level performance of wireless networks that give rise in queueing models with scalable service capacity, as well as in queue-based random access protocols, where the network’s parameters are functions of the queue lengths. A simple numerical illustration, along with some details on the numerical implementation are also presented.

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Acknowledgements

The author would like to thank the Editor in Chief, the Associate Editor and the anonymous Reviewer for the very careful reading of the manuscript and the insightful remarks and input, which helped to improve the original exposition.

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Appendices

Overview of various scheduling policies based on our framework

In the following, we provide an overview of several other scheduling disciplines that are based on our modelling framework, and can be studied by our approach.

  1. (1)

    Queue-based traffic control in telecommunication networks [19] (for example, Asynchronous Transfer Mode networks (ATM), or Broadband Integrated Services Digital Network (B-ISDN)). The authors used a two-dimensional queueing system with two finite capacity queues. Under the employed scheduling scheme, the next customer to be served is of type k with probability \(a_{k}(i,j)\), \(k = 1, 2\), when there are i type-1 and j type-2 jobs, with \((i,j)\in \{0,1,\ldots ,K_{1}\}\times \{0,1,\ldots ,K_{2}\}\) and \(a_{1}(i,j)+a_{2}(i,j)=1\) for all i and j.

  2. (2)

    Head of the line (HOL) priority scheduling policy, for example, [70]. In such a scenario, we have, for eample, \(\beta _{1}(n_{1},n_{2})=1\), for \(n_{1}>0\). i.e., type 1 jobs have HOL priority over type 2 jobs. Our modelling framework is even more general and incorporates a mixed HOL policy, under which priority is assigned to a class based on the inhomogeneous regions, for example, \(\beta _{1}(n_{1},n_{2})=1\) for \((n_{1},n_{2})\in S_{a}\), and \(\beta _{2}(n_{1},n_{2})=1\) for \((n_{1},n_{2})\in S_{d}\).

  3. (3)

    Shortest queue first policy (SQF), for example [44]: The next job to be served is chosen by the least loaded queue. Such a scheduling discipline was recently used to improve the quality of Internet access on high-speed communication links [17]. In such a scenario

    $$\begin{aligned} \beta _{1}(n_{1},n_{2})=\left\{ \begin{array}{ll} 1,&{}n_{1}\le n_{2},\\ 0,&{}n_{1}>n_{2}, \end{array}\right. \,\, \beta _{2}(n_{1},n_{2})=\left\{ \begin{array}{ll} 1,&{}n_{2}\le n_{1},\\ 0,&{}n_{2}>n_{1}. \end{array}\right. \end{aligned}$$
  4. (4)

    Longest queue first policy (LQF), for example [24]: Also known as the greedy maximal scheduling policy, it is recognized for its high performance in practice and chooses the next customer to be served by the longest queue. In such a scenario

    $$\begin{aligned} \beta _{1}(n_{1},n_{2})=\left\{ \begin{array}{ll} 1,&{}n_{1}> n_{2},\\ 0,&{}n_{1}\le n_{2}, \end{array}\right. \,\, \beta _{2}(n_{1},n_{2})=\left\{ \begin{array}{ll} 1,&{}n_{2}> n_{1},\\ 0,&{}n_{2}\le n_{1}. \end{array}\right. \end{aligned}$$
  5. (5)

    Bernoulli scheduling policy [36, 49]: Upon a service completion, a type k customer is chosen with probability \(p_{k}\). In such a scenario, \(\beta _{k}(n_{1},n_{2})=p_{k}\), \(k=1,2\), with \(p_{1}+p_{2}=1\).

  6. (6)

    Queue-length-threshold (QLT) policy [15, 48]: Such a policy is applied for traffic control in telecommunication systems, such as ATM networks. We set a constant L in queue 1, and the next job to be served is chosen according to the following rule:

    $$\begin{aligned} \beta _{1}(n_{1},n_{2})=\left\{ \begin{array}{ll} 1,&{}n_{1}> L,\\ 0,&{}n_{1}\le L, \end{array}\right. \,\, \beta _{2}(n_{1},n_{2})=\left\{ \begin{array}{ll} 1,&{}n_{1}\le L,\\ 0,&{}n_{1}>L. \end{array}\right. \end{aligned}$$

    For \(L=0\), we give HOL priority to queue 1. For \(L=\infty \), then we have the 1-limited discipline, for example [23].

  7. (7)

    QLT scheduling policy with Bernoulli schedule [20]: Refers to a generalization of the QLT policy described above. In such a scenario

    $$\begin{aligned} \beta _{1}(n_{1},n_{2})=\left\{ \begin{array}{ll} 1,&{}n_{1}> L,\\ p,&{}n_{1}\le L, \end{array}\right. \,\, \beta _{2}(n_{1},n_{2})=\left\{ \begin{array}{ll} 1,&{}n_{1}\le L,\\ q:=1-p,&{}n_{1}>L. \end{array}\right. \end{aligned}$$
  8. (8)

    Discrete-time GPS (DGPS) [73]: If both queues are non-empty, the server chooses a job from queue 1 (resp. queue 2) with probability \(\beta \) (resp. \(1-\beta \)). If one of the queues is empty, a job from the other queue is chosen with probability 1:

    $$\begin{aligned} \beta _{1}(n_{1},n_{2})=\left\{ \begin{array}{ll} \beta ,&{}n_{1},n_{2}> 0,\\ 1,&{}n_{1}>0,n_{2}=0, \end{array}\right. \,\, \beta _{2}(n_{1},n_{2})=\left\{ \begin{array}{ll} 1-\beta ,&{}n_{1},n_{2}>0,\\ 1,&{}n_{1}=0,n_{2}>0. \end{array}\right. \end{aligned}$$
  9. (9)

    Modified DGPS: If both queues are non-empty, the server chooses a job from queue 1 (resp. queue 2) with probability \(\beta _{1}^{n_{2}}\) (resp. \(\beta _{2}^{n_{1}}\)), with \(\beta _{1}^{n_{2}}+\beta _{2}^{n_{1}}=1\), i.e., the probability of choosing the next customer of either type is a decreasing function of the number of backlogged jobs of the other type. In such a scenario

    $$\begin{aligned} \beta _{1}(n_{1},n_{2})=\left\{ \begin{array}{ll} \beta _{1}^{n_{2}},&{}n_{1},n_{2}> 0,\\ 1,&{}n_{1}>0,n_{2}=0, \end{array}\right. \,\, \beta _{2}(n_{1},n_{2})=\left\{ \begin{array}{ll} \beta _{2}^{n_{2}},&{}n_{1},n_{2}>0,\\ 1,&{}n_{1}=0,n_{2}>0. \end{array}\right. \end{aligned}$$

Our modelling framework describes the limited versions of the all these service disciplines based on the splitting rule (1), and further generalize them, by allowing

  • an even more general form of the service scheduling function \(\beta _{k}(n_{1},n_{2})\).

  • the arrival and service rates to depend also on the scheme described in (1), (2).

  • combination of scheduling policies according to the splitting rule (1). This means that in each subregion \(S_{j}\), \(j=a,b,c,d,\) we are able to consider different scheduling policies. Such flexibility allows us to provide a quite general service scheduling scheme.

An alternative approach

In the following, we focus on the case \(\Psi (0,0)=0\), i.e., \(p_{-1,-1}=0\), and on the solution of (11), which is now reduced in terms of a solution of a Riemann–Hilbert boundary value problem, by following the method in [33]. The basic steps are briefly summarized below:

  1. Step 1

    We provide a thorough investigation of the kernel equation \(R(x,y)=0\), by identifying its roots as well as their properties (see Sect. B.1).

  2. Step 2

    The functional equation (11) is then used to show that \(g_{0}(x)\), \(h_{0}(y)\) satisfy certain boundary value problems of Riemann–Hilbert type [35], with boundary conditions on closed contours. Information on these curves are obtained from Step 1 (see Sect. B.1, and Lemmas 567). Clearly, the functions \(g_{0}(x)\), \(h_{0}(y)\) are analytic inside the unit disc, but they might have poles in the region bounded by the unit disc and these closed curves. Thus, we have to analytically continue (Appendix C) \(g_{0}(x)\), \(h_{0}(y)\) in the whole interiors of the contours; see [35, Chapter 3]. Similar observations are made also for the functions \(g_{n_{2}}(\cdot )\), \(h_{n_{1}}(\cdot )\). Equation (40) is the boundary condition that satisfies the function \(g_{0}(.)\) on such a closed contour (see Sect. B.2).

  3. Step 3

    Then, we transform (through conformal mappings [38]; see Appendix C) that problem from the closed contour defined in Step 1, into boundary value problem of Riemann–Hilbert type on the unit disc; see (42). Such a conversion is motivated by the fact that the latter problems are more usual and by far more commonly treated in the literature [23].

  4. Step 4

    We solve the new problem by providing an integral expression of the unknown boundary function \(g_{0}(.)\); see (43) in Sect. B.2. Similarly, we obtain the other unknown function \(h_{0}(y)\). Then, substituting in (11) we finally obtain g(xy) in terms of the unknown probabilities: \(\pi (N_{1},n_{2})\), \(n_{2} =0, 1,\ldots ,N_{2} -1\), \(\pi (n_{1},N_{2})\), \(n_{1} =0, 1,\ldots ,N_{1} -1\) and \(\pi (N_{1},N_{2})\). A way to obtain these unknowns is given at end of Sect. B.2.

1.1 Kernel analysis

The kernel R(xy) is a quadratic polynomial with respect to x, y. Indeed,

$$\begin{aligned} R(x,y)={\widehat{a}}(x)y^2+{\widehat{b}}(x)y+{\widehat{c}}(x)=a(y)x^{2}+b(y)x+c(y), \end{aligned}$$

where, for \(p_{i,j}:=p_{i,j}(N_{1},N_{2})=p_{i,j}(n_{1},n_{2})\) for \((n_{1},n_{2})\in S_{d}\),

$$\begin{aligned} \begin{array}{lcl} {\widehat{a}}(x)=-(xp_{0,1}+x^{2}p_{1,1}+p_{-1,1}),&{}&{}a(y)=-(yp_{1,0}+y^{2}p_{1,1}+p_{1,-1}),\\ {\widehat{b}}(x)=x(1-p_{0,0})-p_{-1,0}-x^{2}p_{1,0},&{}&{}b(y)=y(1-p_{0,0})-p_{0,-1}-y^{2}p_{0,1},\\ {\widehat{c}}(x)=-(x^{2}p_{1,-1}+xp_{0,-1}),&{}&{}c(y)=-(y^{2}p_{-1,1}+yp_{-1,0}). \end{array} \end{aligned}$$

In the following, we provide some technical lemmas that are necessary for the formulation of a Riemann–Hilbert boundary value problem.

Lemma 5

For \(|y|=1\), \(y\ne 1\), the kernel equation \(R(x,y)=0\) has exactly one root \(x=X_{0}(y)\) such that \(|X_{0}(y)|<1\). For \(E_{x}=p_{1,0}+p_{1,1}+p_{1,-1}-p_{-1,1}-p_{-1,0}<0\) (i.e., the one-step mean jumps (the drift) from any interior point of \(S_{d}\) with respect to \(x-\)axis), \(X_{0}(1)=1\). Similarly, we can prove that \(R(x,y)=0\) has exactly one root \(y=Y_{0}(x)\), such that \(|Y_{0}(x)|\le 1\), for \(|x|=1\).

Proof

For \(|y|=1\), \(y\ne 1\), the kernel equation \(R(x,y)=0\), or equivalently \(xy=\Psi (x,y)\) has exactly one root \(x=X_{0}(y)\) such that \(|X_{0}(y)|<1\). This is immediately proven by realizing that \(|\Psi (x,y)|<1=|xy|\) and applying Rouché’s theorem. For \(y=1\), \(R(x,1)=0\) implies \((1-x)[x(p_{1,0}+p_{1,1}+p_{1,-1})-(p_{-1,1}+p_{-1,0})]\). Thus, in the case \(E_{x}:=p_{1,0}+p_{1,1}+p_{1,-1}-p_{-1,1}-p_{-1,0}<0\), \(X_{0}(1)=1\). Similarly, we can prove that \(R(x,y)=0\) has exactly one root \(y=Y_{0}(x)\), \(|Y_{0}(x)|\le 1\), for \(|x|=1\). For an alternative derivation see [35, Lemma 5.3.1]. \(\square \)

The next step is to identify the location of the branch points of the two-valued function \(Y_{\pm }(x)=\frac{-{\widehat{b}}(x)\pm \sqrt{D_{Y}(x)}}{2{\widehat{a}}(x)}\) (resp. \(X_{\pm }(y)=\frac{-b(y)\pm \sqrt{D_{X}(y)}}{2a(y)}\)) defined by \(R(x,Y(x))=0\) (resp. \(R(X(y),y)=0\)), where \(D_{Y}(x)={\widehat{b}}(x)^{2}-4{\widehat{a}}(x){\widehat{c}}(x)\) (resp. \(D_{X}(y)=b(y)^{2}-4a(y)c(y)\)). The branch points of \(Y_{\pm }(x)\) (resp. \(X_{\pm }(y)\)) are the roots of \(D_{Y}(x)=0\) (resp. \(D_{X}(y)=0\)).

Lemma 6

The algebraic function Y(x), defined by \(R(x,Y(x)) = 0\), has four real branch points, say \(x_{1},x_{2},x_{3},x_{4}\), such that \(x_{1},x_{2}\) are positive and lie inside the unit disc, and \(x_{3},x_{4}\) lie outside the unit disc. In particular, for \(x_{3}\), \(x_{4}\), the following classification holds:

  1. (1)

    If \(p_{1,0}>2\sqrt{p_{1,1}p_{1,-1}}\), then \(x_{3}\), \(x_{4}\) are positive,

  2. (2)

    If \(p_{1,0}=2\sqrt{p_{1,1}p_{1,-1}}\), then one is positive and the other is infinite,

  3. (3)

    If \(p_{1,0}<2\sqrt{p_{1,1}p_{1,-1}}\), then one is positive and the other is negative.

Moreover, \(D_{Y}(x)<0\), \(x\in (x_{1},x_{2})\cup (x_{3},x_{4})\). Similar arguments hold for X(y).

Proof

See Lemma 2.3.8, pp. 27–28, [33]. \(\square \)

To ensure the continuity of the two-valued function Y(x) (resp. X(y)), we consider the following cut planes: , , where \({\mathbb {C}}_{x}\), \({\mathbb {C}}_{y}\) are the complex planes of x, y, respectively. For (resp. ), denote by \(Y_{0}(x)\) (resp. \(X_{0}(y)\)) the zero of R(xY(x)) (resp. R(X(y), y))) with the smallest modulus, and by \(Y_{1}(x)\) (resp. \(X_{1}(y)\)) the other one. Define also the image contours, \({\mathcal {L}}=Y_{0}([\overrightarrow{\underleftarrow{x_{1},x_{2}}}])={\bar{Y}}_{1}([\overleftarrow{\underrightarrow{x_{1},x_{2}}}])\), \({\mathcal {M}}=X_{0}([\overrightarrow{\underleftarrow{y_{1},y_{2}}}])={\bar{X}}([\overleftarrow{\underrightarrow{y_{1},y_{2}}}])\), where \([\overrightarrow{\underleftarrow{u,v}}]\) stands for the contour (see Appendix C) traversed from u to v along the upper edge of the slit [uv] and then back to u along the lower edge of the slit. The following lemma provides an exact characterization for the contours \({\mathcal {L}}\), \({\mathcal {M}}\), respectively.

Lemma 7

  1. 1.

    For \(y\in [y_{1},y_{2}]\), the algebraic function X(y) lies on a closed contour \({\mathcal {M}}\), which is symmetric with respect to the real line and written as a function of Re(x), i.e.,

    $$\begin{aligned} \begin{array}{l} |x|^{2}=m(Re(x)),\,|x|^{2}\le \frac{c(y_{2})}{a(y_{2})}. \end{array} \end{aligned}$$

    Set \(\beta _{0}:=\sqrt{\frac{c(y_{2})}{a(y_{2})}}\), \(\beta _{1}=-\sqrt{\frac{c(y_{1})}{a(y_{1})}}\), the extreme right and left points of \({\mathcal {M}}\), respectively.

  2. 2.

    For \(x\in [x_{1},x_{2}]\), the algebraic function Y(x) lies on a closed contour \({\mathcal {L}}\), which is symmetric with respect to the real line and written as a function of Re(y) as

    $$\begin{aligned} \begin{array}{l} |y|^{2}=v(Re(y)),\,|y|^{2}\le \frac{{\widehat{c}}(x_{2})}{{\widehat{a}}(x_{2})}. \end{array} \end{aligned}$$

    Set \(\eta _{0}:=\sqrt{\frac{{\widehat{c}}(x_{2})}{{\widehat{a}}(x_{2})}}\), \(\eta _{1}=-\sqrt{\frac{{\widehat{c}}(x_{1})}{{\widehat{a}}(x_{1})}}\), the extreme right and left points of \({\mathcal {L}}\), respectively.

Proof

We will prove the part related to \({\mathcal {L}}\). Similarly, we can also prove part 1. For \(x\in (x_{1},x_{2})\), \(D_{x}(x)={\widehat{b}}^{2}(x)-4{\widehat{a}}(x){\widehat{c}}(x)\) is negative, so \(X_{0}(y)\) and \(X_{1}(y)\) are complex conjugates. Thus, \(|Y(x)|^{2}=\frac{{\widehat{c}}(x)}{{\widehat{a}}(x)}=k(x)\), and together with

$$\begin{aligned} \begin{array}{c} \frac{d}{dx}k(x)=\frac{x^{2}(p_{0,1}p_{1,-1}-p_{1,1}p_{0,-1})+2p_{1,-1}p_{-1,1}x+p_{-1,1}p_{0,-1}}{{\widehat{a}}(x)^{2}} \end{array} \end{aligned}$$

being a non-negative function for \(x\in (0,\infty )\) implies that \(k(x)\le k(x_{2})\). We can further solve \(|Y(x)|^2 = {\widehat{c}}(x)/{\widehat{a}}(x)\) as a function of x, and denote the solution that lies within \([x_1,x_2]\) by

$$\begin{aligned} \begin{array}{l} {\tilde{x}}(y)=\frac{p_{0,-1}-p_{0,1}|y|^{2}-\sqrt{(p_{0,1}|y|^{2}-p_{0,-1})^{2}-4p_{-1,1}|y|^{2}(p_{1,1}|y|^{2}-p_{1,-1})}}{2(p_{1,1}|y|^{2}-p_{1,-1})}. \end{array} \end{aligned}$$

So \({\tilde{x}}(y)\) is in fact the one-valued inverse function of Y(x). For each \(y\in {\mathcal {L}}\) it also follows that

$$\begin{aligned} \begin{array}{l} Re(Y(x))=\frac{-{\widehat{b}}({\tilde{x}}(y))}{2{\widehat{a}}({\tilde{x}}(y))}. \end{array} \end{aligned}$$
(36)

Solving (36) as a function of \(|Y(x)|^{2}\) gives an expression for \(|Y(x)|^{2}\) in terms of Re(y). \(\square \)

1.2 Formulation and solution of a Riemann–Hilbert boundary value problem

We are ready to proceed with the solution of (11). Note that g(xy) is well-defined for \((x, y) = (X_{0}(y), y)\) with \(|y| = 1\), since (i) g(xy) is well-defined for \(|x|\le 1\), \(|y|\le 1\), (ii) \(X_{0}(y)\) is continuous for \(|y| =1\) (note that from Lemma 5 it is known that \(X_{0}(y)\) is analytic in \({\mathbb {C}}_{y} - [y_{1}, y_{2}]\) and that \(0< y_{1}< y_{2} < 1\) so that \(X_{0}(y)\) is continuous for \(|y| = 1\)), (iii) \(|X_{0}(y)| \le 1\) for \(|y|=1\). Thus, the left-hand side of (11) must vanish for all pairs \((X_{0}(y), y)\):

$$\begin{aligned} \begin{array}{c} A(X_{0}(y),y)g_{0}(X_{0}(y))+B(X_{0}(y),y)h_{0}(y)+C(X_{0}(y),y)=0,\,|y|=1. \end{array} \end{aligned}$$
(37)

Let \(y\in {\mathcal {D}}_{y}=\{y\in {\mathbb {C}}_{y}:|y|\le 1,|X_{0}(y)|\le 1\}\). For \(y\in {\mathcal {D}}_{y}-[y_{1},y_{2}]\) both \(g(X_{0}(y))\), \(h_{0}(y)\) are analytic and the left-hand side of (37) can be analytically continued (Appendix C) up to the slit \([y_{1},y_{2}]\) (and similarly for the functions \(g_{n_{2}}(\cdot )\), \(n_{2}=1,\ldots ,N_{2}\), and \(h_{n_{1}}(\cdot )\), \(n_{1}=1,\ldots ,N_{1}\)). For convenience, assume that \(B(X_{0}(y),y)\ne 0\) for any \(y\in [y_{1},y_{2}]\). Therefore, we can divide by \(B(X_{0}(y),y)\) and (37) is written as

$$\begin{aligned} \begin{array}{c} \frac{A(X_{0}(y),y)}{B(X_{0}(y),y)}g_{0}(X_{0}(y))=-h_{0}(y)-\frac{C(X_{0}(y),y)}{B(X_{0}(y),y)},\,y\in [y_{1},y_{2}]. \end{array} \end{aligned}$$
(38)

For \(y\in [y_{1},y_{2}]\), \(X_{0}(y)=x\in {\mathcal {M}}\), so that \(Y_{0}(X_{0}(y))=y\). Thus, (38) can be written as

$$\begin{aligned} \begin{array}{c} \frac{A(x,Y_{0}(x))}{B(x,Y_{0}(x))}g_{0}(x)=-h_{0}(Y_{0}(x))-\frac{C(x,Y_{0}(x))}{B(x,Y_{0}(x))},\,x\in {\mathcal {M}}. \end{array} \end{aligned}$$
(39)

Note that \(g_{0}(x)\) is holomorphic in \(D_{x}=\{x\in {\mathbb {C}}:|x|<1\}\), and continuous in \({\bar{D}}_{x}=\{x\in {\mathbb {C}}:|x|\le 1\}\). However, \(g_{0}(x)\) may have poles in \(S_{x}={\mathcal {M}}^{+}\cap {\bar{D}}_{x}^{c}\), where \({\bar{D}}_{x}^{c}=\{x\in {\mathbb {C}}:|x|>1\}\). These poles (if they exist) coincide with the zeros of \(A(x,Y_{0}(x))\) in \(S_{x}\).

Taking into account the (possible) poles of \(g_{0}(x)\) (say, \(\xi _{1}\),...,\(\xi _{k})\), and noticing that \(h_{0}(Y_{0}(x))\) is real for \(x\in {\mathcal {M}}\) we conclude,

$$\begin{aligned} \begin{array}{c} Re(iU(x)f(x))=w(x),\,x\in {\mathcal {M}}, \end{array} \end{aligned}$$
(40)

where

$$\begin{aligned} U(x)=\frac{A(x,Y_{0}(x))}{\prod _{i=1}^{k}(x-\xi _{i})B(x,Y_{0}(x))},f(x)=\prod _{i=1}^{k}(x-\xi _{i})g_{0}(x),w(x)=Im\left( \frac{C(x,Y_{0}(x))}{B(x,Y_{0}(x))}\right) . \end{aligned}$$

In order to solve (40), we must first conformally transform it from \({\mathcal {M}}\) to the unit circle \({\mathcal {C}}\). Let the mapping, \(z=\gamma (x):{\mathcal {M}}^{+}\rightarrow {\mathcal {C}}^{+}\), and its inverse \(x=\gamma _{0}(z):{\mathcal {C}}^{+}\rightarrow {\mathcal {M}}^{+}\). Then, we have the following problem: Find a function \({\tilde{T}}(z)=f(\gamma _{0}(z))\) regular for \(z\in {\mathcal {C}}^{+}\), and continuous for \(z\in {\mathcal {C}}\cup {\mathcal {C}}^{+}\) such that

$$\begin{aligned} Re(iU(\gamma _{0}(z)){\tilde{T}}(z))=w(\gamma _{0}(z)),\,z\in {\mathcal {C}}. \end{aligned}$$
(41)

To obtain the conformal mappings, we need to represent \({\mathcal {M}}\) in polar coordinates, i.e., \({\mathcal {M}}=\{x:x=\rho (\phi )\exp (i\phi ),\phi \in [0,2\pi ]\}.\) This procedure is described in detail in [23]. We briefly summarized the basic steps: Since \(0\in {\mathcal {M}}^{+}\), for each \(x\in {\mathcal {M}}\), a relation between its absolute value and its real part is given by \(|x|^{2}=m(Re(x))\) (see Lemma 6). Given the angle \(\phi \) of some point on \({\mathcal {M}}\), the real part of this point, say \(\delta (\phi )\), is the solution of \(\delta -\cos (\phi )\sqrt{m(\delta )}\), \(\phi \in [0,2\pi ].\) Since \({\mathcal {M}}\) is a smooth, egg-shaped contour, the solution is unique. Clearly, \(\rho (\phi )=\frac{\delta (\phi )}{\cos (\phi )}\), and the parametrization of \({\mathcal {M}}\) in polar coordinates is fully specified. Then, the mapping from \(z\in {\mathcal {C}}^{+}\) to \(x\in {\mathcal {M}}^{+}\), where \(z = e^{i\phi }\) and \(x= \rho ({\tilde{\psi }}(\phi ))e^{i{\tilde{\psi }}(\phi )}\), satisfying \(\gamma _{0}(0)=0\) and \(\gamma _{0}(z)=\overline{\gamma _{0}({\overline{z}})},\) is uniquely determined by (see [23], Section I.4.4)

$$\begin{aligned} \begin{array}{rl} \gamma _{0}(z)=&{}z\exp [\frac{1}{2\pi }\int _{0}^{2\pi }\log \{\rho ({\tilde{\psi }}(\omega ))\}\frac{e^{i\omega }+z}{e^{i\omega }-z}{\text {d}}\omega ],\,|z|<1,\\ {\tilde{\psi }}(\phi )=&{}\phi -\int _{0}^{2\pi }\log \{\rho ({\tilde{\psi }}(\omega ))\}\cot (\frac{\omega -\phi }{2}){\text {d}}\omega ,\,0\le \phi \le 2\pi , \end{array} \end{aligned}$$
(42)

i.e., \({\tilde{\psi }}(.)\) is uniquely determined as the solution of a Theodorsen integral equation with \({\tilde{\psi }}(\phi )=2\pi -{\tilde{\psi }}(2\pi -\phi )\). Due to the correspondence-boundaries theorem (see Appendix C), \(\gamma _{0}(z)\) is continuous in \({\mathcal {C}}\cup {\mathcal {C}}^{+}\).

The solution of the boundary value problem depends on its index \(\chi =\frac{-1}{\pi }[arg\{U(x)\}]_{x\in {\mathcal {M}}}\) (see Appendix C). If \(\chi \le 0\), our problem has at most one linearly independent solution. The solution of the problem defined in (40) is given for \(z\in {\mathcal {C}}_{x}^{+}\) by

$$\begin{aligned} \begin{array}{l} g_{0}(\gamma _{0}(z))=\prod _{i=1}^{k}(\gamma _{0}(z)-\xi _{i})^{-1}e^{i\sigma (z)}z^{\chi }[iK+\frac{1}{2\pi i}\int _{|t|=1}e^{\omega _{1}(t)}\delta (t)\frac{t+z}{t-z}\frac{{\text {d}}t}{t}], \end{array} \end{aligned}$$
(43)

where K is a constant to be determined, \(\omega _{1}(z)=Im(\sigma (z))\), \(\delta (z)=\frac{w(\gamma _{0}(z))}{|U(\gamma _{0}(z))|}\), \(U(\gamma _{0}(z))=b_{1}(z)+ia_{1}(z)\) and

$$\begin{aligned} \begin{array}{rl} \sigma (z)=&\frac{1}{2\pi i}\int _{|t|=1}(\arctan \frac{b_{1}(t)}{a_{1}(t)}-\chi \arg t)\frac{t+z}{t-z}\frac{{\text {d}}t}{t}. \end{array} \end{aligned}$$

Note that \(g_{0}(x)=g_{0}(\gamma _{0}(\gamma (x)))\). When \(\chi =0\), K can be determined from the solution to \(g_{0}(0)\). If \(\chi <0\), then \(K=0\) and a solution exists if [38]

$$\begin{aligned} \frac{1}{2\pi i}\int _{|t|=1}e^{\omega _{1}(t)}\delta (t)t^{-k-1}{\text {d}}t=0, \end{aligned}$$
(44)

for \(k=0,1,\ldots ,-\chi -1.\) Similarly, we can obtain \(h_{0}(y)\), by solving another Riemann–Hilbert problem on \({\mathcal {L}}\). Having obtained \(g_{0}(x)\) and \(h_{0}(y)\), we can obtain g(xy) from (11) in terms of the unknown probabilities \(\pi (N_{1},n_{2})\), \(n_{2}=0,\ldots ,N_{2}-1\), \(\pi (n_{1},N_{2})\), \(n_{1}=0,\ldots ,N_{1}-1\), \(\pi (N_{1},N_{2})\), as we did in Sect. 3.3.

The following steps summarize the way we fully determine the stationary distribution:

  1. (1)

    The \(N_{1}\times N_{2}\) equations for \(S_{a}\) involve \((N_{1}+1)\times (N_{2}+1)\) unknowns: \(\pi (n_{1},n_{2})\) for \(n_{1}=0,1,\ldots ,N_{1}\), \(n_{2}=0,1,\ldots ,N_{2}\), excluding \(\pi (N_{1},N_{2})\). Thus, we further need \(N_{1}+N_{2}\) equations that involve the unknowns \(\pi (N_{1},n_{2})\), \(n_{2}=0,1,\ldots ,N_{2}-1\), and \(\pi (n_{1},N_{2})\), \(n_{1}=0,1,\ldots ,N_{1}-1\).

  2. (2)

    As in Sect. 3, we have expressed the generating functions of equilibrium probabilities that are associated with the states of subregions \(S_{b}\), \(S_{c}\) in terms of \(g_{0}(x)\), \(h_{0}(y)\), which are obtained in terms of the solution of a Riemann–Hilbert boundary value problem. That solution obtains \(g_{0}(x)\), \(h_{0}(y)\) in terms of A(xy), B(xy) and C(xy). The first two are known, and the third one contains the \(N_{1}+N_{2}+1\) unknown probabilities. The \(N_{1}+N_{2}\) probabilities mentioned in point 1 and \(\pi (N_{1},N_{2})\). These additional equations are derived from the (integral) expressions of \(g_{0}(x)\), \(h_{0}(y)\) as follows at steps 3 and 4.

  3. (3)

    The solution of (5), (8) is similar to those in Sect. 3, but now the coefficients are

    $$\begin{aligned} \begin{array}{cccc} e_{n_{2}}(x)=\frac{q_{n_{2}}(x)}{x^{n_{2}}},&t_{n_{2}}(x)=\frac{l_{n_{2}}(x)}{x^{n_{2}}},&{\tilde{e}}_{n_{1}}(y)=\frac{{\tilde{q}}_{n_{1}}(y)}{y^{n_{1}}},&{\tilde{t}}_{n_{1}}(y)=\frac{{\tilde{l}}_{n_{1}}(y)}{y^{n_{1}}}, \end{array} \end{aligned}$$

    where \(q_{n_{2}}(x)\), \(l_{n_{2}}(x)\), \({\tilde{q}}_{n_{1}}(y)\), \({\tilde{l}}_{n_{1}}(y)\) are polynomials. Therefore, to derive additional equations for the unknown probabilities in point 1, we use (6), (9) and take the derivatives of \(g_{0}(x)\), \(h_{0}(y)\) at point 0, i.e.,

    $$\begin{aligned} \begin{array}{rl} n_{2}!\pi (N_{1},n_{2})=&{}\frac{{\text {d}}^{n_{2}}}{{\text {d}}x^{n_{2}}}[e_{n_{2}}(x)g_{0}(x)+t_{n_{2}}(x)]|_{x=0},\,n_{2}=1,\ldots ,N_{2},\\ n_{1}!\pi (n_{1},N_{2})=&{}\frac{{\text {d}}^{n_{1}}}{{\text {d}}y^{n_{1}}}[{\tilde{e}}_{n_{1}}(y)h_{0}(y)+{\tilde{t}}_{n_{1}}(y)]|_{y=0},\,n_{1}=1,\ldots ,N_{1}.\end{array} \end{aligned}$$
    (45)

    This procedure will provide the \(N_{1}+N_{2}\) equations referred to at step 1.

  4. (4)

    The last additional equation for the determination of the last unknown \(\pi (N_{1},N_{2})\) is found by the use of the normalization equation:

    $$\begin{aligned} 1=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}\pi (n_{1},n_{2})+\sum _{n_{1}=0}^{N_{1}-1}h_{n_{1}}(1)+\sum _{n_{2}=0}^{N_{2}-1}g_{n_{2}}(1)+g(1,1). \end{aligned}$$
    (46)

Having obtained the equilibrium probabilities associated with subregion \(S_{a}\), and the pgfs reffering to the equilibrium probabilities in subregions \(S_{i}\), \(i=b,c,d\), we can obtain useful performance metrics; i.e., the expression (29) in Sect. 3.4 remains valid in this section. The only difference relies on the way we solved (11), by following an analysis in [33].

Overview of definitions and theorems used in this paper

Definition 1

([58]) A function \(f:{\mathbb {C}}\rightarrow {\mathbb {C}}\) is called regular at a point \(z_{0}\in {\mathbb {C}}\) if there exists a neighbourhood of \(z_{0}\) in every point of which f is complex differentiable; f is called regular (equivalently analytic of holomorphic) in a domain D if it is regular at every point of D.

Definition 2

([58]) A complex function \(z=f(t)\) of a real variable, defined on a closed interval \(a\le t\le b\) is said to define a (continuous) curve. If the same point z corresponds to more than one parameter value in the half-open interval \(a\le t<b\), we say that z is a multiple point of the curve \(z=f(t)\), \(a\le t\le b\). A curve with no multiple points is called a Jordan curve. A closed curve is called a contour. A continuous curve \({\mathcal {L}}\) is called a smooth curve if among its various parametric representations there is at least one, i.e., \(z=f(t):=\gamma (t)+i\delta (t)\), such that f(t) has a continuous non-vanishing derivative \(f^{\prime }(t)\) at every point of the interval [ab].

Definition 3

([38]) Consider a function G, defined on a contour \({\mathcal {L}}\). The increment of the argument of G(t) as t traverses \({\mathcal {L}}\) once in the positive direction, divided by \(2\pi \), is called the index of G(.) on \({\mathcal {L}}\).

Definition 4

([60]) A continuous function f is said to be univalent in the domain D if \(z_{1}\ne z_{2}\) implies \(f(z_{1})\ne f(z_{2})\) for \(z_{1},z_{2}\in D\). A regular function that is univalent is called a conformal mapping. For more information see [60, 58, Section I.8].

Definition 5

Let D a domain, \(E\subset D\), and f(z) a function defined on E. A function F(z) which is regular in the domain D and coincides with f(z) on the set E is called an analytic continuation of f(z) into the domain D. Analytic continuation deals with the problem of properly redefining an analytic function so as to extend its domain of analyticity.

Theorem 5

(The corresponding boundaries theorem [60]) Denote by \({\mathcal {L}}^{+}\) the interior of a contour \({\mathcal {L}}\). If \({\mathcal {L}}_{1}^{+}\), \({\mathcal {L}}_{2}^{+}\) are two domains bounded by smooth contours, then the conformal mapping \(\omega : {\mathcal {L}}_{1}^{+}\rightarrow {\mathcal {L}}_{2}^{+}\) is continuous in \({\mathcal {L}}_{1}^{+}\cup {\mathcal {L}}_{1}\) and establishes a one-to-one correspondence among the points of \({\mathcal {L}}_{1}\) and \({\mathcal {L}}_{2}\).

Theorem 6

(The principle of corresponding boundaries [23, Section I.4.2, p. 67]) Let \({\mathcal {L}}_{1}^{+}\), \({\mathcal {L}}_{2}^{+}\) be two domains bounded by piecewise smooth contours \({\mathcal {L}}_{1}\), \({\mathcal {L}}_{2}\). If f(z) is regular in \({\mathcal {L}}_{1}^{+}\) and continuous in \({\mathcal {L}}_{1}^{+}\cup {\mathcal {L}}_{1}\), and maps \({\mathcal {L}}_{1}\) one-to-one onto \({\mathcal {L}}_{2}\), then f(z) is univalent in \({\mathcal {L}}_{1}^{+}\cup {\mathcal {L}}_{1}\). If f(z) preserves the positive directions on \({\mathcal {L}}_{1}\) and \({\mathcal {L}}_{2}\), then f(z) maps \({\mathcal {L}}_{1}^{+}\) conformally onto \({\mathcal {L}}_{2}^{+}\), otherwise onto \({\mathcal {L}}_{2}^{-}\).

Proof of Theorem 2

It is readily seen that \(\Psi (gs,gs^{-1})\) is for every fixed \(|s|=1\) regular in \(|g|<1\), continuous in \(|g|\le 1\), and, for \(|g|=1\)

$$\begin{aligned} |\Psi (gs,gs^{-1})|\le 1=|g^{2}|, \end{aligned}$$

and the proof of the first statement is a straightforward application of Rouché’s theorem. Moreover, for \(s=1\), it is readily seen that \(g(1)=1\) is a zero of multiplicity one provided that \(E_{x}+E_{y}<0\). Similarly, for \(s=-1\), \(g(-1)=-1\) is also a simple zero if \(E_{x}+E_{y}<0\). The second statement follows from (14) when we first sum in the denominator the terms for the indices i, j for which \(i+j\) and \(i-j\) are both even, and then when both are odd; for more details see also [23, Lemma 2.1, II.3.2, p. 153], and [25, Theorem II.2.2.1, p. 65].\(\square \)

Proof of Lemma 2

For assertion 1, it is readily seen that \(R(gs,gs^{-1})=0\) is written as

$$\begin{aligned}&g^{2}=g^{2}p_{0,0}+g^{3}sp_{1,0}+gs^{-1}p_{-1,0}+g^{4}p_{1,1}+g^{3}s^{-1}p_{0,1}\nonumber \\&\quad +g^{2}s^{-2}p_{-1,1}+gsp_{0,-1}+g^{2}s^{2}p_{1,-1}. \end{aligned}$$
(47)

It is clear that \(g:=g(s)=0\) is a zero of \(R(gs,gs^{-1})\). For \(|s|=1\), \(0<|g|\le 1\), (47) is reduced to \(m_{1}(g)=m_{2}(g,s)\), where \(m_{1}(g)=g\), \(m_{2}(g,s)=gp_{0,0}+g^{2}sp_{1,0}+s^{-1}p_{-1,0}+g^{3}p_{1,1}+g^{2}s^{-1}p_{0,1}+gs^{-2}p_{-1,1}+sp_{0,-1}+gs^{2}p_{1,-1}.\) We now show that if \(E_{x}+E_{y}<0\), and for fixed s with \(|s|=1\), \(s\ne \pm 1\), \(m_{1}(g)=m_{2}(g,s)\) has a unique root in \(|g|\le 1\).

Clearly, \(m_{1}(g)\) has a single zero in the complex unit disk, and we wish to establish that \(|m_{1}(g)|>|m_{2}(g,s)|\) for \(|g|=1\), \(|s|=1\), \(s\ne \pm 1\). Note that \(|m_{1}(g)|=|g|=m_{1}(|g|)\), and \(|m_{2}(g,s)|\le m_{2}(|g|,1)\). Thus, we require \(m_{1}(|g|)>m_{2}(|g|,1)\) for \(|g|=1\). However, for \(|g|=1\), \(m_{1}(|g|)=m_{1}(1)=1=m_{2}(1,1)=m_{2}(|g|,1)\). To resolve this issue, we evaluate \(m_{1}(|g|)\), \(m_{2}(|g|,1)\) on the circle \(|g|=1+\epsilon \) with \(\epsilon \) small and positive. To accomplish this task, we use the Taylor expansion \(m_{2}(1+\epsilon ,1)=m_{2}(1,1)+\epsilon m_{2}^{\prime }(1,1)+o(\epsilon )\) (where \(m_{2}^{\prime }(1,1)=\frac{d}{d|g|}m_{2}(|g|,1)|_{|g|=1}\)), and similarly for \(m_{1}(1+\epsilon )\). So we need to show that \(m_{1}(1+\epsilon )>m_{2}(1+\epsilon ,1)\). Since \(m_{1}(1)=m_{2}(1,1)\) we are left to show that \(m_{1}^{\prime }(1)>m_{2}^{\prime }(1,1)\), or equivalently

$$\begin{aligned} p_{-1,0}+p_{0,-1}>p_{1,0}+2p_{1,1}+p_{0,1}. \end{aligned}$$
(48)

It is easy to see that (48) is related to the mean drifts in region \(S_{d}\), and it is always true as \(E_{x}+E_{y}<0\). So, for sufficiently small \(\epsilon >0\) we have that \(m_{1}(|g|)>m_{2}(|g|,1)\) for \(|g|\in (1,1+\epsilon ]\), which proves assertion 1. If \(s=1\), then \(g(1)=1\) is the only root of \(m_{1}(g)=m_{2}(g,1)\) in \(|g|\le 1\), and that \(m_{2}^{\prime }(1,1)<1\), i.e., that \(E_{x}+E_{y}<0\), implies that this is a simple root. Similarly, for \(s=-1\), \(g(-1)=-1\) is the unique root of \(m_{1}(g)=m_{2}(g,1)\) in \(|g|\le 1\) with multiplicity 1 if \(E_{x}+E_{y}<0\). Equation (30) follows directly from (14). Assertion 2 also follows from (30); see also [23, Lemma 10.1, II.3.10].\(\square \)

On the induced Markov chains in Sect. 4.2

For \(Q_{1,n}>N_{1}\), \(Q_{2,n}\) evolves as a one-dimensional RW with one step transition probabilities \(w^{(2)}_{j}(n_{2})=P(Q_{2,n+1}=n_{2}+j|Q_{2,n}=n_{2})\), \(j=0,\pm 1\), for \(n_{2}=0,1,\ldots ,\) given by

$$\begin{aligned} w^{(2)}_{1}(n_{2})= & {} p_{0,1}(N_{1},n_{2})+p_{1,1}(N_{1},n_{2})+p_{-1,1}(N_{1},n_{2})\\&=\lambda _{2}(N_{1},n_{2})[1-{\bar{a}}_{1}(N_{1},n_{2})a_{2}(N_{1},n_{2})],\\ w^{(2)}_{-1}(n_{2})= & {} p_{1,-1}(N_{1},n_{2})+p_{0,-1}(N_{1},n_{2})={\bar{\lambda }}_{2}(N_{1},n_{2}){\bar{a}}_{1}(N_{1},n_{2})a_{2}(N_{1},n_{2}),\\ w^{(2)}_{0}(n_{2})= & {} p_{1,0}(N_{1},n_{2})+p_{-1,1}(N_{1},n_{2})+p_{0,0}(N_{1},n_{2}). \end{aligned}$$

Note that for \(n_{2}\ge N_{2}\), \(w^{(2)}_{j}(n_{2}):=w^{(2)}_{j}\), \(j=0,\pm 1\). Recall \(\psi :=(\psi _{0},\psi _{1},\ldots )\), its stationary distribution. Then, simple calculations yield

$$\begin{aligned} \begin{array}{rl} \psi _{n_{2}}=&{}\psi _{0}\prod _{j=0}^{n_{2}-1}\frac{w^{(2)}_{1}(j)}{w^{(2)}_{-1}(j+1)},\,1\le j\le N_{2},\\ \psi _{n_{2}}=&{}\psi _{N_{2}}\left( \frac{w^{(2)}_{1}}{w^{(2)}_{-1}}\right) ^{n_{2}-N_{2}},\,j\ge N_{2}+1,\\ \psi _{0}=&{}\left[ 1+\sum _{n_{2}=1}^{N_{2}-1}\prod _{j=0}^{n_{2}-1}\frac{w^{(2)}_{1}(j)}{w^{(2)}_{-1}(j+1)}+\frac{w^{(2)}_{-1}}{w^{(2)}_{-1}-w^{(2)}(1)}\prod _{j=0}^{N_{2}-1}\frac{w^{(2)}_{1}(j)}{w^{(2)}_{-1}(j+1)}\right] ^{-1}, \end{array} \end{aligned}$$

where \(w^{(2)}_{-1}-w^{(2)}(1)={\bar{a}}_{1}a_{2}-\lambda _{2}<0\). Similar argumentation follows for the component \(Q_{1,n}\), which evolves as a one-dimensional RW for \(Q_{2,n}>N_{2}\).

One-step transition probabilities for the model in Sect. 4.1

The one-step transition probabilities for the model in Sect. 4.1 are given below:

$$\begin{aligned} p_{-2,0}({\underline{n}})= & {} \theta _{1}(n_{1}){\bar{\theta }}_{2}(n_{2})\beta _{1}({\underline{n}}-{\underline{1}}_{1})\mu _{1}(n_{1}-1)d_{0,0}({\underline{n}})\\&+\theta _{1}(n_{1})\theta _{2}(n_{2})\beta _{1}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2})\mu _{1}(n_{1}-1)d_{0,1}({\underline{n}}),\\ p_{0,-2}({\underline{n}})= & {} \theta _{2}(n_{2}){\bar{\theta }}_{1}(n_{1})\beta _{2}({\underline{n}}-{\underline{1}}_{2})\mu _{2}(n_{2}-1)d_{0,0}({\underline{n}})\\&+\theta _{1}(n_{1})\theta _{2}(n_{2})\beta _{2}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2})\mu _{2}(n_{2}-1)d_{1,0}({\underline{n}}),\\ p_{-2,1}({\underline{n}})= & {} \theta _{1}(n_{1}){\bar{\theta }}_{2}(n_{2})\beta _{1}({\underline{n}}-{\underline{1}}_{1})\mu _{1}(n_{1}-1)d_{0,1}({\underline{n}}),\\ p_{1,-2}({\underline{n}})= & {} \theta _{2}(n_{2}){\bar{\theta }}_{1}(n_{1})\beta _{2}({\underline{n}}-{\underline{1}}_{2})\mu _{2}(n_{2}-1)d_{1,0}({\underline{n}}),\\ p_{-2,-1}({\underline{n}})= & {} \theta _{1}(n_{1})\theta _{2}(n_{2})\beta _{1}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2})\mu _{1}(n_{1}-1)d_{0,0}({\underline{n}}),\\ p_{-1,-2}({\underline{n}})= & {} \theta _{1}(n_{1})\theta _{2}(n_{2})\beta _{2}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2})\mu _{2}(n_{2}-1)d_{0,0}({\underline{n}}),\\ p_{1,-1}({\underline{n}})= & {} \theta _{2}(n_{2}){\bar{\theta }}_{1}(n_{1})[\beta _{2}({\underline{n}}-{\underline{1}}_{2})\mu _{2}(n_{2}-1)d_{1,1}({\underline{n}})+(\beta _{1}({\underline{n}}-{\underline{1}}_{2})\bar{\mu _{1}}(n_{1})\\&+\beta _{2}({\underline{n}}-{\underline{1}}_{2}){\bar{\mu }}_{2}(n_{2}-1))d_{1,0}({\underline{n}})] +{\bar{\theta }}_{1}(n_{1}){\bar{\theta }}_{2}(n_{2})\beta _{2}({\underline{n}})\mu _{2}(n_{2})d_{1,0}({\underline{n}}),\\ p_{1,1}({\underline{n}})= & {} {\bar{\theta }}_{1}(n_{1}){\bar{\theta }}_{2}(n_{2})(\beta _{1}({\underline{n}})\bar{\mu _{1}}(n_{1})+\beta _{2}({\underline{n}}){\bar{\mu }}_{2}(n_{2}))d_{1,1}({\underline{n}}),\\ p_{-1,-1}({\underline{n}})= & {} \theta _{2}(n_{2})\theta _{1}(n_{1})\left[ \beta _{1}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2})\mu _{1}(n_{1}-1)d_{1,0}({\underline{n}})\right. \\&\left. +\beta _{2}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2})\mu _{2}(n_{2}-1)d_{0,1}({\underline{n}})\right. \\&\left. +(\beta _{1}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2})\bar{\mu _{1}}(n_{1}-1)+\beta _{2}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2}){\bar{\mu }}_{2}(n_{2})-1)d_{0,0}({\underline{n}})\right] \\&+\theta _{1}(n_{1}){\bar{\theta }}_{2}(n_{2})\beta _{2}({\underline{n}}-{\underline{1}}_{1})\mu _{2}(n_{2})d_{0,0}({\underline{n}})\\&+\theta _{2}(n_{2}){\bar{\theta }}_{1}(n_{1})\beta _{1}({\underline{n}}-{\underline{1}}_{2})\mu _{1}(n_{1})d_{0,0}({\underline{n}}),\\ p_{-1,0}({\underline{n}})= & {} \theta _{1}(n_{1}){\bar{\theta }}_{2}(n_{2})\left[ \beta _{1}({\underline{n}}-{\underline{1}}_{1})\mu _{1}(n_{1}-1)d_{1,0}({\underline{n}})+\beta _{2}({\underline{n}}-{\underline{1}}_{1})\mu _{2}(n_{2})d_{0,1}({\underline{n}})\right. \\&\left. +(\beta _{1}({\underline{n}}-{\underline{1}}_{1})\bar{\mu _{1}}(n_{1}-1)+\beta _{2}({\underline{n}}-{\underline{1}}_{1})\mu _{2}(n_{2}))d_{0,0}({\underline{n}})\right] \\&+\bar{\theta _{1}}(n_{1}){\bar{\theta }}_{2}(n_{1})\beta _{1}({\underline{n}})\mu _{1}(n_{1})d_{0,0}({\underline{n}})+\theta _{2}(n_{2}){\bar{\theta }}_{1}(n_{1})\beta _{1}({\underline{n}})\mu _{1}(n_{1})d_{0,1}({\underline{n}})\\&+\theta _{1}(n_{1})\theta _{2}(n_{2})[\beta _{1}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2})(\mu _{1}(n_{1}-1)d_{1,1}({\underline{n}})+{\bar{\mu }}_{1}(n_{1}-1)d_{0,1}({\underline{n}}))\\&+\beta _{2}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2}){\bar{\mu }}_{2}(n_{2}-1)d_{0,1}({\underline{n}})],\\ p_{1,0}({\underline{n}})= & {} \theta _{2}(n_{2}){\bar{\theta }}_{1}(n_{1})(\beta _{1}({\underline{n}}-{\underline{1}}_{2})\bar{\mu _{1}}(n_{1})+\beta _{2}({\underline{n}}-{\underline{1}}_{2}){\bar{\mu }}_{2}(n_{2}-1))d_{1,1}({\underline{n}})\\&+{\bar{\theta }}_{1}(n_{1}){\bar{\theta }}_{2}(n_{2})[\beta _{2}({\underline{n}})\mu _{2}(n_{2})d_{1,1}({\underline{n}})\\&+(\beta _{1}({\underline{n}})\bar{\mu _{1}}(n_{1})+\beta _{2}({\underline{n}}){\bar{\mu }}_{2}(n_{2}))d_{1,0}({\underline{n}})],,\\ p_{-1,1}({\underline{n}})= & {} \theta _{1}(n_{1}){\bar{\theta }}_{2}(n_{2})[\beta _{1}({\underline{n}}-{\underline{1}}_{1})\mu _{1}(n_{1}-1)d_{1,1}({\underline{n}})+(\beta _{1}({\underline{n}}-{\underline{1}}_{1})\bar{\mu _{1}}(n_{1}-1)\\&+\beta _{2}({\underline{n}}-{\underline{1}}_{1}){\bar{\mu }}_{2}(n_{2}))d_{0,1}({\underline{n}})]+{\bar{\theta }}_{1}(n_{1}){\bar{\theta }}_{2}(n_{2})\beta _{1}({\underline{n}})\mu _{1}(n_{1})d_{0,1}({\underline{n}}),\\ p_{0,-1}({\underline{n}})= & {} \theta _{2}(n_{2}){\bar{\theta }}_{1}(n_{1})\left[ \beta _{1}({\underline{n}}-{\underline{1}}_{2})\mu _{1}(n_{1})d_{1,0}({\underline{n}})+\beta _{2}({\underline{n}}-{\underline{1}}_{2})\mu _{2}(n_{2}-1)d_{0,1}({\underline{n}})\right. \\&\left. +(\beta _{1}({\underline{n}}-{\underline{1}}_{2})\bar{\mu _{1}}(n_{1})+\beta _{2}({\underline{n}}-{\underline{1}}_{2})\mu _{2}(n_{2}-1))d_{0,0}({\underline{n}})\right] \\&+\bar{\theta _{1}}(n_{1}){\bar{\theta }}_{2}(n_{2})\beta _{2}({\underline{n}})\mu _{2}(n_{2})d_{0,0}({\underline{n}})+\theta _{1}(n_{1}){\bar{\theta }}_{2}(n_{2})\beta _{2}({\underline{n}}-{\underline{1}}_{1})\mu _{2}(n_{2})d_{1,0}({\underline{n}})\\&+\theta _{1}(n_{1})\theta _{2}(n_{2})[\beta _{2}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2})(\mu _{2}(n_{2}-1)d_{1,1}({\underline{n}})+{\bar{\mu }}_{2}(n_{2}-1)d_{1,0}({\underline{n}}))\\&+\beta _{1}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2}){\bar{\mu }}_{1}(n_{1}-1)d_{1,0}({\underline{n}})],\\ p_{0,1}({\underline{n}})= & {} \theta _{1}(n_{1}){\bar{\theta }}_{2}(n_{2})(\beta _{1}({\underline{n}}-{\underline{1}}_{1})\bar{\mu _{1}}(n_{1}-1)+\beta _{2}({\underline{n}}-{\underline{1}}_{1}){\bar{\mu }}_{2}(n_{2}))d_{1,1}({\underline{n}})\\&+{\bar{\theta }}_{1}(n_{1}){\bar{\theta }}_{2}(n_{2})[\beta _{1}({\underline{n}})\mu _{1}(n_{1})d_{1,1}({\underline{n}})\\&+(\beta _{1}({\underline{n}})\bar{\mu _{1}}(n_{1})+\beta _{2}({\underline{n}}){\bar{\mu }}_{2}(n_{2}))d_{0,1}({\underline{n}})],\\ p_{0,0}({\underline{n}})= & {} \theta _{1}(n_{1})\theta _{2}(n_{2})(\beta _{1}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2})\bar{\mu _{1}}(n_{1}-1)\\&+\beta _{2}({\underline{n}}-{\underline{1}}_{1}-{\underline{1}}_{2}){\bar{\mu }}_{2}(n_{2}-1))d_{1,1}({\underline{n}})\\&+\theta _{1}(n_{1}){\bar{\theta }}_{2}(n_{2})[\beta _{2}({\underline{n}}-{\underline{1}}_{1})\mu _{2}(n_{2})d_{1,1}({\underline{n}})+(\beta _{1}({\underline{n}}-{\underline{1}}_{1})\bar{\mu _{1}}(n_{1}-1)\\&+\beta _{2}({\underline{n}}-{\underline{1}}_{1}){\bar{\mu }}_{2}(n_{2}))d_{1,0}({\underline{n}})]\\&+\theta _{2}(n_{2}){\bar{\theta }}_{2}(n_{2})[\beta _{1}({\underline{n}}-{\underline{1}}_{2})\mu _{1}(n_{1})d_{1,1}({\underline{n}})+(\beta _{1}({\underline{n}}-{\underline{1}}_{2})\bar{\mu _{1}}(n_{1})\\&+\beta _{2}({\underline{n}}-{\underline{1}}_{2}){\bar{\mu }}_{2}(n_{2}-1))d_{0,1}({\underline{n}})]\\&+{\bar{\theta }}_{1}(n_{1}){\bar{\theta }}_{2}(n_{2})[\beta _{1}({\underline{n}})\mu _{1}(n_{1})d_{1,0}({\underline{n}})+\beta _{2}({\underline{n}})\mu _{2}(n_{2})d_{0,1}({\underline{n}})\\&+(\beta _{1}({\underline{n}})\bar{\mu _{1}}(n_{1})+\beta _{2}({\underline{n}}){\bar{\mu }}_{2}(n_{2}))d_{0,0}({\underline{n}})], \end{aligned}$$

where \(d_{i,j}({\underline{n}})\), \(i,j=0,1\), are as in (32).

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Dimitriou, I. On partially homogeneous nearest-neighbour random walks in the quarter plane and their application in the analysis of two-dimensional queues with limited state-dependency. Queueing Syst 98, 95–143 (2021). https://doi.org/10.1007/s11134-021-09705-y

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