# Stability of Polling Systems for a Large Class of Markovian Switching Policies

Konstantin Avrachenkov, Kousik Das, Veeraruna Kavitha, Vartika Singh

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Konstantin Avrachenkov, Kousik Das, Veeraruna Kavitha, Vartika Singh. Stability of Polling Systems for a Large Class of Markovian Switching Policies. CDC 2025 - IEEE Conference on Decision and Control, Dec 2025, Rio de Janeiro, Brazil. hal-05458169

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Stability of Polling Systems for a Large Class of Markovian Switching Policies

Konstantin Avrachenkov<fnref n="1" />, Kousik Das<fnref n="2" />, Veeraruna Kavitha<fnref n="2" /> and Vartika Singh<fnref n="3" />

*Abstract*— We consider a polling system with two queues, where a single server is attending the queues in a cyclic order and requires non-zero switching times to switch between the queues. Our aim is to identify a fairly general and comprehensive class of Markovian switching policies that renders the system stable. Potentially a class of policies that can cover the Pareto frontier related to individual-queue-centric performance measures like the stationary expected number of waiting customers in each queue; for instance, such a class of policies is identified recently for a polling system near the fluid regime (with large arrival and departure rates), and we aim to include that class. We also aim to include a second class that facilitates switching between the queues at the instance the occupancy in the opposite queue crosses a threshold and when that in the visiting queue is below a threshold (this inclusion facilitates design of ‘robust’ polling systems). Towards this, we consider a class of two-phase switching policies, which includes the above mentioned classes. In the maximum generality, our policies can be represented by eight parameters, while two parameters are sufficient to represent the aforementioned classes. We provide simple conditions to identify the sub-class of switching policies that ensure system stability. By numerically tuning the parameters of the proposed class, we illustrate that the proposed class can cover the Pareto frontier for the stationary expected number of customers in the two queues.

# I. INTRODUCTION

Polling systems are a special class of queueing systems where a single server cyclically visits multiple queues, potentially incurring non-zero switching or walking times between the queues (see [1], [2], [3]). These switching times introduce “vacations,” making polling systems non-work-conserving. Polling systems are useful for modeling a variety of applications like, traffic signals, manufacturing systems, logistics, railway crossings etc. (see [4], [5], [6], [7], [8]). The analytical complexity of the polling systems heavily depends on the service discipline adopted (e.g., exhaustive or gated, see [9], [10]). Tools like pseudo-conservation laws have proven to be invaluable in analyzing the total workload of such systems, providing insights into system-wide performance metrics (see [11] and references therein). However, the performance of the individual queues such as stationary average waiting times (or stationary average occupancy levels) remain difficult to be computed.

The authors are listed in alphabetical order. This work is financially supported by DST-Inria CEFIPRA LION Project. The paper is accepted at IEEE CDC 2025.

To obtain individual-queue-centric analysis (like achievable region of all possible stationary average occupancy levels in the individual queues), one needs to consider a large class of state dependent or Markovian switching policies; for example, Kleinrock in [12] consider Markov priority based switching policies in the context of queueing systems without vacations. For such work conserving systems, stability is ensured once the load factor is less than one. However, with non-zero switching times, that is for non-work-conserving systems, stability can further depend upon the parameters of the switching policies. In other words, stability can become sensitive to the switching policies, and thus it is important to identify the subclass (among any given class) of policies that are stable.

To the best of our knowledge, the focus of the polling system based literature is on the system performance. Further, except for a few strands (like [13], [14], [15]), the switching policies considered in the current literature are not exactly Markovian (specifically, the switching decisions do not depend on elaborate details such as queue lengths of the individual queues at the time of decision). For instance, in [1], [4], [6], [7], [10], the authors consider variants of exhaustive, gated, or, globally gated disciplines at individual queues to determine when to switch from the visiting queue; some of them (see [6]) choose the next queue to be visited using more sophisticated rules (instead of traditional cyclic routing). However, they do not consider Markovian switching rules in the strict sense.

The authors in [14] consider Markovian (occupancy-driven) switching rules; however, they restrict to finite buffer systems which are automatically stable, whereas we consider infinite buffer capacity systems. In [13], the authors study Pareto frontier of polling systems under a class of Markovian switching policies; however, they focus on fluid polling systems. In [15], the authors consider threshold-based switching policies (based on the occupancy of the unattended queue). However, they assume that one of the queues have finite buffer capacity and that the switching times are exponentially distributed. Due to these assumptions, they have a continuous time Markov jump process which is stable, once the combined load factor is strictly less than one. Essentially, the threshold parameters do not impact the stability of the system due to finite buffer considerations. We do not make any such assumptions and consider a big class of Markovian two-phase switching policies. In our case, the system stability further depends upon the parameters of the switching policy (apart from the load factor being strictly less than one).

In all, our work is a step towards individual-queue-centric considerations by defining a sufficiently large class of two-

<footnote n="1">¹Inria Centre at Université Côte d'Azur, Sophia Antipolis, France k.avrachenkov@inria.fr</footnote>

<footnote n="2">²Department of IEOR, IIT Bombay, Mumbai, India {20004007, vkavitha}@iitb.ac.in</footnote>

<footnote n="3">³Vartika Singh was with University of Colorado, Colorado Springs, United States when this research was conducted singh.vsvartika@gmail.com</footnote>
phase Markovian policies, for systems with infinite buffers at both the queues. Our switching policy is characterized by affine functions of queue-occupancy in both the queues and is parameterized by eight coefficients. The affine functions are piecewise constant, changing only at arrival or departure epochs, thereby allowing potential switching decisions at these precise epochs. The proposed class of policies includes all the above mentioned classes (e.g., threshold policies of [14], [15], priority-factor based policies of [13]). We also numerically illustrate that our policies can cover the Pareto frontier, defined in terms of the stationary expected numbers in the individual queues.

The main aim of this initial work is to identify the subclass of proposed switching policies that are stable (further theoretical investigation with respect to control and optimization is planned for future exploration). The stochastic process representing the system evolution in our case is not even Markovian. The idea is to study the stability of the chain, observed at the server-visit-epochs to one of the queues. For Poisson arrivals, this forms a discrete-time Markov chain. Using the Foster-Lyapunov stability criterion, we show that a single inequality, defined in terms of two parameters of the switching policy determines the stability. We then prove the time stationarity of the system by defining appropriate regeneration epochs and by showing that the expected value of the corresponding regeneration time is finite; the latter statement follows from the stability of the discrete chain.

## II. POLLING SYSTEM AND SWITCHING POLICIES

We consider a polling system with two queues $Q_1$ and $Q_2$, where a single server attends the customers of the two queues in a cyclic order and switches between the queues with non-zero switching times. The arrival of customers to $Q_i$ (for $i \in \{1, 2\}$) is modeled as a Poisson process with rate $\lambda_i$, independent of that in the other queue indexed by $j := 3 - i$ (when $i = 1, j = 2$ and when $i = 2, j = 1$). The queues have infinite buffer capacity and follow the non-resume preemptive service discipline; basically if the server switches to the other queue while in the middle of a service, then it has to restart that service upon its next visit (see [3], [16]). The jobs are served at both the queues according to independent and identically distributed (i.i.d.) service times and with common rate $\mu$. The customers leave the system only after their service is completed. We define the load factors $\rho_i := \lambda_i/\mu \ \forall i$ and assume $\rho := \rho_1 + \rho_2 < 1$. Upon visiting a queue, the server attends it until the switching instance governed by a given two-phase switching policy described in Section II-A. The switching times $\{S_{ik}\}_{k \ge 1}$ from $Q_i$ to $Q_j$ at various switching instances indexed by $k$ are assumed to be i.i.d. random variables with mean $s_i$; these are independent of other (previous) events.

In this work, we aim to identify a sufficiently generic class of stable switching policies for polling systems. Our further aim is to identify a big class that can be parametrized compactly (using a small set of parameters). For example, a class general enough to obtain an appropriate and approximate achievable region of the polling system (more details in

Section III-A) or a class general enough to produce required collection of robust policies (see Section III-B).

### A. A Family of Two-phase Switching Policies, $\mathcal{T}$

We consider stationary two-phase switching policies that are employed at each (customer) arrival and departure epoch. Any such policy depends upon the system state, the number of waiting customers (say $N_1(t)$, $N_2(t)$) in the two queues, at the decision epochs. While visiting (say) $Q_i$, the policies are defined using two affine linear functions $\Pi_{iB}(\cdot)$ and $\Pi_{iC}(\cdot)$,

$$ \Pi_{iB}(N_i(t), N_j(t)) := \alpha_i^B N_i(t) + N_j(t) + \mu \beta_i^B \quad (1) $$

$$ \Pi_{iC}(N_i(t), N_j(t)) := N_i(t) + \alpha_i^C N_j(t) + \mu \beta_i^C, \quad (2) $$

which respectively determine the beginning and the concluding phase of the server-visit time. Here $\{\alpha_i^m, \beta_i^m\}_{m \in \{B,C\}}$ for both $i$ are non-positive real constants and represent the parameters of the policy. Basically the parameters $\{\alpha_i^m, \beta_i^m\}_{m \in \{B,C\}}$ for any $i$ represent the priorities given to the visiting queue ($Q_i$)—the higher values of $B$ parameters and/or lower values of $C$ parameters for $Q_i$ imply longer visit times and hence a higher priority to $Q_i$.

Now, say the server arrives to $Q_i$ at time epoch $\Phi$. During the subsequent $Q_i$-visit time, the server only attends to $Q_i$ and hence $N_j(t)$ keeps increasing with each arrival to $Q_j$. On the other hand, the number $N_i(t)$ can increase/decrease with time, but will decrease eventually (as $\rho_i < 1$). These fluctuations eventually determine the end of server $Q_i$-visit phase; the precise details of the two-phase switching policy are in the following:

**Beginning Phase:** The server continues serving the $Q_i$ as long as the function $\Pi_{iB}(N_i(t), N_j(t)) < 0$. Once $\Pi_{iB}(N_i(t), N_j(t))$ becomes zero or positive, the beginning phase, characterized by $\Pi_{iB}(\cdot)$ ends. Hence, the length of the beginning phase at $Q_i$, represented by $B_i$, equals

$$ B_i := \inf\{t \ge \Phi : \Pi_{iB}(N_i(t), N_j(t)) \ge 0\} - \Phi. \quad (3) $$

**Concluding Phase:** The concluding phase starts immediately after $B_i$, the server continues serving $Q_i$ as long as $\Pi_{iC}(N_i(t), N_j(t)) > 0$. Once $\Pi_{iC}(N_i(t), N_j(t))$ becomes zero or negative, the server switches to (immediately starts moving towards) $Q_j$ and the concluding phase at $Q_i$ ends. The length of the concluding phase at $Q_i$ is represented by $C_i$,

$$ C_i := \inf\{t \ge \Phi + B_i : \Pi_{iC}(N_i(t), N_j(t)) \le 0\} - (\Phi + B_i). \quad (4) $$

In any cycle (say $k$-th cycle), the total visit time of the server at $Q_1$ equals $M_{1k} := B_{1k} + C_{1k}$. The server leaves $Q_1$ after this visit time $M_{1k}$, arrives to $Q_2$ further after the random switching time $S_{1k}$ and then spends time $M_{2k}$ at $Q_2$ (again employing a similar switching policy) now at $Q_2$. And this continues.

Note that the functions $\Pi_{iB}(\cdot)$ and $\Pi_{iC}(\cdot)$ are piece-wise constant and their value changes only at an arrival or a

<footnote n="1">¹The scaling of $\beta_i^C, \beta_i^B$ factors with $\mu$ helps in comparative analysis across different $\mu$ systems, however this is not meant for asymptotic analysis.</footnote>
departure epoch. Thus the arrival and departure epochs form the potential switching epochs. Let $\mathcal{T}$ represent the collection of two-phase policies, each determined by eight parameters:

$$ \mathcal{T} := \{\pi = (\alpha_i^m, \beta_i^m) : \alpha_i^m, \beta_i^m \le 0, i \in \{1, 2\}, m \in \{B, C\}\}. $$

We will observe that the family $\mathcal{T}$ is fairly general and includes various policies considered in the literature; some interesting examples are discussed in section III; for now, observe that the proposed class can include one phase policies with only C phase being active, by setting $\alpha_i^B = 0 = \beta_i^B$ for both $i$. We next describe the arrival and service process for the polling system and the required assumptions.

## B. Arrival and Uninterrupted Service Processes

The arrival process to any queue (say $Q_i$) is modeled using a Poisson arrival process with rate $\lambda_i$. Let $\Lambda_i(t)$ represent the number of arrivals to $Q_i$ until time $t$.

When the server attends a queue continuously (without interruptions), the departure process is referred to as uninterrupted service process, which is a part of a renewal process. The uninterrupted service process at $Q_i$ during the k-th visit of the server is modeled as renewal process<fnref n="2" /> defined via the i.i.d. service-times $\{J_{ik,n}\}_{n \ge 1}$ (observe here $E[J_{ik,n}] = 1/\mu$ for each $i, k, n$). The counting process representing the number of departures in $Q_i$ during the k-visit of the server is represented by $\Gamma_{ik}(\cdot)$ and is given by

$$ \Gamma_{ik}(t) = \sup \left\{ n : \sum_{v=1}^{n} J_{ik,v} \le t \right\} \quad \text{for any } t \ge 0. \quad (5) $$

Observe here that, in each visit of the server to any queue, a new uninterrupted service process begins that lasts till the switching epoch.

# III. SOME INTERESTING CLASSES OF POLICIES

## A. Achievable Region of [13] and Pareto Frontier

In [13], in a quest to identify the achievable region of the polling system, defined in terms of customer-average waiting times (or equivalently in terms of stationary expected number of customers in individual queues by Little's law), the authors analyzed a fluid model under a special class of priority schedulers. We will show that the two-phase switching policies $\mathcal{T}$ include this class. In section V, we discuss the approximate achievable region using this class and also illustrate numerically that the corresponding Pareto frontier converges to the one derived in [13, Theorem 8].

The achievable region in terms of customer-average waiting times is defined as (see [13]):

$$ \mathcal{A} = \{ (\bar{w}_1(\pi), \bar{w}_2(\pi)) : \pi \text{ is any stationary policy} \}, $$

where $\bar{w}_i(\pi)$ is the customer-average of the waiting times (if exists) of the customers of $Q_i$ under switching policy $\pi$. A class B of switching policies is said to be complete, if every

pair $(\bar{w}_1, \bar{w}_2) \in \mathcal{A}$ is achieved by a switching policy $\pi \in \mathcal{B}$, i.e., if,

$$ \mathcal{A} = \{ (\bar{w}_1(\pi), \bar{w}_2(\pi)) : \pi \in \mathcal{B} \}, $$

If such an achievable region is shown to exist and if a complete class B is identified, any optimization problem related to the polling systems can be solved using the feasible performance pairs of $\mathcal{A}$. For example, the authors in [13] consider a constrained optimization as below:

$$ \text{Optimize}_{(\bar{w}_1, \bar{w}_2) \in \mathcal{A}} \mathcal{F}(\bar{w}_1, \bar{w}_2) \text{ s.t. } \bar{w}_1 \le v_1, $$

and observe that it is equivalent to

$$ \text{Optimize}_{\pi \in \mathcal{B}} \mathcal{F}(\bar{w}_1(\pi), \bar{w}_2(\pi)) \text{ s.t. } \bar{w}_1(\pi) \le v_1. $$

In [13] authors identified one such class of complete schedulers at an appropriate fluid regime. These policies are parameterized by constants, say $\alpha_1, \alpha_2$. Here, the server switches from $Q_i$ to $Q_j$, at the first time instance $t$ (after its visit to $Q_i$), when $N_j(t) > N_i(t)\alpha_i$.

Such a switching policy can be described by our two-phase policies upon considering

$$ \alpha_i^C = -\frac{1}{\alpha_i}, \quad \alpha_i^B = \beta_i^B = \beta_i^C = 0 \quad \text{for each } i, $$

in (1) and (2). Hence, for these policies, the length of the beginning phase $B_i$ equals zero. Authors in [13] also consider exhaustive (ex) policies where the server visiting $Q_i$ switches only when the queue becomes empty and $Q_j$ is non-empty. This policy is attained by two-phase policies when all the parameters are zero in (1) and (2). The beginning phase equals zero even in case of exhaustive policy, see (3). One can also have exhaustive policy at one queue and a policy with non-zero parameters at other queue (such policies are shown to obtain weak Pareto frontier in [13, Theorem 6] and these switching policies are discussed in section V); one can achieve $\pi_{ex} = (ex, \alpha)$ policy, for any $\alpha$, by setting in (1)-(2):

$$ \alpha_1^B = \alpha_1^C = \beta_1^B = \beta_1^C = 0 \text{ and } \alpha_2^C = -\frac{1}{\alpha}, \alpha_2^B = \beta_2^B = \beta_2^C = 0. $$

## B. Robust Policies [14], [15]

The focus of the previous class of the switching policies is on expected or customer-average performance. However, many a times, the focus is on robust design. For example, once a queue is above a prescribed (danger) level, one may require to switch immediately and serve that queue.

Authors in [14], [15] consider switching policies that can be useful in this context. They consider a threshold policy, where the server in $Q_i$ considers switching to $Q_j$ once the queue length at $Q_j$ exceeds a threshold $\beta_j$, and waits till the queue length in $Q_i$ falls below a threshold $\beta_i$, and then switches. Such policies can be captured by our two-phase policies when $\alpha_i^B = \alpha_i^C = 0$, $\beta_i^B = -\beta_j$ and $\beta_i^C = -\beta_i$ for all $i$. Basically, the functions defining these two-phase policies are now given by:

$$ \begin{aligned} \Pi_{iB}(N_i(t), N_j(t)) &= N_j(t) - \mu\beta_j, \\ \Pi_{iC}(N_i(t), N_j(t)) &= N_i(t) - \mu\beta_i. \end{aligned} \quad (6) $$

<footnote n="2">²For mathematical convenience the uninterrupted service process is defined using an infinite sequence, but observe only the first $v$ (i.e., $n \le v$) of them are used if $v$ customers are served during the k-th visit; such a definition does not alter anything as the job sizes are i.i.d. across cycles also.</footnote>
Hence while the server is visiting $Q_i$, the first phase ends when the queue length at the unattended $Q_j$ is above $\mu\beta_j$ and the second phase ends (and the server switches) when the queue length at $Q_i$ is below $\mu\beta_i$. There is a possibility that the phase C ends before phase B; however it is guaranteed that the queue length $N_i$ is at maximum $\mu\beta_i$ at all the switching epochs from $Q_i$.

Thus this class of policies allow the server to immediately jump to the other queue, of course as long as the number of customers in the visiting queue is below a manageable threshold; hence they exhibit robustness.

Authors in [14] actually consider a sub-class of policies defined above, where $\beta_i^C = \beta_{-i}^B$ for both $i$; hence their policies are parametrized by two parameters while ours are parametrized by four parameters, but offer more generality.

#### IV. RANDOM SYSTEMS

We analyze the polling systems for any given arrival and departure rates $(\lambda_1, \lambda_2, \mu)$. The particular focus is on deriving the stability conditions and thereby to identify the sub-class among $\mathcal{T}$ that renders the polling system stable for given set of rates $(\lambda_1, \lambda_2, \mu)$ and the switching times.

Our discussions will also include comparison of polling systems with different values of $\mu \in (0, \infty)$. Towards this second purpose, we consider one system for each $\mu$, while the load factors $\rho_i$, $i = 1, 2$ are kept fixed. Henceforth, we represent all the quantities with superscript $\mu$ to explicitly show this dependency and call it $\mu$-system. We represent the arrival process $\Lambda_i^{\lambda_i}(\cdot) = \Lambda_i^{\lambda_i(\mu)}(\cdot) = \Lambda_i^{\rho_i\mu}(\cdot)$ at $Q_i$ by $\Lambda_i^{\mu}(\cdot)$ and the uninterrupted service process for the $k$-th visit of the server$^3$ by $\Gamma_{ik}^{\mu}(\cdot)$. Thus, for any $\mu$, we have the following almost surely,

$$E\left[\left.\frac{\Lambda_i^{\mu}(T)}{\mu}\right|T\right] = \rho_i T, \quad E\left[\left.\frac{\Gamma_i^{\mu}(T)}{\mu}\right|T\right] = T \quad \text{for all } i, \quad (7)$$

for any random time $T$ independent of the stochastic processes $\Lambda^{\mu}(\cdot)$ and $\Gamma^{\mu}(\cdot)$.

##### A. Evolution of the Polling system

We first describe the evolution of the polling system with service rate $\mu$ and introduce some of the notations to present the main results of this work. Without loss of generality, we define a cycle to be the time period between subsequent server-visit epochs of $Q_1$. We represent the duration of $k$-th cycle by $\Psi_k^\mu$. Let $\Phi_k^\mu := \sum_{n<k} \Psi_n^\mu$ represent the starting epoch of $k$-th cycle. Recall $N_i^\mu(t)$ represents the number of customers in $Q_i$ at time $t$ and let $N_{ik}^\mu := N_i^\mu(\Phi_k)$ represent this number at $\Phi_k^\mu$. Let $S_{ik}^\mu$ be the random switching time from $Q_i$ to $Q_j$ in the $k$-th cycle with mean $s_i^\mu$. Similarly, let $M_{ik}^\mu, B_{ik}^\mu$, and $C_{ik}^\mu$ be the total visit time, and the durations of the beginning and the concluding phases of the visit time, respectively. Hence, $M_{ik}^\mu = B_{ik}^\mu + C_{ik}^\mu$.

With general i.i.d. service and switching times, it is clear that our process is not Markovian. However our

switching policies are Markovian: basically the decisions depend upon the 'state' at the decision epoch (say $t$), $(N_1^\mu(t), N_2^\mu(t), P^\mu(t))$ with $P^\mu(t)$ taking values in the set $\{(1, B_1), (1, C_1), S_1, (2, B_2), (2, C_2), S_2\}$ that represents various phases of the system (see Fig. 1); here $(i, B_i)$ and $(i, C_i)$ respectively represent the beginning and concluding phases in $Q_i$ while $S_i$ represents the switching phase from $Q_i$.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Polling system with two-phase switching policy</figcaption></figure>

We now present the evolution of the number of customers in any queue. For ease of notation, define

$$D_{1k}^{\mu} := \Phi_{k}^{\mu} + M_{1k}^{\mu}, \quad \text{and, } D_{2k}^{\mu} := \Phi_{k}^{\mu} + M_{1k}^{\mu} + S_{1k}^{\mu} + M_{2k}^{\mu}, \quad (8)$$

respectively as the departure epochs from $Q_1$ and $Q_2$ in $k$-th cycle. Then the number of customers in $Q_1$ evolve as:

$$N_1^\mu(t) = \begin{cases} N_{1k}^\mu + \Lambda_1^\mu(t) - \Lambda_1^\mu(\Phi_k^\mu) - \Gamma_{1k}^\mu(t - \Phi_k^\mu) & \text{if } \Phi_k^\mu \le t \le D_{1k}^\mu, \\ N_1^\mu(D_{1k}^\mu) + \Lambda_1^\mu(t) - \Lambda_1^\mu(D_{1k}^\mu) & \text{if } D_{1k}^\mu < t < \Phi_{k+1}^\mu, \end{cases} \quad \text{for some } k. \quad (9)$$

In the above, $\Lambda_1^\mu(t) - \Lambda_1^\mu(s)$ represents the number of arrivals in $Q_1$ in the time duration $(t-s)$ for $s \le t$. Since the uninterrupted service process is assumed to be non-resume preemptive, $\Gamma_{1k}^\mu(t-s)$ equals the number of departures from $Q_1$ in time duration $(t-s)$; recall we have one such process for each $k$ which ends at $D_{1k}^\mu$. Once the server departs from $Q_1$ at epoch $D_{1k}^\mu$, there are only arrivals at $Q_1$ (and no departures). Similarly, the number of customers in $Q_2$ evolve as below:

$$N_2^\mu(t) = \begin{cases} N_{2k}^\mu + \Lambda_2^\mu(t) - \Lambda_2^\mu(\Phi_k^\mu) & \text{if } \Phi_k^\mu \le t \le D_{1k}^\mu + S_{1k}^\mu, \\ N_2^\mu(D_{1k}^\mu + S_{1k}^\mu) + \Lambda_2^\mu(t) - \Lambda_2^\mu(D_{1k}^\mu + S_{1k}^\mu) \\ - \Gamma_{2k}^\mu(t - D_{1k}^\mu - S_{1k}^\mu) & \text{if } D_{1k}^\mu + S_{1k}^\mu < t \le D_{2k}^\mu, \\ N_2(D_{2k}^\mu) + \Lambda_2^\mu(t) - \Lambda_2^\mu(D_{2k}^\mu) & \text{if } D_{2k}^\mu < t < \Phi_{k+1}^\mu, \end{cases} \quad \text{for some } k. \quad (10)$$

In the above, $\Lambda_2^\mu(t) - \Lambda_2^\mu(s)$ and $\Gamma_{2k}^\mu(t-s)$ represent the number of arrivals and departures in $Q_2$ in duration of length $t-s$, as before. Note that we have departures in $Q_2$ only during $D_{1k}^\mu + S_{1k}^\mu < t \le D_{2k}^\mu$, i.e., during its server-visit phase.

##### B. Markov Chain at Server Visit Epochs

With Poisson arrival process, $\{(N_{1k}^\mu, N_{2k}^\mu)\}_{k \in \mathbb{N}}$ is a discrete-time Markov chain for every $\mu$ and so is the case with the normalized chain, $\{\mathbf{X}_k^\mu\}_{k \in \mathbb{N}}$, where

$$\mathbf{X}_k^\mu := (\Theta_{1k}^\mu, \Theta_{2k}^\mu), \quad \Theta_{ik}^\mu = \Theta_i^\mu(\Phi_k), \quad \Theta_i^\mu(t) := \frac{N_i^\mu(t)}{\mu}, \quad i = 1, 2.$$

<footnote>$^3$Note that due to notational complexity, in the remaining part of this section, we use the notation $\Gamma_i^\mu(\cdot)$ instead of $\Gamma_{ik}^\mu(t)$ and correspondingly simplify the notation for service-times.</footnote>
We refer to {$\mathbf{X}_k^\mu$}$_{k \in \mathbb{N}}$ as the Palm Markov chain. In this section, we aim to derive the stability of this chain and then eventually that of the joint stochastic process $\mathbf{X}^\mu := (\mathbf{X}^\mu(t))_{t \ge 0}$, defined for all $t$, where $\mathbf{X}^\mu(t) := (\Theta_1^\mu(t), \Theta_2^\mu(t))$.

From (4), the affine functions $\Pi_{iC}$ with $i = 1, 2$ determine the concluding phases $C_{ik}^\mu$ of the k-th visit time. Let random variables, $\mathcal{H}_{1k}^\mu$ and $\mathcal{H}_{2k}^\mu$ respectively represent the values of this function at the switching instances in the two queues. Using the departure times of (8) we have

$$ \Pi_{1C}(N_1^\mu(D_{1k}^\mu), N_2^\mu(D_{1k}^\mu)) = N_1^\mu(D_{1k}^\mu) + \alpha_1^C N_2^\mu(D_{1k}^\mu) + \mu\beta_1^C = \mathcal{H}_{1k}^\mu, \quad (11) $$

$$ \Pi_{2C}(N_2^\mu(D_{2k}^\mu), N_1^\mu(D_{2k}^\mu)) = N_2^\mu(D_{2k}^\mu) + \alpha_2^C N_1^\mu(D_{2k}^\mu) + \mu\beta_2^C = \mathcal{H}_{2k}^\mu. \quad (12) $$

We now assume the following about these residual values:

**R.1:** Assume $E[\mathcal{H}_{ik}^\mu]$, is bounded uniformly in $k$, for each $i$.

If the concluding C phase is strictly positive for both the queues at all the departure epochs, then the above is trivially true, as explained in the immediate next. Let $\Pi_{iC}(t^+)$ and $\Pi_{iC}(t^-)$ respectively represent the value of $\Pi_{iC}$ function in (2) immediately after and before the departure epoch $D_{ik}^\mu$. Observe that for strictly positive concluding phases, by definition, $\Pi_{iC}(t^-) < 0$ and $\Pi_{iC}(t^+) \ge 0$ and such a transition is triggered either by an arrival or by a departure. Thus in fact $\mathcal{H}_{ik}^\mu$ are bounded almost surely by $|\alpha_i^C| + 1$, and hence, **R.1** is trivially true.

Therefore **R.1** is true for the class of policies considered in Section III-A, the ones that include the Pareto complete class at fluid limit.

From (6), it is clear that $E[\mathcal{H}_{ik}^\mu] \le \mu|\beta_i^C|$ for all $k$ and thus **R.1** is also satisfied for the robust class described in Section III-B.

The assumption **R.1** can be a restriction only for the subclass of policies for which the concluding phases in some cycles are zero (possible if C phase ends before B phase sometimes); Section III-B describes one such class, but the assumption **R.1** is satisfied for this class directly using the uniform boundedness of the visiting queue at the server-departure epochs. It is also intuitive that one cannot anticipate stability without assumption **R.1**.

Now to determine the stability of the Palm Markov chain {$\mathbf{X}_k^\mu$}$_{k \in \mathbb{N}}$, we use the Foster stability criteria by constructing an appropriate Lyapunov function (see e.g., [17]). In particular, we consider the following as Lyapunov function

$$ V(\mathbf{X}_k^\mu) = \rho_2 \Theta_{1k}^\mu + (1 - \rho_1) \Theta_{2k}^\mu. \quad (13) $$

Then, the mean drift is given by

$$ \begin{aligned} \Delta V(\mathbf{x}^\mu) &= E[V(\mathbf{X}_{k+1}^\mu) - V(\mathbf{X}_k^\mu)|\mathbf{X}_k^\mu = \mathbf{x}^\mu] \\ &= E[\rho_2(\Theta_{1(k+1)}^\mu - \Theta_{1k}^\mu) \\ &\quad +(1 - \rho_1)(\Theta_{2(k+1)}^\mu - \Theta_{2k}^\mu)|\mathbf{X}_k^\mu = \mathbf{x}^\mu], \quad (14) \end{aligned} $$

where $\mathbf{x}^\mu := (\theta_1^\mu, \theta_2^\mu)$.

Now at any k-th cycle, we have the following relation for the number of customers at two consecutive cycles in $Q_i$ of the random system (see (9) and (10)):

$$ \begin{aligned} N_{i(k+1)}^\mu = & N_{ik}^\mu + \Lambda_i^\mu (M_{1k}^\mu + S_{1k}^\mu + M_{2k}^\mu + S_{2k}^\mu) \\ & - \Gamma_i^\mu (M_{ik}^\mu), \quad i = 1, 2. \end{aligned} \quad (15) $$

Hence, by normalizing the above equations for $i = 1, 2$ with $\mu$ and taking the expectations $E[\cdot|\mathbf{X}_k^\mu = \mathbf{x}^\mu]$, we obtain the following, using (7),

$$ \begin{aligned} E[\Theta_{i(k+1)}^\mu | \mathbf{X}_k^\mu = \mathbf{x}^\mu] &= \theta_i^\mu + \rho_i (E[M_{1k}^\mu + M_{2k}^\mu | \mathbf{X}_k^\mu = \mathbf{x}^\mu] + s_1^\mu + s_2^\mu) \\ &\quad - E[M_{ik}^\mu | \mathbf{X}_k^\mu = \mathbf{x}^\mu]. \end{aligned} $$

Consequently, from (14), we have (recall $\rho = \rho_1 + \rho_2$)

$$ \Delta V(\mathbf{x}^\mu) = (\rho - 1)E[M_{2k}^\mu | \mathbf{X}_k^\mu = \mathbf{x}^\mu] + \rho_2(s_1^\mu + s_2^\mu). \quad (16) $$

We now provide the conditions under which the two-phase switching policy $\pi \in \mathcal{T}$ renders the Palm chain stable.

**Theorem 1:** [Palm stability] Assume **R.1**. Then the sufficient condition for stability of the Markov chain {$\mathbf{X}_k^\mu$}$_{k \in \mathbb{N}}$ with switching policy $\pi \in \mathcal{T}$ is that

$$ \rho_1 + \rho_2 < 1 \quad \text{and} \quad \alpha_1^C \alpha_2^C < 1. $$

*Proof:* The proof is in Appendix.

<figure><img src="image_3.png" /></figure>

The intuition behind taking such a Lyapunov function is the following. Any k-th cycle, by our definition, starts with server-visit of $Q_1$ and length $\Psi_k^\mu$ of the cycle is the time lapsed between two consecutive visits to $Q_1$. The total visit time at $Q_2$ within this $\Psi_k^\mu$, is given by $M_{2k}^\mu$. This depends upon the number of customers in both the queues at the start of k-th cycle, i.e., on $\Theta_{1k}^\mu$ and $\Theta_{2k}^\mu$ and also upon the visit time to $Q_1$, given by $M_{1k}^\mu$. Hence, it was sufficient to construct Lyapunov function (13) in such a way that the drift of the function would be influenced only by the change in the second visit time of the cycle (i.e., visit time of $Q_2$), see (16).

Let $\mathcal{T}_s$ represent the subclass of stable switching policies:

$$ \mathcal{T}_s := \{\pi \in \mathcal{T} : \alpha_1^C \alpha_2^C < 1\}. \quad (17) $$

Interestingly, once $\rho < 1$, the stability depends only on two parameters ($\alpha_1^C, \alpha_2^C$) of the policies.

We now prove that the first moments related to Palm chain converge as the cycle number increases.

**Theorem 2:** [Palm First moments] Assume **R.1**, $\rho < 1$. For any $\pi \in \mathcal{T}_s$, as the cycle number $k \to \infty$, we have:

$$ \begin{aligned} E[\Theta_{ik}^\mu] &\rightarrow \theta_i^* + o(1/\mu) \quad \text{for each } i \in 1, 2, \text{ and} \\ E[\Psi_k^\mu] &\rightarrow \psi^* < \infty \quad \text{and hence, } \sup_k E[\Psi_k^\mu] < \bar{\psi} < \infty, \end{aligned} $$

where $\theta_1^*, \theta_2^*$ are defined in (27), (22), and $\psi^*$ and $\bar{\psi}$ are some finite ($\mu$ dependent) constants.

*Proof:* The proof is in Appendix.

<figure><img src="image_4.png" /></figure>
### C. Time Stationarity

The time epochs $R_m^\mu$ are called regeneration epoch for a stochastic process $\mathbf{X}^\mu(t)$, if the following two stochastic processes are stochastically identical (e.g., [18], [19]):

$$ \mathbf{X}^{\mu}(R_{1}^{\mu} + \cdot) \stackrel{d}{=} \mathbf{X}^{\mu}(R_{m}^{\mu} + \cdot) \text{ for each } m \geq 1. $$

Observe that the regeneration times $T_m^\mu := R_m^\mu - R_{m-1}^\mu$, for $m > 1$, are i.i.d. by definition. The process $\mathbf{X}^\mu(\cdot)$ is called positive recurrent if $E[T_m^\mu] < \infty$ and in such cases (e.g., [19]) the time stationary limit distribution ($\nu_x$) exists—for each $x$ the following is true almost surely:

$$ \lim_{t \to \infty} \frac{\int_0^t \mathbf{1}_{\{\mathbf{X}^\mu(t)=x\}} dt}{t} = \frac{E\left[\int_{R_{m-1}^\mu}^{R_m^\mu} \mathbf{1}_{\{\mathbf{X}^\mu(t)=x\}} dt\right]}{E[T_m^\mu]} =: \nu_x. \quad (18) $$

Towards proving time stationarity of our polling system $\mathbf{X}^\mu(t)$, representing the normalized versions of (9)-(10), we begin with identifying a regeneration epoch using Theorem 1.

Define the following server visit epochs (to $Q_1$), recursively (assuming that the server starts at $Q_1$, i.e., $\Phi_0^\mu = 0$)

$$ R_1^\mu := \inf \{ \Phi_k^\mu : k \geq 1, \Theta_{ik}^\mu = \theta_i^* \text{ for both } i \}, \text{ and,} $$

$$ R_m^\mu := \inf \{ \Phi_k^\mu > R_{m-1}^\mu : \Theta_{ik}^\mu = \theta_i^* \text{ for both } i \}, \forall m > 1. $$

By Theorem 1, $(\theta_1^*, \theta_2^*)$ are recurrent states of the stationary Palm chain, and hence, clearly $R_m^\mu$ with $m > 1$, defines the regeneration epochs for the system, due to Poisson arrivals (see e.g., [19]). Define the $m$-th regeneration cycle $T_m^\mu := R_m^\mu - R_{m-1}^\mu$, and let $\xi_m^\mu$ represent the 'Palm cycle' number accounting to $m$-th regeneration epoch, $R_m^\mu$. Then the regeneration epoch $R_m^\mu$ and regeneration cycle $T_m^\mu$ can be rewritten as:

$$ R_m^\mu = \Phi_{\xi_m^\mu}^\mu \text{ and } T_m^\mu = \sum_{k=\xi_{m-1}^\mu}^{\xi_m^\mu} \Psi_k^\mu. $$

Thus, by Theorems 1-2, with $\nu^P$ representing the stationary distribution of the Palm chain, for any $m > 1$ we have:

$$ E[T_m^\mu] \leq \sup_k E[\Psi_k^\mu] E[\xi_m^\mu - \xi_{m-1}^\mu] < \frac{\bar{\psi}}{\nu_{\theta_1^*, \theta_2^*}^P} < \infty. \quad (19) $$

We thus have proved the following using the results of [19, Section 3.5.1]:

**Theorem 3:** [Time stationarity] Assume R.1, $\rho < 1$ and consider any $\pi \in \mathcal{T}_s$. Then the stochastic process $\mathbf{X}^\mu(t)$ representing the two-queue polling system is positive recurrent and satisfies (18).

## V. NUMERICAL ILLUSTRATION

In this section we would like to obtain individual-queue-centric performance measures like the stationary expected number of waiting customers in each (say, $i$-th) queue for any $\pi \in \mathcal{T}_s$:

$$ \lim_{K \to \infty} \frac{1}{K} \int_0^K N_i^\mu(t) dt = E_{\nu_\pi}[N_{i*}^\mu] \text{ (when it exists).} \quad (20) $$

In the above, $N_{i*}^\mu$ denotes the number of customers in $Q_i$ in steady-state and $\nu_\pi$ denotes the stationary distribution under the given stable switching policy $\pi \in \mathcal{T}_s$. In the strict sense, one first needs to prove the existence of stationary moments in (20), before discussing them. The precise proof of this result is postponed to journal version, however we will now provide partial theoretic and partial heuristic arguments towards the conjecture that such moments exists for $\pi \in \mathcal{T}_s$. It is not difficult to observe the following (recall $\Phi_k^\mu$ is the server-visit epoch to $Q_1$) for any $k$:

$$ \sup_{t \in [\Phi_k^\mu, \Phi_{k+1}^\mu]} \Theta_1^\mu(t) \leq \Theta_1^\mu(\Phi_k^\mu) + \Lambda_1^\mu(\Psi_k^\mu) \text{ almost surely, (21)} $$

and thus $E[\Theta_1^\mu(t)] \leq \sup_k (E[\Theta_{ik}^\mu] + E[\Psi_k^\mu])$ for any $t$. By Theorem 2, these moments are bounded and so we expect to prove (recall $T$ is a typical regeneration epoch):

$$ E \left[ \int_{0}^{T} \Theta_{1}^{\mu}(t) dt \right] < \infty; $$

if the above is proved, then the results of regenerative processes confirm the existence of the stationary moments in (20) (similarly for $Q_2$ consider a Palm chain at server-visit epochs to $Q_2$). For now we assume the conjecture and proceed towards numerical observations.

Now, among the stable class $\mathcal{T}_s$, we are interested in finding a suitable subclass of policies that generates the Pareto frontier for the achievable region (the policies under which the performance vectors are strongly or weakly efficient, in that one can't improve one of them without hampering the other or one cannot strictly improve both of them respectively). By [13, Theorem 6], the Pareto-complete sub-class at fluid limit is given by:

$$ \mathcal{P}_s = \{ \pi_{ex} : \pi_{ex} = (ex, \alpha^C) \text{ or } (\alpha^C, ex), \alpha^C \leq 0 \}. $$

We now examine numerically if $\mathcal{P}_s$ sub-class generates Pareto frontier (among $\mathcal{T}_s$) specially for sufficiently large $\mu$.

We use (20) to numerically obtain the stationary performance measures using Monte Carlo simulations. The simulated events are the arrivals and the departures of the customers and the switching of the server from the visiting queue to the other.

<figure><img src="image_5.png" /><figcaption>Fig. 2: Left figure: Performance comparison of Pareto class $\mathcal{P}_s$ with $\mathcal{T}_s$. Right figure: Pareto frontier for different $\mu$.</figcaption></figure>

In the left sub-figure of Fig. 2, we plot the estimates of the stationary expected number of customers in each queue for
some switching policies drawn randomly from class $\mathcal{T}_s$ (see black small circles). In the same figure and for the same set of parameters, we also plot the performance measures for some policy $\pi_{ex}$ from supposed Pareto class $\mathcal{P}_s$ (see red squares). As seen from the figure the performance of the sub-class $\mathcal{P}_s$ is indeed on the Pareto frontier.

In the right sub-figure of Fig. 2, we plot the performance measures for some policy $\pi_{ex} \in \mathcal{P}_s$ for different values of $\mu$. This supposedly represents the Pareto frontier for different $\mu$, and as seen from the figure, the Pareto frontier indeed approaches the weak frontier obtained in [13, Theorem 8] for the fluid limit.

## VI. FUTURE DIRECTIONS

One of the most interesting future direction of the current work is to analyze our polling setup for large $\mu$-systems. By Theorem 2, $\lim_{\mu \to \infty} \lim_{k \to \infty} E[\Theta_{ik}^{\mu}] = \theta_i^*$ for both $i$. In fact by using the results of [20] and examining the small set (26) used for proving stability in Appendix, one can observe that as $\mu \to \infty$ the random sequence $\Theta_{1k}^{\mu}$ in steady state concentrates near $(\theta_1^*, \theta_2^*)$ for large $\mu$ (recall scaling with $\mu$). These results can probably be used in constructing a robust achievable region (see (21)), defined probably in terms of probabilities of queue-lengths exceeding some given thresholds. Working further on regenerative arguments (second moments), one can directly derive the time-average or stationary performance and can derive their limit in fluid regime, or when $\mu \to \infty$. We anticipate such limits to have simple closed-form expressions which can provide much clearer insights for systems with large $\mu$.

## VII. CONCLUSION

This work introduces a general class of two-phase Markovian switching policies in a two-queue polling system with infinite buffer, general service times and non-zero general switch-over times. We identify the subclass of the policies that are stable; interestingly once the load factor is fixed at a value less than one, the stability condition remains the same irrespective of other system parameters (like service and arrival rates).

Eight parameters can represent our policies in the most generic form, while the stability of the policy is determined by only two parameters. We observe that the time limit of some first order moments (related to quantities at server visit epochs) converge as the service rate $\mu$ increases to $\infty$. Using such observations, we discuss potential connections to different varieties of achievable regions.

## APPENDIX

**Proof of Theorem 1:** Let $\rho := (\rho_1, \rho_2)$. We begin with obtaining the conditional expectation $E[M_{2k}^{\mu}|\mathbf{X}_k^{\mu} = \mathbf{x}^{\mu}]$, conditioned on initial state $\mathbf{x}^{\mu} = (\theta_1^{\mu}, \theta_2^{\mu})$ towards analysing (16). Taking conditional expectations $E[\cdot|\mathbf{X}_k^{\mu} = \mathbf{x}^{\mu}]$ of equation (11) (after normalizing with $\mu$) and rearranging we have (also using (7), (8), (9), and (10)):

$$ E[M_{1k}^{\mu}|\mathbf{X}_k^{\mu} = \mathbf{x}^{\mu}] = \frac{\theta_1^{\mu} + \alpha_1^C \theta_2^{\mu} + \beta_1^C}{\eta_1(\boldsymbol{\rho})} - \frac{E[\mathcal{H}_{1k}^{\mu}|\mathbf{X}_k^{\mu} = \mathbf{x}^{\mu}]}{\mu \eta_1(\boldsymbol{\rho})}, \quad (23) $$
where, $\eta_1(\boldsymbol{\rho}) := (1 - \rho_1) - \alpha_1^C \rho_2,$

as from (9), we (for example) have:

$$ E \left[ \frac{N_1(D_{1k}^\mu)}{\mu} \middle| \mathbf{X}_k^\mu = \mathbf{x}^\mu \right] = \theta_1^\mu + (\rho_1 - 1) E[M_{1k}^\mu | \mathbf{X}_k^\mu = \mathbf{x}^\mu]. $$

Observe $\eta_1(\boldsymbol{\rho}) > 0$, as $\alpha_1^C \le 0$. Now considering equation (12), proceeding as above and using (23), we have

$$ \begin{aligned} E[M_{2k}^{\mu}|\mathbf{X}_k^{\mu} = \mathbf{x}^{\mu}] &= \frac{(1 - \alpha_1^C \alpha_2^C)((1 - \rho_1)\theta_2^{\mu} + \rho_2\theta_1^{\mu}) + \beta_1^C(\rho_2 + \alpha_2^C(\rho_1 - 1))}{\eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \\ &\quad - \frac{E[\mathcal{H}_{1k}^{\mu}|\mathbf{X}_k^{\mu} = \mathbf{x}^{\mu}]}{\mu \eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} + \frac{(\rho_2 + \alpha_2^C \rho_1)s_1^{\mu} + \beta_2^C}{\eta_2(\boldsymbol{\rho})} \\ &\quad - \frac{E[\mathcal{H}_{2k}^{\mu}|\mathbf{X}_k^{\mu} = \mathbf{x}^{\mu}]}{\mu \eta_2(\boldsymbol{\rho})}, \end{aligned} $$
where, $\eta_2(\boldsymbol{\rho}) = (1 - \rho_2) - \alpha_2^C \rho_1$. \quad (24)

Observe that $\eta_2(\boldsymbol{\rho}) > 0$. Now substituting the above value of $E[M_{2k}^{\mu}|\mathbf{X}_k^{\mu} = \mathbf{x}^{\mu}]$ in (16), we have

$$ \begin{aligned} \Delta V(\mathbf{x}^{\mu}) &= \rho_2(s_1^{\mu} + s_2^{\mu}) + \frac{(\rho_1 + \rho_2 - 1)((\rho_2 + \alpha_2^C \rho_1)s_1^{\mu} + \beta_2^C)}{\eta_2(\boldsymbol{\rho})} \\ &\quad (1 - \alpha_1^C \alpha_2^C)((1 - \rho_1)\theta_2^{\mu} + \rho_2\theta_1^{\mu}) \\ &\quad + (\rho_1 + \rho_2 - 1) \frac{+ \beta_1^C(\rho_2 + \alpha_2^C(\rho_1 - 1))}{\eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \\ &\quad + \frac{(1 - \rho_1 - \rho_2)E[\mathcal{H}_{1k}^{\mu} + \eta_1(\boldsymbol{\rho})\mathcal{H}_{1k}^{\mu}|\mathbf{X}_k^{\mu} = \mathbf{x}^{\mu}]}{\mu \eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \\ &\quad (1 - \alpha_1^C \alpha_2^C)((1 - \rho_1)(\theta_2^{\mu} - \theta_2^*) \\ &\stackrel{L}{=} (\rho_1 + \rho_2 - 1) \frac{+ \rho_2(\theta_1^{\mu} - \theta_1^*)}{\eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \\ &\quad + \frac{(1 - \rho_1 - \rho_2)E[\mathcal{H}_{1k}^{\mu} + \eta_1(\boldsymbol{\rho})\mathcal{H}_{1k}^{\mu}|\mathbf{X}_k^{\mu} = \mathbf{x}^{\mu}]}{\mu \eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})}; \end{aligned} \quad (25) $$

by Lemma 1, the equality 'L' is true using a unique $(\theta_1^*, \theta_2^*)$ that satisfy (27) and (22). By assumption **R.1** the magnitude of the last two terms in the (25) can be bounded by $\Omega$ using some $\Omega < \infty$, Now choose $\Upsilon_i$, $i = 1, 2$ big enough such that

$$ \Omega < \frac{(1 - \alpha_1^C \alpha_2^C)(1 - \rho_1 - \rho_2)((1 - \rho_1)\Upsilon_2 + \rho_2\Upsilon_1)}{\eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} $$

possible, as $(1 - \alpha_1^C \alpha_2^C) > 0$ and $\rho_1 + \rho_2 < 1$. Now define

$$ \mathcal{C} = \{\mathbf{x}^{\mu} : \theta_i^{\mu} - \theta_i^* \le \Upsilon_i, i = 1, 2\}, \quad (26) $$

which has finitely many values as the state space

$$ \{ (\theta_1^\mu, \theta_2^\mu) : \theta_i^\mu = \frac{1}{\mu} z_i, z_i \in \{0, 1, 2, \dots\} \text{ for each } i \} $$

is countable. Hence, for any $\mu$ and $\forall \mathbf{x}^{\mu} \notin \mathcal{C}$, we can have
$$
\begin{align}
a_{11} &= \frac{(\rho_1 + \rho_2 - 1) - (\rho_1 - 1)\rho_1\alpha_2^C - \rho_1\rho_2\alpha_1^C\alpha_2^C}{\eta_1(\boldsymbol{\rho})\eta_2(\boldsymbol{\rho})}, & a_{12} &= \frac{(1 - \rho_1)\rho_1 - (1 - \rho_1)(1 - \rho_2)\alpha_1^C}{\eta_1(\boldsymbol{\rho})\eta_2(\boldsymbol{\rho})} \nonumber \\
a_{21} &= \frac{-\rho_1\rho_2\alpha_2^C - (\rho_2 - 1)\rho_2\alpha_1^C\alpha_2^C}{\eta_1(\boldsymbol{\rho})\eta_2(\boldsymbol{\rho})}, & a_{22} &= \frac{(1 - \rho_1)(\rho_2 - 1) + \rho_2(1 - \rho_2)\alpha_1^C - (\rho_1 + \rho_2 - 1)\alpha_1^C\alpha_2^C}{\eta_1(\boldsymbol{\rho})\eta_2(\boldsymbol{\rho})} \nonumber \\
a_{13} &= \frac{(1 - \rho_1 - \rho_2)\beta_1^C}{\eta_1(\boldsymbol{\rho})\eta_2(\boldsymbol{\rho})} - \frac{\rho_1\beta_2^C + \rho_1s_1^\mu}{\eta_2(\boldsymbol{\rho})} - \rho_1s_2^\mu, & a_{23} &= \frac{\alpha_2^C(\rho_1 + \rho_2 - 1)\beta_1^C}{\eta_1(\boldsymbol{\rho})\eta_2(\boldsymbol{\rho})} + \frac{(1 - \rho_2)\beta_2^C + \rho_1s_1^\mu\alpha_2^C}{\eta_2(\boldsymbol{\rho})} - \rho_2s_2^\mu. \tag{22}
\end{align}
$$

from (25)

$$
\begin{align*}
\Delta V(\mathbf{x}^{\mu}) &= \frac{(1 - \alpha_1^C \alpha_2^C)(\rho_1 + \rho_2 - 1)}{\eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \\
&\quad + \frac{((1 - \rho_1)(\theta_2^\mu - \theta_2^*) + \rho_2(\theta_1^\mu - \theta_1^*))}{\mu \eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \\
&\quad + \frac{E[\mathcal{H}_{1k}^\mu + \eta_1(\boldsymbol{\rho})\mathcal{H}_{1k}^\mu | \mathbf{X}_k^\mu = \mathbf{x}^\mu]}{\eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \\
&< -\zeta
\end{align*}
$$

if $\alpha_1^C \alpha_2^C < 1$ and $\rho_1 + \rho_2 < 1$. Here $\zeta > 0$ is a real constant. Therefore, by Foster stability criteria (see Thorem 2.6.4 in [17]) we can say that the Markov chain in random system $\{\mathbf{X}_k^\mu\}_{k \in \mathbb{N}}$ is stable if $\alpha_1^C \alpha_2^C < 1$ and $\rho_1 + \rho_2 < 1$. $\square$

**Lemma 1:** The equality 'L' in (25) is satisfied.

*Proof:* To begin with, we first obtain the solution $(\theta_1^*, \theta_2^*)$ of the following pair of equations where (coefficients $a_{mn}$ for different $m, n$ are in (22) at the top of the this page):

$$a_{11}\theta_{1}^{*} + a_{12}\theta_{2}^{*} = a_{13} \text{ and, } a_{21}\theta_{1}^{*} + a_{22}\theta_{2}^{*} = a_{23}; \quad (27)$$

The solution exists as the determinant

$$d = \frac{(1 - \rho_1 - \rho_2)(1 - \alpha_1^C \alpha_2^C)}{\eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \neq 0, \quad (28)$$

because $\alpha_1^C \alpha_2^C \neq 1$ and $\rho_1 + \rho_2 \neq 1$. Now combine $(\theta_1^*, \theta_2^*)$ as below and observe that equality L is satisfied:

$$
\begin{align}
& \frac{(1 - \rho_1 - \rho_2)(1 - \alpha_1^C \alpha_2^C) ((1 - \rho_1)\theta_2^* + \rho_2\theta_1^*)}{\eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \nonumber \\
&= \rho_2(s_1^\mu + s_2^\mu) + \frac{(\rho_1 + \rho_2 - 1)((\rho_2 + \alpha_2^C \rho_1)s_1^\mu + \beta_2^C)}{\eta_2(\boldsymbol{\rho})} \nonumber \\
&\quad + \frac{(\rho_1 + \rho_2 - 1)\beta_1^C (\rho_2 + \alpha_2^C (\rho_1 - 1))}{\eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})}. \tag{29}
\end{align}
$$

equations (12) and (30) and proceeding as above,

$$
\begin{align}
E[M_{2k}^{\mu}] &= \frac{(1 - \alpha_1^C \alpha_2^C) ((1 - \rho_1)E[\Theta_{2k}^{\mu}] + \rho_2 E[\Theta_{1k}^{\mu}])}{\eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \nonumber \\
&\quad + \frac{\mu \beta_1^C (\rho_2 + \alpha_2^C (\rho_1 - 1)) - E[\mathcal{H}_{1k}^{\mu}]}{\mu \eta_1(\boldsymbol{\rho}) \eta_2(\boldsymbol{\rho})} \nonumber \\
&\quad + \frac{(\rho_2 + \alpha_2^C \rho_1) s_1^\mu + \beta_2^C}{\eta_2(\boldsymbol{\rho})} - \frac{E[\mathcal{H}_{2k}^{\mu}]}{\mu \eta_2(\boldsymbol{\rho})}. \tag{31}
\end{align}
$$

By normalizing the equations (15) for $i = 1, 2$ with $\mu$ and taking the expectation, we obtain the following using (7),

$$E[\Theta_{i(k+1)}^{\mu}] = E[\Theta_{i}^{\mu}] + \rho_i (E[M_{1k}^{\mu} + [M_{2k}^{\mu}] + s_1^{\mu} + s_2^{\mu}] - E[M_{i}^{\mu}]).$$

Using the values of $E[M_{i}^{\mu}]$ in the above equation, we have

$$E[\Theta_{1(k+1)}^{\mu}] = (1 + a_{11})E[\Theta_{1k}^{\mu}] + a_{12}E[\Theta_{2k}^{\mu}] - a_{13} + \frac{r_{1k}}{\mu}$$

$$E[\Theta_{2(k+1)}^{\mu}] = a_{21}E[\Theta_{1k}^{\mu}] + (1 + a_{22})E[\Theta_{2k}^{\mu}] - a_{23} + \frac{r_{2k}}{\mu}$$

where the terms $r_{ik}/\mu$ are constructed using $E[\mathcal{H}_{ik}^\mu]$ of (30)-(31).

Define $m_{ik} := E[\Theta_{ik}^\mu - \theta_i^*]$ for each $i, k$ and using (27) in above equations we have matrix equation:

$$m_{1(k+1)} = (1 + a_{11})m_{1k} + a_{12}m_{2k} + \frac{r_{1k}}{\mu}$$

$$m_{2(k+1)} = a_{21}m_{1k} + (1 + a_{22})m_{2k} + \frac{r_{2k}}{\mu}$$

Define $\bar{\mathbf{m}}_k := [m_{1k}, m_{2k}]^\mathrm{tr}$, $\bar{\mathbf{r}}_k := [r_{1k}, r_{2k}]^\mathrm{tr}$ for each $k$. Therefore,

$$\bar{\mathbf{m}}_{k+1} = \mathbf{A}\bar{\mathbf{m}}_k + \frac{1}{\mu}\bar{\mathbf{r}}_k, \quad \mathbf{A} = \begin{bmatrix} 1 + a_{11} & a_{12} \\ a_{21} & 1 + a_{22} \end{bmatrix}.$$

Hence, we have

$$\bar{\mathbf{m}}_{k+1} = \mathbf{A}^k \bar{\mathbf{m}}_1 + \frac{1}{\mu} \sum_{l=1}^{k} \mathbf{A}^{k-l} \bar{\mathbf{r}}_l.$$

Now, as $\mathbf{Id} - \mathbf{A}$ has the determinant $d$ given in (28), which is non-zero from the condition $\alpha_1^C \alpha_2^C \neq 1$ and $\rho_1 + \rho_2 \neq 1$. Thus $\mathbf{A}^k \to \mathbf{O}$, the zero matrix.

Also, from $\mathbf{R}$.1, we can have $|r_{ik}| < r_i$ uniformly in $k$, for each $i$, where $r_i$ is some constant. Therefore, $\bar{\mathbf{r}}_k$ is bounded uniformly in $k$. Thus $E[\Theta_{ik}^\mu] \to \theta_i^* + r_i/\mu$ for each $i \in 1, 2$, when $k \to \infty$. Substituting this in (30)-(31) we obtain the rest of the proof as, $E[\Psi_k^\mu] = E[M_{1k}^\mu + M_{2k}^\mu] + s_1^\mu + s_2^\mu$. $\square$

**Proof of Theorem 2:** In the following we compute the expectations $E[\Theta_{ik}^\mu]$ and $E[\Psi_{ik}^\mu]$. Taking expectations of equation (11) (after normalizing with $\mu$) and rearranging we have (also using (7), (8), (9) and (10)):

$$E[M_{1k}^{\mu}] = \frac{E[\Theta_{1k}^{\mu}] + \alpha_1^C E[\Theta_{2k}^{\mu}] + \beta_1^C}{\eta_1(\boldsymbol{\rho})} - \frac{E[\mathcal{H}_{1k}^{\mu}]}{\mu \eta_1(\boldsymbol{\rho})}, \quad (30)$$

These steps are as in the proof of Theorem 1. Now using
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<table><caption>TABLE I: Summary of Notations</caption><thead><tr><td>Notation</td><td>Description</td></tr></thead><tbody><tr><td>λ, μ, Q<sub>i</sub>, ρ<sub>i</sub></td><td>Arrival rate, Service rate, i-th Queue, traffic load at Q<sub>i</sub></td></tr><tr><td>Π<sub>iB</sub>, Π<sub>iC</sub></td><td>Switching functions relevant while visiting Q<sub>i</sub></td></tr><tr><td>α<sup>B</sup><sub>i</sub>, β<sup>B</sup><sub>i</sub>, α<sup>C</sup><sub>i</sub>, β<sup>C</sup><sub>i</sub></td><td>Switching parameters relevant while visiting Q<sub>i</sub></td></tr><tr><td>S<sup>μ</sup><sub>i</sub></td><td>Switching time from Q<sub>i</sub> to Q<sub>-i</sub> in μ-system</td></tr><tr><td>Λ<sup>μ</sup><sub>i</sub></td><td>Arrival process for Q<sub>i</sub> in μ-system</td></tr><tr><td>Γ<sup>μ</sup><sub>ik</sub></td><td>Uninterrupted service process at k-th visit for Q<sub>i</sub> in μ-system</td></tr><tr><td>B<sup>μ</sup><sub>ik</sub>, C<sup>μ</sup><sub>ik</sub></td><td>Length of Beginning phase, Concluding phase for k-th visit of Q<sub>i</sub> in μ-system</td></tr><tr><td>M<sup>μ</sup><sub>ik</sub></td><td>Total visit time for k-th visit of Q<sub>i</sub> in μ-system</td></tr><tr><td>N<sup>μ</sup><sub>i</sub>(t)</td><td>Number of waiting customers at Q<sub>i</sub> at time t in μ-system</td></tr><tr><td>Ψ<sup>μ</sup><sub>k</sub></td><td>Duration of k-th cycle in μ-system</td></tr><tr><td>Φ<sup>μ</sup><sub>k</sub></td><td>Starting epoch of k-th cycle in μ-system</td></tr><tr><td>N<sup>μ</sup><sub>ik</sub></td><td>Number of waiting customers for Q<sub>i</sub> at Φ<sup>μ</sup><sub>k</sub></td></tr><tr><td>Θ<sup>μ</sup><sub>ik</sub></td><td>Normalized number of waiting customers for Q<sub>i</sub> at Φ<sup>μ</sup><sub>k</sub></td></tr><tr><td>X<sup>μ</sup><sub>k</sub></td><td>Joint Palm Markov chain (discrete-time) for normalized number of customers at both queues in μ-system</td></tr><tr><td>X<sup>μ</sup>(t)</td><td>Joint stochastic process for normalized number of customers at both queues in μ-system</td></tr><tr><td>R<sup>μ</sup><sub>m</sub></td><td>Regeneration epochs for X<sup>μ</sup>(t)</td></tr><tr><td>T<sub>s</sub></td><td>Class of stable switching policies</td></tr><tr><td>P<sub>s</sub></td><td>Subclass of stable switching policies generating Pareto frontier</td></tr></tbody></table>