In this paper, we consider a spatial queueing network consisting of an infinite collection of processor sharing queues
interacting with each other through translation invariant coupling rules. In our model, there is a queue located at each grid
point of , for some . The queues evolve in continuous time and serve the customers according
to a generalized processor-sharing discipline. The arrivals to the queues form a collection of i.i.d. Poisson Point
Processes (PPPs) of rate . Thus, the total arrival rate to the network is infinite since there is an infinite
number of queues. The different queues interact through their departure rates. We model the interactions through
an interference sequence that we denote by . It is such that
and for all . We also assume that this sequence is finitely supported,
i.e., . For ease of exposition,
we also assume that in certain sections of the paper, although our model and its analysis can be carried out for any non-zero value
of . For any , let
denote the queue lengths at time in the network, i.e., the state of the system at time .
Then the interference experienced by a customer located in queue at time is defined as
, i.e., some weighted sum of queue lengths of the
neighbors of queue . Observe that the neighborhood definition is translation invariant.
Conditional on the queue lengths at time , the instantaneous
departure rate from any queue at time is given by ,
with interpreted as being equal to . Note that since the interference sequence
is non-negative, and , for all and all ,
the instantaneous departure rate from queue at time is
and is hence bounded. Since is non-negative, the rate of service at a queue is reduced
if its ‘neighbors’ have larger queue lengths. This is meant to capture the fundamental spatio temporal dynamics
in wireless networks where the instantaneous rate of a link is reduced if there are a lot of other links
accessing the spectrum nearby, due to an increase of interference.
This model is motivated by fundamental design questions in wireless networks. The motivation for this particular model comes from certain mathematical questions about such wireless dynamics left open in 
. In our model, we view the queues to represent ‘regions of space’ and the customers in each queue to be the wireless links in that region of space. One can interpret a link or customer to be a transmitter–receiver pair, with the transmitter transmitting a file to its intended receiver. For simplicity, we assume that the links are very tiny, i.e., a single customer represents both the transmitter and receiver. The links share the wireless spectrum in space and hence they impact each other’s performance due to interference. We assume that links arrive ‘uniformly’ in space, and each transmitter has a file whose length is exponentially distributed to transmit to its receiver. A link departs and leaves the network once the transmitter has finished sending the file to its receiver. We model the instantaneous rate of communication any transmitter can send to its own receiver as being inversely proportional to the interference seen at the receiver, i.e., as. This can be viewed as the low ‘Signal-to-Noise-and-Interference-Ratio (SINR)’ channel capacity of a point-to-point Gaussian channel (see ). Since there are links simultaneously transmitting, and each of them has an independent unit mean exponentially distributed file, the rate at which a link departs is then . The instantaneous rate of transmission of a link is lowered if it is in a ‘crowded’ area of space, due to interference, and hence it takes longer for this link to complete the transmission of its file. In the meantime, it is more likely that a new link will arrive at some point nearby before it finishes transmitting, further reducing the rate of transmission. Understanding how the network evolves due to such spatio temporal interference dynamics is crucial in designing and provisioning of wireless systems (see discussions in ).
The central thrust of this paper is to understand when the above described model is stable. Note that the Markovian dynamics of our model is non-reversible and that the model does not fall under the class of generalized Jackson networks. By stability, we mean stabilization in time
of the distribution of the infinite-dimensional queue-length vector. Traditionally, this means that the distribution of any finite-dimensional restriction of the vector converges weakly to the limiting one. In fact, in this paper, we introduce an appropriate coupling construction to investigate a stronger version of the sample-path stability (or boundedness). We show the coupling-convergence of finite-dimensional vectors (that imply convergence in the total variation norm), using the so-called Loynes backward representation of the system dynamics (see, e.g.,). The latter means that we fix initial (non-random) values of the queue-length process, start with this values at time and observe the queue lengths at time . Then we let tend to infinity. We begin with all-zero initial values. We establish certain monotonicity properties to conclude that, in the case of zero initial values, increases a.s. with , for any . Therefore, the limit
exists a.s. It may be either finite or infinite (where each occurs with probability either zero or one). This is theminimal stationary regime: any other stationary regime, say must satisfy , for all . Then we identify a sufficient condition for stability, i.e., for the finiteness of the minimal stationary regime. Remarkably, we are able to provide an exact formula for the mean queue length of the minimal stationary solution.
If , then the system is stable. Furthermore, for all and , the minimal stationary solution satisfies
The proof of this theorem is carried out in Section 7, with some accompanying calculations in Section 6. In the rest of the paper, the condition will be referred to as the stability criterion for the system. In this theorem, we only considered whether there exists a stationary solution to the dynamics. However, as our network consists of infinitely many queues, uniqueness of stationary solutions is not guaranteed. The following Proposition sheds light on this matter.
If , then is the unique stationary solution with finite second moment.
This proposition is proved in Section 8. This result relies on the finiteness of second moment of the stationary queue length, which does not follow immediately from the conclusions of Theorem 1.1. In this regard, we have the following proposition, that establishes finiteness of second moment under further restrictive conditions than stability.
If , where , then we have .
The proof of this proposition is carried out in Section 7, with some accompanying calculations in Section 6. Note that under our assumption of , the value of the constant can be simplified as . The above Proposition does not cover the full range of stability, since for any valid interference sequence , we have , with if and only if for all , in which case the model is uninteresting as there are no interactions among the queues. For the simplest non-trivial case of one dimensions and the interference sequence being if and if , the second moment is finite for . From Propositions 1.2 and 1.3, we have the following immediate corollary.
If , where is given in Proposition 1.3, then is the unique stationary solution with finite second moment.
Our next set of results assesses whether queue length process converges to any stationary solution when started from different starting states. Observe that we deemed the system stable if when started with all queues empty, the queue lengths converge to a proper random variable. Thus, stability alone does not imply convergence from other initial conditions. In this regard, our main results are stated in Theorems1.5 and 1.7 which show the sensitivity of the dynamics to the starting conditions. In particular, we show in Theorem 1.5, that if is sufficiently small and the initial conditions are uniformly bounded, then the queue lengths converge to the minimal stationary solution. Surprisingly, in Theorem 1.7 below, we exhibit both deterministic and random initial conditions for all , such that the queue lengths diverge, even though the stability criterion is met. This is a new type of result which holds primarily since the network consists of an infinite collection of queues.
Let , where is given in Proposition 1.3. If the initial condition satisfies , then the queue length process converges weakly to the minimal stationary solution.
This theorem is proved in Section 9. As the queue lengths are positive integer valued, and the dynamics admits a form of monotonicity, every fixed finite collection of coordinates also converges to the minimal stationary solution in the total variation norm in the above Theorem, which is stronger than just weak convergence. We further examine sensitivity to initial conditions in Theorem 1.7 by constructing examples where the queue lengths diverge, even though the stability criterion is met. To state the result, we need a natural ‘irreducibility’ condition on the interference sequence .
The interference sequence is said to be irreducible if, for all , there exists and , not necessarily distinct, such that and for all .
This is a natural condition which ensures that we cannot ‘decompose’ the grid into many sets of queues, each of which does not interact with the queues in the other group. In the extreme case, this disallows the case when for all , in which case the network can be decomposed into an infinite collection of independent Processor Sharing queues.
For all , , and irreducible interference sequences , and even when the stability criterion holds, there exists
A deterministic sequence such that if the initial condition satisfied for all , then the queue length of satisfies almost surely.
A distribution on such that if the initial condition is an i.i.d. sequence with each , being distributed as independent of everything else, then the queue length of (or any finite collection of queues) satisfies almost surely.
This theorem is proved in Section 10. Based on the proof of this theorem, we make the following remark.
For all , the support of in statement above can be made arbitrarily sparse, i.e. for any sequence such that , the initial conditions can be chosen, such that , yet the queue lengths converge almost surely to infinity.
The above theorem is qualitative in nature, as it only establishes the existence of bad initial conditions, but does not provide estimates for how large this initial condition must be. In this regard, we include Proposition1.9, which pertains to the deterministic starting state in the simplest non-trivial system, namely the case of , and the interference sequence being such that for and otherwise. This simplest non-trivial example already contains the key ideas and hence we present the computations involved explicitly here. In principle, one can provide a quantitative version of the above theorem in full generality. However, we do not pursue this here as they involve heavy calculations without additional insight into the system.
Consider the system with and the interference sequence if and otherwise. Let be arbitrary deterministic non-negative integer valued sequence such that . If the initial condition, for , and otherwise, then almost surely.
This proposition is proved in Appendix LABEL:appendix:bad_quant_proof. Regarding the converse to stability, we prove the following result in Theorem 1.11, which establishes that the phase-transition at the critical is sharp, at-least in certain cases, and we conjecture it to be sharp for all cases. In order to state the result about transience, we require the following definition about the monotonicity of the interference sequence.
The interference sequence for the dynamics on the one dimensional grid is said to be monotone if for all , holds true.
The following theorem is the main result regarding instability.
For the system with and monotone interference sequence, if , then the system is unstable.
1.1 Open Questions and Conjectures
We now list some conjectures and questions that are left open by the present paper. The first one concerns the finiteness of the second moment of the minimal stationary solution in Proposition 1.3. Our techniques cannot yet answer whether the second moment of queue lengths is bounded in the entire stability region. Based on some numerical evidence, we put forth the following conjecture.
If , then .
Thus, if Conjecture 1.12 is verified, this would imply from Proposition 1.2 that the minimal stationary solution is indeed the unique stationary solution to the dynamics that admits finite second moments. Furthermore, from all bounded initial conditions, the queue length process will converge to this unique stationary solution if . In this regard, three natural interesting questions arise - one concerning what other moments of stationary queue lengths are finite, one regarding correlation decay and another on existence of other stationary solutions.
For each , what moments of are finite ?
How does the correlation decay as ?
Does the dynamics admit stationary solutions other than the minimal one ? If so, do there exist initial conditions such that the law of the queue lengths converge to them ?
In regard to the above question, an interesting sub-problem is to asses whether there exists solutions that are almost surely finite but having no first or second moments. In regard to establishing transience, a natural open question in light of Theorem 1.11 is to extend this result to higher dimensions and non monotone interference sequence. We make the following conjecture.
For all and interference sequence , if , then the system is unstable.
1.2 Main Ideas in the Analysis
The key technical challenge in analyzing our model is the positive correlation between queue
lengths, which persist even in the model with infinitely many queues
(see also Figure LABEL:fig:queue_corr). As mentioned, our system of queues is neither reversible,
nor falls under the category of generalized Jackson networks. Thus, our model does not admit a product
form stationary distribution, even when there are finitely many queues.
In particular, the model has no asymptotic independence properties as those encountered in
“mean-field models” (such as the supermarket model ).
The correlations across queues is intuitive, since if a queue has a large number of customers,
then its neighboring queues will receive lower rates, and thus they will in turn build up.
Therefore in steady state, if a particular queue is large, most likely, its neighboring queues
are also large (see also Figure LABEL:fig:queue_corr).
To prove the sufficient condition for stability,
we first study finite space-truncated torus systems in Section 5. In words, we restrict
the dynamics to a large finite set , and study its stability by employing fluid-like
and Lyapunov arguments. For this model, we write down rate conservation equations in Section 6 and solve for the mean
queue-length of this dynamics. This section contains the key technical innovations in this paper. The rate conservation equations turn out to be surprisingly fruitful, as we are able to obtain an
exact formula for the mean queue length. This formula also gives as a corollary, that the queue length
distributions are tight, as the size of the truncation increases to .
In Section 7, we then show that we can take a limit as increases to all of
and consider the stationary solution as an appropriate
limit of the stationary solutions of the space-truncated system. The central argument in this section
is to exploit the many symmetries, the monotonicity of the dynamics and the aforementioned tightness
to arrive at the desired conclusion. We furthermore, apply a similar rate conservation equation
for the infinite system, which along with monotonicity
arguments, establishes the uniqueness of stationary solutions with finite second moments.
An open question in our model is the existence of other stationary solutions with infinite mean, but being almost surely finite.
To study the convergence from different initial conditions, we employ different arguments, again exploiting the symmetry and monotonicity in the model. To show that stability implies convergence from bounded initial conditions, we define a modified -shifted system in Section 4.2. It is a model having the same dynamics as our original model, except that the queue lengths do not go below , for some . We carry out the same program of identifying a bound on the first moment on the minimal stationary solution to the shifted dynamics by analyzing similar rate conservation equations as for the original system. We then exploit the monotonicity and the fact that a stationary solution with finite mean is unique, to conclude that stability implies convergence to the minimal stationary solution from bounded initial conditions. In order to identify initial conditions from where the queue length can diverge even though the stability condition holds, we first consider a simple idea of ‘freezing’ a boundary of queues at a large distance , to a ‘large value’ around a typical queue, say , and then consider its effect on the queue length at the origin. By freezing, we mean, there are no arrivals and departures in those queues, but a constant number of customers that cause interference. We see that by choosing sufficiently large, this wall can influence the stationary distribution at queue . We leverage this observation, along with monotonicity, to construct both deterministic and random translation invariant initial conditions such that queue lengths diverge to even though the stability condition holds. This proof technique is inspired by similar ideas developed to establish non-uniqueness of Gibbs measures in the case when the state space of a particle is finite, while our methods and results bear on the case when the state space is countable.
1.3 Organization of the Paper
The rest of the paper is organized as follows. In Section 2, we survey related work on infinite queueing dynamics and place our model in context. We then start the technical part of the paper by providing the complete mathematical framework in Section 3, where we formalize the model and the questions studied. We also state the monotonicity properties satisfied by the model, which are crucial throughout. We discuss certain generalizations of the model in Section 4. We subsequently then proceed to state and prove the main results in this paper. In Section 5, we introduce the space truncated finite system version of our model and analyze it using fluid-like arguments. The space truncated system can be viewed as a certain finite dimensional approximation of our infinite dimensional dynamics. The key technical part in that section is in writing and analyzing certain rate-conservation equations in Section 6, which give an explicit formula for the mean queue length in steady state. Based on the results in this section, we complete the proof of Theorem 1.1 in Section 7, where we establish that the minimal stationary solution of our dynamics is a limit of the stationary solutions of the finite approximations in an appropriate sense. Subsequently in Section 8, we prove Proposition 1.2. In Section 8, we prove Theorem 1.5. The proof of Theorem 1.7 which establishes the presence of bad initial conditions is then done in Section 10. The proof of Theorem 1.11 establishing the converse to stability is carried out in Section 11. For ease of exposition, we delegate many details of the proof to the Appendix while outlining the key ideas in the body of the paper. For instance, the details on construction of the process are forwarded to the Appendix.
2 Related Work
Our study is motivated by the performance analysis of wireless networks which has a large and rich literature
(see for ex.  ,  and the references therein).
Our model is an adaptation of the Spatial Birth-Death model proposed in , where a dynamics
of this type was introduced on a compact subset of the Euclidean space. Although that paper has a phase-transition
result similar to ours for stability, the analysis sheds no light on whether the result holds true
for an infinite network. In this paper, we answer in the affirmative in Theorem 1.1, that the same result indeed holds in the infinite
discrete network case. From a mathematical point of view, the tools and techniques of ,
which rely on fluid limits, are very different from those discussed in the present paper.
The results are quite different too, with new quantitative results (like the closed
form for the mean queue size) and new qualitative phenomena such as the existence of
multiple stationary solutions being reachable depending on the initial conditions.
Since some of the new properties are directly linked to the fact that there are infinitely many queues, we thought it appropriate to briefly survey the mathematical literature on queueing models consisting of infinitely many queues interacting through some translation invariant dynamics. A model related to ours is the so called Poisson Hail model which has been studied in a series of papers ,,. The discrete version of this model consists of a collection of queues on , where the queues interact through their service mechanism in a translation invariant manner. In this model, the customer at a queue occupies a ‘footprint’ and when being served, no other customer in the queues belonging to its footprint is served. In contrast, in our model a customer slows down the customers in neighboring queues, but does not block them. Another set of papers close to ours is , , and . These papers analyze an infinite collection of queues in series. The main results are connections with last passage percolation on grids. A similar model to this is studied by , where analogues of Burke’s theorem are established for a network of infinite collection of queues on the integers. There is also a series of papers on infinite polling systems. The paper  considers a polling model with an infinite collection of stations, and addresses questions about ergodicity and positive recurrence of such models. In a similar spirit,  considers infinite polling models and establishes the presence of many stationary solutions leveraging the fact that the Markov process is not finite-dimensional. The dynamics in these polling systems are however very different from ours. The paper of  also introduced a nice problem with translation invariant dynamics, but only analyzed the setting with finitely many queues. The paper of  introduced an elegant problem on Jackson queueing networks on infinite graphs. However, the stationary distribution there admits a product-form representation, which is very different from our model in the present paper. The paper 
studies translation-invariant dynamics on infinite graphs arising from combinatorial optimization, which again falls broadly in the same theme, but for a fundamentally different class of problems. Queueing like dynamics on an infinite number of nodes are also studied, though under different names, in the interacting particle system literature in the sense of. The most well known instance of interacting particle system connected to queueing is probably the TASEP. Another fundamental class of interacting particle system exhibiting a positive correlation between nodes (like our model) is the ferromagnetic Ising model. The first difference is that the state-space of a node is not compact (i.e., , since the state is the number of customers in the queue) in our model, whereas it is finite in these models. Another fundamental difference between our model and these is the lack of reversibility. The common aspects are the infinite dimensional Markovian representation of the dynamics, the non uniqueness of stationary solutions, and the sensitivity to initial conditions. Infinite queueing models are also central in mean-field limits. In the literature on mean-field queueing systems ([31, 18, 13]) the finite case exhibits correlations among the queue lengths thereby making them difficult to analyze. However, in the large number of node limit, one typically shows that there is ‘propagation of chaos’. This then gives that the queue lengths become independent in the limit. This independence can then be leveraged to write evolution equations for the limiting dynamics which can be analyzed. Such mean-field analysis have recently become very popular in the applied literature (for ex. ,). Our model differs fundamentally from the above models in many aspects. First, unlike the mean-field models described above, we can directly define the limiting infinite object, i.e., a model with infinitely many queues. Secondly and more crucially, our infinite model does not exhibit any independence properties in the limit, i.e., queue lengths are positively correlated even in the infinite model. This is why we need different techniques to study this model. Our main technical achievement in this context is to introduce coupling and rate conservation techniques not relying on any independence properties.
3 Problem Setup
In this section, we give a precise description of our model in subsection 3.1 and demonstrate certain useful monotonicity properties it satisfies in Subsection 3.3. We then precisely state the definition of stability in Section 3.4 and the notion of stationary solutions to the dynamics in Section 3.5.
Our model is parametrized by and an interference sequence which is a non-negative sequence. This sequence satisfies , for all and , i.e., is finitely supported. We also impose the sequence to be irreducible, which gives that for all , there exists and not necessarily distinct, such that and for all . To describe the probabilistic setup, we assume there exists a probability space that
contains the stationary and ergodic driving sequences .
For each , is a PPP of intensity
on , independent of everything else and is a PPP of intensity on , independent of everything else.
Our stochastic process denoting the queue lengths
will be constructed as a factor of the process .
The process encodes the fact that,
at times , there is an arrival of a customer in queue . Thus the arrivals to queues form PPPs of intensity and are independent of everything else.
The process encodes
that there is a possible departure from queue at time , with an additional independent
random variable provided by . To precisely describe the departures, we define the interference at a customer in queue at time as equal to . A customer, if any, is removed from queue at times if and only if .
In other words, conditionally on the state of the network at time , we remove a customer from queue at time with probability
, independently of everything else.
Thus we see that conditionally on the network state at time , the instantaneous rate of departure from any queue at time is , independently of everything else. Observe that since , if , then necessarily, .
We further assume (without loss of generality) that the probability space equipped with a group of measure preserving functions from to itself where denotes the ‘time shift operator’ by . More precisely is the same driving sequence where each of the arrivals and departures are shifted by time in all queues, i.e., if and , then and , for all . We also assume that the system is ergodic, i.e. if for some event , if for all , then .
3.2 Construction of the Process
Before analyze the above model, one needs to ensure that it is ‘well-defined’. We mean that our model is well defined if given the initial network state , any time and any index , we are able to construct the queue length unambiguously and exactly. In the case of finite networks (i.e., networks with finitely many queues), the construction is trivial: almost surely, one can order all possible events in the network with increasing time, and then update the network state sequentially using the evolution dynamics described above. Such a scheme works unambiguously since, almost surely, all event times will be distinct and in any interval , there will be finitely many events. The main difficulty in the case of infinite networks is that there is no first-event in the network. In other words, in any arbitrarily small interval of time, infinitely many events will occur almost surely and hence we cannot construct by ordering all the events in the network. However we show in Appendix LABEL:appendix_construction that in order to determine the value of any arbitrary queue at any time , we can effectively restrict our attention to an almost surely finite subset and determine by restricting the dynamics to to the interval . This is then easy to construct as it is a finite system. Thereby we can determine unambiguously. Such construction procedures are common in Interacting Particle systems setup (for example, the book of ). Nevertheless, we present the entire details of construction in Appendix LABEL:appendix_construction for completeness.
We establish an obvious but an extremely useful property of path-wise monotonicity satisfied by the dynamics. Note that our model is not monotone separable in the sense of  since the dynamics does not satisfy the external monotonicity condition. Nonetheless, the model still enjoys certain restricted forms of monotonicity, which we state below. We only highlight the key idea for the proof and defer the details to Appendix LABEL:appendix_monotonicity.
If we have two initial conditions and such that for all , , then there exists a coupling such that for all and all almost surely.
The proof is by a path-wise coupling argument, where the two different initial conditions are driven by the same arrival and potential departures. The key idea the following. At arrival times, the ordering will trivially be maintained. Consider some queue and time where there is a potential departure. If , then, since at most one departure occurs, the ordering will be maintained. But if , then the rates and hence the ordering will again be maintained. This observation can be leveraged again to have the following form of monotonicity.
For all initial conditions , for all , all , and all , is coordinate-wise larger in the true dynamics than in the dynamics constructed by setting for all .
3.4 Stochastic Stability
We establish a law stating that either all queues are transient or all queues are recurrent (made precise in Lemma 3.3 in the sequel). Thus, we can then claim that the entire network is stable if and only if any (say queue indexed without loss of generality) is stable (made precise in Definition 3.4 in the sequel). To state the lemmas, we set some notation. Let and be arbitrary and finite. Denote by the value of the process seen at time when started with the empty initial state at time , i.e., with the initial condition of for all . Lemma 3.1 implies that for every queue , and for almost-every , we have is non-decreasing for every fixed . Thus, for every , and every , there exists an almost sure limit . From the definition, this limit is shift-invariant, i.e., for all , we have almost surely.
We have either or .
The proof follows from standard shift-invariance arguments which we present here for completeness. Since for all and all , , we have that this lemma implies for all , either or .
It suffices to first show that for any fixed , we have
. Assume that we have established for some
(say without loss of generality that) .
From the translation invariance of the dynamics, it follows that, for all ,
we have . Thus,
if , then .
Similarly, if , then .
Thus to prove the lemma, it suffices to prove that .
The key observation is that the event is invariant under for all . To show this, first notice that from elementary properties of PPP, we have that for every and every compact set , a.s.. Now for any , we have , which is finite almost surely if almost surely. Similarly for every , , which again implies that is almost surely finite if . Thus, for all , we have , which from ergodicity of implies and thus the lemma is proved. ∎
The following definition of stability follows naturally.
The system is stable if almost surely. Conversely, we say the system is unstable if almost surely.
Observe that the definition of stability does not require to be finite. In words, we say that our model is stable if when starting with all queues being empty at time in the past, the queue length of any queue stays bounded at time when letting go to infinity. This definition of stability is similar to the definition introduced for example by  in the single server queue case. A nice account of such backward coupling methods can be found in .
The main result in this paper is to prove that if , then the system is stable (Theorem 1.1). Moreover, in this case, we compute exactly the mean queue length in steady state, i.e., an explicit formula for (Theorem 1.1) and by shift-invariance it is equal to . We also conjecture this condition to be necessary, i.e., if , then almost surely. We are unable to prove this conjecture yet, but prove it for the special case of in Theorem 1.11.
3.5 Stationary Solutions
A probability measure on is said to be invariant for the dynamics if, whenever is distributed according to independently of everything else, then, for all , the random variables are also distributed as .
From the definition of our dynamics, it follows that any stationary solution must be translation invariant in space. Notice that the driving sequence is translation invariant on , i.e., for all , is equal in distribution to . For every , let be the bijection that for all , sends to . From translation invariance of the driving data, any invariant measure for the dynamics must be invariant under the transformation of by , for all .
Moreover, as our network is not finite-dimensional, stability in the sense of Definition 3.4
does not imply ergodicity in the usual Markov chain sense. In particular, it does not imply that stationary distributions are unique, and starting from any initial condition on, the queue lengths converge in some sense to the minimal stationary distribution considered in Definition 3.4. Stability only implies the existence of a stationary solution, namely the law of is an invariant measure for the dynamics. However, uniqueness is not granted and one of our main results in Proposition 1.2 bears on this. Moreover, convergence to stationary solutions from different starting states is more delicate as evidenced in Theorems 1.5 and 1.7.
4 Model Extensions
In this section, we introduce two natural extensions to the model not considered in Section 3. We show that similar results as for our original model hold, albeit with a little bit more notation. Hence we separate this discussion from the main body of the paper with proofs deferred to the Appendix as the key ideas are the same as for the model described earlier.
4.1 Infinite Support for the Interference Sequence
We consider here a system where is such that for all with having infinite cardinality but being summable, i.e., . In this case as well, we can uniquely construct the system in a sense as a limit of finite systems with finite truncation. The following proposition encapsulates the main results
Consider such that has infinite cardinality, and . Then the dynamics is well defined.
To show the existence of the dynamics, we introduce a sequence of systems, with the th system evolving according to the dynamics described in Section 3 with the interference sequence being . This interference sequence satisfies all the conditions specified in Section 3 and hence the dynamics can be constructed. We now construct the infinite dynamics sequentially as follows. Consider any arbitrary initial conditions . For this system, for every , we can define the process , , which is the process corresponding to the truncated interference sequence . Now, it suffices to assert that at each arrival and potential departure event at queue , we can unambiguously decide on how the system with infinite interference support evolves. The evolution due to an arrival event is easy, we just add an customer to queue . At the first potential departure event at queue , with the independent mark given by , we have to decide whether to remove a customer or not. Now, at this time, we can do this unambiguously by deciding whether or not. The existence of the almost sure limit is guaranteed by monotonicity. In words, is non-decreasing in and the queue lengths are non-decreasing in for each and . The numerator can be deduced without resorting to limits as this is the first potential departure after time in queue . Hence we can unambiguously decide on the outcome of the first potential departure event at queue after time . Now, by induction, we can construct the sample path of any queue over any finite time interval, thereby establishing that the dynamics is well defined. ∎
Based on the construction described above, it is not immediately clear that a stability region even exists for the case with infinitely supported interference sequence. The following proposition gives an alternative representation of the dynamics as a point-wise limit of dynamics with truncated interference sequence.
Consider an initial condition and interference sequence with being infinite and such that . Consider the sequence of processes each driven by the -truncated interference sequence dynamics. Then for each and finite, we have almost surely.
For every queue and finite time , there are only finitely many potential departure events almost surely in the interval . From Proposition 4.1, we know that at each instance of a potential departure at queue , we take a limit in , the truncation length to determine whether or not to remove a customer. However, since there are only finitely many events in the time interval , one can make the limit uniform to conclude that for all and all , almost surely. ∎
Based on the construction outlined above, one can extend the existence of a stationary solution to the case when the interference sequence has an infinite support. Indeed, it is not a corollary, as, by the construction of the infinite support dynamics as a point-wise limit of the truncated interference systems’ dynamics, the existence of a stability region is not granted.
Suppose that the interference sequence is such that is infinite with . Under this conditions, if , then there exists a minimal stationary solution with .
The proof is deferred to Appendix LABEL:appendix:infinite_support_proof. However, establishing uniqueness of stationary regime in this case is slightly more delicate and we leave it to future work. The main difficulty being that writing down rate-conservation equations as done in Section 6 when the interference support is infinite is not obvious.
4.2 -Shifted System
In this subsection, we introduce a model of queues which ‘reflect’ at level . In other words, we consider a dynamic which will forbid any departures from a queue if it has or more customers at any point of time. Note that the original model we describe is the shifted, or the model reflected at . Thus, if is the stochastic process corresponding to the shifted dynamics for some , then the instantaneous rate of departure from any queue at time is then given by
For the purposes of this section, we assume that , although one could extend this definition to include the case of as well by the ideas introduced in Section 4.1. In this case of finitely supported interference sequence, the process can be formally defined through a Poisson clock similar to that used in Section 3. The main result for the general shifted system is the following.
If , then for all , the -shifted dynamics is stable. Moreover, the minimal stationary solution satisfies
If , where the constant , then we have .
The proofs are deferred to Appendix LABEL:appendix:K_shifted_proof. The -shifted dynamics is introduced as it will later be used to show convergence from bounded initial conditions to the stationary regime of the original initial dynamics, i.e., it is used as a tool to prove Theorem 1.5 in Section 8. One can also naturally extend the -shifted dynamics to accommodate the case when the interference sequence has infinite support satisfying , but we do not do so here.
5 Space Truncated Finite Systems
In this section, we study a finite version of the aforementioned infinite queueing network. For any , we consider two -truncated systems, both of which are obtained by restricting the dynamics to the set , the ball of radius centered at . For notational convenience, we shall drop and denote by the ball of radius centered at . For every , we define two truncated dynamics, and . The process evolves with the set ‘wrapped around’ to form a torus. More precisely, the process is driven by . The arrival dynamics is the same as for the infinite system described in Section 3 wherein, for all
, at each epoch of, a customer is added to queue . The departure dynamics is driven by as before, but we treat the set as a torus. More precisely, given any , define , where the modulo operation is coordinate-wise. Thus, at any time , and any , the rate at which a departure occurs from queue at time in the process is . Since is finite, the stochastic process is a continuous time Markov process on a countable state-space, i.e., on . Moreover, since the jumps are triggered by a finite number of Poisson processes, this chain has almost surely no-explosions. Similarly, the process is driven by the arrival data as before, but this time the set is viewed as a subset of and in particular the ‘edge effects’ are retained. From the monotonicity in the dynamics, we have the following proposition.
For all , there exists a coupling of the processes , and such that for all , and all , we have and almost surely.
The key result in this section on the space truncated systems is the following theorem.
For all and , the Markov process is positive recurrent. Let denote the stationary queue length distribution on of any queue and let be distributed as . Then there exists a possibly depending on such that .
The symmetry in the torus implies that the marginal stationary queue length distribution of any queue , , is the same for all .
The existence of an exponential moment yields that all power moments of are finite.
In view of Proposition 5.1, if , then for all , the process is positive recurrent. Moreover, for all , the stationary distribution of , denoted by , is such that there exists a possibly depending on satisfying , where is distributed according to .
Proof of Theorem 5.2
In order to carry out the proof, we define a modified dynamics
which is coupled with the evolution of .
For notational brevity, in this section, we will drop the superscript since all systems of interest
are on a fixed torus . In particular, we write and
We construct the modified dynamics such that it satisfies