Flow level convergence and insensitivity for multi-class queueing networks

We consider a multi-class queueing network as a model of packet transfer in a communication network. We define a second stochastic model as a model document transfer in a communication network where the documents transferred have a general distribution. We prove the weak convergence of the multi-class queueing process to the document transfer process. Our convergence result allows the comparison of general document size distributions, and consequently, we prove general insensitivity results for the limit queueing process.


Introduction
We present a result that formally demonstrates the separation of timescales between a communication model, where discrete packets are transferred, and a second model, where documents are transferred elastically. Such convergence results are distinct from the fluid and diffusion limit results which are typically applied to queueing processes. The result applies to the transfer of documents with generally distributed sizes and quasi-reversible queues. This extends the convergence proof which was previously applied to the simpler case of exponential file sizes and processor sharing queues [10]. By generalising this result, we can formally prove insensitivity results about the limit queueing system.
Our prelimit model is a quasi-reversible multi-class queueing network as considered by Baskett et al. [1], Kelly [6]. We endow this model with a specific routing structure. Documents for transfer on different routes of the network, arrive as a Poisson process. A document consists of a number of packets which are transferred one-by-one across their route.
Our limit model is a bandwidth sharing model. These stochastic processes model the elastic transfer of documents in a communication network. Bandwidth sharing models were first introduced by Roberts and Massoulié [7]. In this paper, we are particularly interested in bandwidth sharing models associated with multi-class queueing networks. Models of this form were first considered by Bonald and Proutiere [3]. Bonald and Proutiere demonstrated that these models were insensitive to documents with a phase-type distributions.
Our convergence result will allow us to extend this result to documents of any non-atomic distribution.
The formal convergence proof uses a coupling argument. The proof demonstrates weak convergence in the Skorohod topology of the number of documents in transfer of the multi-class queueing network to that of a bandwidth sharing model. The prelimit models considered here are well understood product form queueing networks. Even so such explicit product form results are not required, the arguments used to prove this separation of timescales result are general and could be applied in the analysis of a diverse range of queueing models.

An informal description of the results
We consider a multi-class queueing network. The queueing network processes documents along different routes. Arriving documents are divided into packets which are sent across the network one-by-one. Once all the packets in a document are sent the document departs.
We could describe the state of this queueing network in several ways: we could consider the explicit behaviour of the network, Q, by storing residual document sizes and the location of packets on their routes; or, we could consider the states of packets onlyQ, in doing so, we would ignore information about the residual sizes of documents; or, finally, we could consider the residual document sizes only Y , and thus ignore precise information about the positions of packets on their routes. These descriptions form an explicit description, a packet-level description and a flow-level description of our network.
We are interested in the interactions of a queueing network at these levels. In particular, the rate at which packets are transferred in a modern communication network is often an order of magnitude larger than the time it takes to transfer a document. Thus, given the number of documents in transfer, we should be able to abstract away the packet level behaviour of the network. More formally, conditional on the number of documents in transfer, the quick transition of packets within the network should imply that the distribution of packets with in the network converges quickly to its stationary distribution, and thus the processing of documents is best described by the stationary behaviour of the packet-level queueing network.
We mathematically demonstrate this by taking a sequence of multi-class queueing networks Q (c) . Along this sequence, we increase the rate that queues process packets by a factor c and accordingly increase the sizes of documents by a factor c. In this regime, packet transitions occur on a time scale of order O(1/c), whilst document transfers remain at a timescale of O(1). Thus in this limit, in between document arrival-departure events, the packet-level stateQ will converge to its stationary distribution and document transfers will receive a linear rate of transfer given by the stationary throughput of the packet-level network.
In this paper, we show that in this limit. If we let the initial state and document sizes converge, then the times at which documents arrive and depart the network will converge. The formal result, Theorem 5.1 proves that the flow-level state of queueing networks converges in the Skorohod topology to a flow-level model of the network.
With this result we can consequently prove a general insensitivity result for these flow-level networks. In this context, insensitivity means that the stationary distribution of number of documents in transfer only depends on the mean size of documents. We can demonstrate this property for discrete document sizes in the prelimit networks. The Skorohod convergence implies that the stationary distribution for the prelimit network converges to the stationary distribution for the limit network. Consequently in Corollary 6.1, we demonstrate that the limit queueing network is insensitive amougst all non-atomic document size distributions.

Organisation
In Section 2, we give basic notation used throughout the paper. In Section 3, we review results on some well understood product form queueing networks. In Section 4, we define the bandwidth sharing networks which will be the limit of our mutli-class queueing networks. We, also, define what it means for these networks to be insensitive. In Section 5, we prove the main convergence result Theorem 5.1. Finally, in Section 6, we prove the insensitivity of these queueing networks.

Notation and network structure
We let the finite set J index the set of queues in a network. Let J = |J |. A route through the network is a non-empty set of queues. Let I ⊂ 2 J be the set of routes. Let I = |I|. For each route i = {j i 1 , ..., j i ki } ∈ I, we associate an order (j i 1 , ..., j i ki ). We allow for queues to be repeated in our route order. For i ∈ I and j ∈ i, we let ζ ji ∈ N be the number of times queue j is included in ordering (j i 1 , ..., j i ki ). Also we define the set of queue-route incidences, K := {(j, i) : i ∈ I, j ∈ J , j ∈ i} and let K = |K|. We will view a multi-class queueing network model as transferring a number of documents across the different routes of the network. The vector n = (n i : i ∈ I) ∈ Z I + will denote the number of documents in transfer across the routes of the network. We also let the vector m = (m ji : (j, i) ∈ K) ∈ Z K + refer to the number of packets in transfer across each route at each queue. That is m ji is the number of packets on route i at queue j. We also define the number of packets in transfer at a queue to be For each n ∈ Z I + , we define S(n) = {m ∈ Z K + : j:j∈i m ji = n i ∀ i ∈ I}, that is the set of queue sizes with n documents in transfer on each route.

Multi-class queueing networks
In this section, we present some well understood queueing networks that will be studied subsequently. In order to model the transfer of documents across a packet switching network, we define a special case of these queueing networks where packets have a specific routing structure. We then the define closed queueing networks as described in [6,Section 3.4].

Multi-class queue
First, we define what we will call a multi-class queue. We consider a single queue j from a set of queues J . We call the customers of this queue packets. The queue will receive packet arrivals from different classes. The set of classes will consist of a set of packet route choices C. 1 Packets occupy different positions within a queue. Given there are m j ∈ Z + packets at queue j packets may occupy positions 1, 2, ..., m j . Packets of each route at the queue require an independent exponentially distributed service requirement with mean 1. Given there are m j ∈ Z + packets at queue j, the total service devoted to packets is given by φ j (m j ). We assume φ j (m j ) > 0 if m j > 0. This service is then divided amongst packets within the queue. Given there are m j ∈ Z + packets at queue j, a proportion γ j (l, m j ) of service is devoted to the packet in position l ∈ {1, ..., m j } of queue j. Since γ j (·, m j ) represents a proportion, mj l=1 γ j (l, m j ) = 1, m j ∈ N.
Upon completing its service the packet at position l will leave the queue and the packets at positions l + 1, ..., m j will move to positions l, ..., m j − 1, respectively. We assume packets of class c ∈ C will arrive at the queue from independent Poisson processes of rate ρ jc . Given there are m j packets at the queue an arriving packet will move to position l ∈ {1, ..., m j + 1} with probability δ j (l, m j + 1). Once again as δ j (·, m j + 1) represents a proportion mj +1 l=1 δ j (l, m j + 1) = 1, When a packet arrives at position l the packets in positions l, ..., m j will move to positions l + 1, ..., m j + 1, respectively. Let q j = (c j 1 , ..., c j mj ) ∈ I mj , for m j > 0, give the state of queue j. Let function, T c ·,(j,l) denote the arrival of a class c packet to position l in queue j and let function T c (j,l),· denote the departure of a class c packet in position l. Thus the state of this queue forms a continuous-time Markov chain with transition rates given by, otherwise.
The queue itself will not discriminate between different packet's classes and thus the stationary distribution of the queue size will be oblivious to different packets' route type. Ignoring packet classes, when stationary M j the Markov chain recording the total number of packets at the queue is reversible. Given m j , routes of the packets in positions 1, 2, ..., m j are independent. The probability a packet in a given position is from route i is ζjiρi r∋j ζjr ρr . Thus letting Markov chain Q j record the position and routes of packets at queue j and letting Q j = ∪ ∞ mj =1 I mj gives all possible states of the queue, we can calculate the stationary distribution of the queue.  [5]). A stationary multi-class queue is quasi-reversible and the stationary distribution of Q j must be Moreover, for j ∈ J , the process (M jc : i ∈ C) giving the number of packets of each route type at queue j, has stationary distribution The combinatorial term in (3) is required as the probability distribution (2) ignores the order of packets within the queue.

Multi-class queueing networks
A multi-class queueing network with spinning (MQNwS) is a multi-class network of quasi-reversible queues with the following class routing structure. The class of a packet is of the form c = (i, k, y) where i ∈ I records the route the packet is on, k ∈ N records the stage of the packet on route i and y ∈ N records the packet's residual document size, that is the remaining number of times the packet must traverse its route. Routing through classes occurs in the following way. If route i has associated route order (j i 1 , ..., j i ki ) then as a Poisson process of rate ν i P(X i = x) class (i, 1, x) packets arrive at queue j i 1 . Here X i is a random variable with values in N and with mean µ −1 i < ∞. For k = 1, ..., k i − 1, a class (i, k, y) packet on departing queue j i k will join queue j i k+1 and become a class (i, k + 1, y) packet. For a packet that has completed service at the final queue on route i ∈ I and has not been fully processed through the network, that is a packet of class (i, k i , y) with y > 1, the packet will join queue j i 1 as a class (i, 1, y − 1) packet. For a route i ∈ I packet that has completed its service at the final queue k i and has been fully processed through the network, that is of class (i, k i , 1), the packet will depart the network. In addition, we let the constant ζ ji ∈ Z + give the number of times a packet visits queue j each time it traverses route i. Finally, we define traffic intensities ρ i = νi µi for each i ∈ I. We can interpret this routing structure in two ways. First, we could consider each packet on route i to arrive as a Poisson process and to repeat its route a number of times that is independent and with distribution equal to X i . This interpretation leads us to think of a packet as spinning around its route a random number of times. Second, we could consider the network to be transferring documents. Documents which require to be transferred across route i arrive as a Poisson process as of rate ν i . Each document consists of a number of packets, that is independent and with distribution equal to X i . These packets are then sent across the network one by one until the document is transferred.
For a Markov process description of a MQNwS we record its explicit state: gives the class of each customer in each occupied position in queue j. Here the class c j (l) = (i j (l), k j (l), y j (l)) records the route, stage and residual document size associated with the l-th packet in queue j. We let Q define the set of all possible states for this explicit description of our queueing network.
Recalling that m ji is the number of route i packets in transfer at queue j and that n i is the number of route i documents in transfer. As each document has one packet in transfer in the network at any point in time We define two further descriptions of the state of a MQNwS: the packet level state and the flow level state.
We define the packet level state of a multi-class queueing network with spinning to be,q = (q j : j ∈ J ), whereq j = (c j (1), ...,c j (m j )) and wherẽ c j (l) = (i j (l), k j (l)) records the route and stage associated with the l-th packet in queue j. We letQ define the set of all possible packet level states for this description of our queueing network. The packet level state of a MQNwS is concerned with the position and route of packets but not of the state of document transfer. Similarly the flow level state is interested in the state of document transfer and not in the specific position of packets.
We define the flow level state of a MQNwS to be, given by vector imsart-generic ver. 2011/01/24 file: insenscov.tex date: February 16, 2022 Here, we order elements so that y ik ≤ y ik+1 for all k = 1, ..., n i − 1. Note, we record no information about each packet's position on its route. As described above n i refers to the number of route i documents in transfer on route i and k indexes each specific packet in transfer on route i.The number y ik is the residual document size of the k-th document in route i, that is the number of packets yet to be transferred from the document. We let Y be the set of flow level states achievable by a MQNwS. The processes associated with the packet level or flow level state of a multiclass queueing network with spinning need not be Markov. However, these state descriptions will be useful for proving weak convergence results. For this purpose we define on Y the norm

Stationary behaviour
We now calculate certain quantities associated with the stationary distribution of a MQNwS. As a direct consequence of known reversibility results [6, Theorem 3.1], we can calculate the stationary distribution of a MQNwS.
Theorem 3.1. The explicit state of an ergodic multi-class queueing network with spinning has stationary distribution, provided Proof. A multi-class queueing network with spinning is a network of quasireversible queues with a deterministic routing structure. It is known, [6, Theorem 3.1], that a network of quasi-reversible queues has a stationary distribution where, and where β jc solves the traffic equations imsart-generic ver. 2011/01/24 file: insenscov.tex date: February 16, 2022 Here ν jc is the arrival rate of class c customers at queue j and p jc,ld gives the packet routing probabilities, which in our case are, for c = (i, k, y), In this way, packets are transferred between queues, the next packet is injected at the ingress and document departures occur.
So, all that is needed is to verify thatβ j,c solves the traffic equations along our deterministic path. Observe that, for k > 1,β j,(i,k,y) =β j,(i,k−1,y) and, for This verifies the traffic equations are satisfied and hence gives the result.
The condition (6) is the necessary and sufficient for a multi-class queueing network with spinning to be ergodic and thus is equivalent to the assumptions ergodicity in subsequent results. We encapsulate this in the following assumption: Assumption 1. Unless stated otherwise we assume a multi-class queueing network with spinning satisfies the following necessary and sufficient condition for ergodicity: The following three corollaries are a consequence of Theorem 3.1. Each of these results require summing over an appropriate set of states. For example, from Theorem 3.1, we can calculate the stationary distribution of the number of packets in transfer along each route at each queue.
Proof. We know from Theorem 3.1 that our queueing network has a stationary distribution equal to that of a simpler queueing network. In this simpler queueing network, each queue j behaves independently in isolation and where class c = (i, k, y) packets, with j i k = j, arrive at queue j as a Poisson process of rate ν i P(X i ≥ y). Let us work with this simpler but equivalent queueing model. In this model, route i packets arrive into queue j as a Poisson process of rate So queue j will have an independent stationary distribution exactly of the form of (1). So, as in Section 3.1, by ignoring packet positions we can gain distribution (2) for each queue. Thus by independence equation (8) holds for the network.
Remark 1. Observe that distribution (8) only depends on the distribution of X i through its mean 1 µi . Thus the stationary distribution of a MQNwS only depends on the distribution of document sizes through their mean size. This suggests a form of insensitivity holds. This point is noted by Massoulié and in the thesis of Proutiere [9]. Similar observations are made earlier in [6] for individual queues. We will discuss this observation in more detail in Chapter 3.
We can express the stationary distribution of the number of documents in transfer on each route, N = (N i : i ∈ I). We define Q(n) andQ(n) be the set of explicit states and packet level states a MQNwS that occur with positive probability given there are n ∈ Z I + documents in transfer on each route. We also let S(n) = {m ∈ Z K + : j:j∈i m ji = n i , ∀ i ∈ I} be the set of route-queue states achievable given there are n ∈ Z I + documents in transfer.
where we define Finally, we give the stationary distribution for the packet level state of a MQNwS,Q = (Q j : j ∈ J ). And, we also give the stationary distribution the flow level state of a MQNwS, Y = (Y ik : k = 1, ..., N i , i ∈ I). Corollary 3.3. Given the stability condition Assumption 1 is satisfied, the stationary packet level state of a MQNwS,Q = (Q j : j ∈ J ), has distribution ,q ∈Q.
The stationary flow level state of a MQNwS, Y = (Y ik : k = 1, ..., N i , i ∈ I), has distribution Here n iy is the number of route i packets with residual file size y. Also we define, .
The above two corollaries simply involve summing distribution (5) over the specified set of states. We omit the explicit calculation in their proof.

Closed multi-class queueing network
A closed multi-class queueing network behaves as an MQNwS except that document arrivals and departures are forbidden, see [6,Section 3.4]. In effect the network behaves as if there are a fixed number of infinitely large documents in transfer. We now more formally define a closed multi-class queueing network.
Given there are n ∈ Z I + packets on each route, a closed multi-class queueing network is a packet level Markov process on the statesQ(n) We now define the class and routing structure of this queueing network. The class of a packet is of the formc = (i, k) where i ∈ I records the route a packet is on and k ∈ N records the stage of the packet on its route i. Routing through classes occurs in the following way. For k = 1, ..., k i − 1, a class (i, k) packet on departing queue j i k will join queue j i k+1 and become a class (i, k + 1) packet. A class (i, k i ) packet that has completed service will join queue j i 1 as a class (i, 1) packet. This description is sufficient to give a Markov chain description of a closed multi-class queueing network, but is not sufficient for this Markov chain to be irreducible. For example, a network consisting of a single last-come-first-served queue would reducible. For this reason, we require the following assumption to hold throughout this paper.

Assumption 2.
We assume for all closed queueing networks in this thesis that the set of statesQ(n) is irreducible.
It is worth noting that if Assumption 2 is broken then there need not be a unique stationary distribution or a unique stationary throughput for the closed queueing network. Note due to the finite state space of these Markov chains we do not require any stability condition to hold.
As is proven in Section 3.4 of [6], we now give the stationary distribution for this queueing network. transfer on each route at each queue has stationary distribution for each m ∈ S(n), where B n is defined by (11).
Finally, we can characterise the stationary throughput of packets in a closed multi-class queueing networks.
Corollary 3.5. Given Assumption 2, for a closed multi-class queueing network with n ∈ Z I + documents in transfer across routes and with n i > 0, the stationary throughput of each route i packet, at stage k and at queue j = j i k is where B n is defined by (11) and e i is the i-th unit vector in R I + . Proof. The probability the network is in state m ∈ Z K + is given by (13). Given the network is in state m, by Corollary 3.4 of [6] or from stationary distribution (13), the probability at queue j the packet position k ′ ∈ {1, ..., m j } is traversing route i at stage k is 1 ζji mji mj . The throughput of the packet in position k ′ of queue j is γ j (k ′ , m j )φ j (m j ). By our irreducibility assumption, all arrangements of the n i route i packets are equally likely. Thus the probability this packet is any specific route i packet is (ni−1)! ni! = 1 ni . Thus, the stationary throughput of this route i packet is In the first inequality, we used the fact that mj l=1 γ j (l, m j ) = 1, ∀ m j ∈ N. In the second equality, we cancelled terms and substituted m ′ lr = m lr − 1 if (l, r) = (j, i) and m ′ lr = m lr otherwise. In subsequent chapters, an important quantity will be the stationary rate packets are transferred on route i of a closed multi-class queueing network.

Bandwidth sharing networks
In this section, we consider a flow level bandwidth sharing model introduced by Massoulié and Roberts [7,8]. We call these models stochastic flow level models (SFLM). SFLMs model the dynamic, elastic transfer rate received by document transfers in a communication network. Multi-class queueing networks with spinning (MQNwS) model packet level dynamics SFLMs model document level dynamics. We think of MQNwSs as a microscopic model of a communication network. We think of SFLMs as a macroscopic model of a communication network. We will formally relate MQNwS and SFLMs. Massoulié and Roberts [8] discuss the separation of time scales between a certain SFLM and MQNwS. In the next section, we will give a proof that a SFLM is the limit of a sequence of MQNwS, and thus, we formally justify a separation of time scales. We call our limit flow level model a "spinning network". The models of this type are considered by Bonald and Proutiere [3] under the name the "Store-Forward Network".
In performing this analysis, we are able to prove general insensitivity results for the spinning network. As cited by Proutiere [9, Section 3.4] the spinning network was first considered by Massoulié because of its insensitivity. Bonald and Proutiere [3] proved insensitivity for spinning networks with documents with size given by phase type distributions.
In this section, we introduce the stochastic flow level models and we define the spinning network. In the next section, and specifically in Theorem 5.1, we prove the main result of this chapter: the weak convergence of a sequence multiclass queueing networks to its spinning network. In Section 6 and specifically in Corollary 6.1, we prove insensitivity results which hold as a consequence of Theorem 5.1.

Bandwidth allocations and stochastic flow level models
A bandwidth allocation policy is a map Λ : Z I + → R I + . For n ∈ Z I + , the vector Λ(n) = (Λ i (n) : i ∈ I) is a bandwidth allocation. Here Λ i (n) represents the rate that route i documents are transferred through each route of a communication network, given there are n = (n i : i ∈ I) documents in transfer on each route.
The stochastic model we describe represents the randomly varying number of document transfers within a network. The model is studied as a flow level model of Internet congestion control. We first assume that documents have a size that is exponentially distributed. We will then generalise to document sizes that are independent and of a general distribution.
For document sizes that are independent exponentially distributed, a stochastic flow level model operating under bandwidth allocation policy Λ (SFLM) is a imsart-generic ver. 2011/01/24 file: insenscov.tex date: February 16, 2022 continuous-time Markov chain on Z I + with rates for n, n ′ ∈ Z I + , where e i is the i-th unit vector in Z I + . This model can be interpreted as follows: documents wishing to be transferred across route i arrive as a Poisson process of rate ν i . These documents have a size that is independent and exponentially distributed with mean µ −1 i . If currently the number of documents in transfer across routes is given by vector n ∈ Z I + then each document on route i is transferred at rate Λi(n) ni . Documents are then processed at this rate until there is a change in the network's state, either by a document transfer being completed and thus leaving the network, or by a document arrival occurring. Thanks to the memoryless property of our process we need not record residual document sizes when an arrival or departure event occurs.
The key distinction between this model of document transfer and our previous queueing models is that we do not consider packet level dynamics. These dynamics are abstracted away, and instead, we only consider the flow-level descriptions of the network's state.
We can generalise SFLMs to allow the transfer of documents with an independent arbitrarily distributed size. In this case, similar to the flow level state of a multi-class queueing network with spinning, we record the flow level state of a generalised SFLM. For each document in transfer, we will record the documents residual size, that is the amount of the document that is still to be processed. Given there are n = (n i : i ∈ I) documents in transfer, the flow level state of a generalised stochastic flow level model is given by the vector y = (y ik : k = 1, ..., n i , i ∈ I). ( Here y ik ∈ (0, ∞) is the residual document size of the k-th document in transfer on route i. We order elements so that y ik ≤ y ik+1 for k = 1, ..., n i − 1.
The dynamics of this generalised SFLM are morally the same as our previous definition: documents arrive as a Poisson process; documents are transferred at an elastic rate depending on the number of documents in transfer along different routes and documents depart once transferred.
More explicitly, the dynamics of this model are defined as follows. Documents for transfer on route i arrive as a Poisson process of rate ν i . An arriving document on route i will then have a residual document size X ′ i added to the flow level description (15). We assume X ′ i is an independent positive random variable with finite mean µ −1 i , and we assume X ′ i is equal in distribution to some positive random variable X i . In between a document arrival or departure event, the residual document size of a route i document decreases linearly at rate Λi(n) ni . A document on route i departs the network at the instant its residual document size equals 0. At this point, the corresponding document is removed from the network's flow level state description.
Given the network's state y, all future events are a function of y and independent random variables, thus the state description describes this process as a Markov process. As described in Section 3.2, we let Y be the set of flow level states.

Insensitive stochastic flow level models
A stochastic flow level model, as described above, has stationary distribution π Y when is the Borel σ-field defined on the set of flow level states Y from norm (4). We say that a random variable X with values in R + is non-atomic if P(X = x) = 0 for all x ∈ R + .
We say that a stochastic flow level model is insensitive to non-atomic distributions with stationary distribution π N = (π N (n) : n ∈ Z I + ), if every generalised SFLM with non-atomic document size distributions, mean document sizes ( 1 µi : i ∈ I) has a stationary distribution π Y satisfying π N (n) = P πY (N (0) = n), ∀ n ∈ Z I + .
In other words, the distribution π N only depends on the document size distribution through its mean (µ −1 i : i ∈ I).

Spinning networks
Bandwidth allocations represent the stationary rate of document transfer, given the number of documents in transfer on each route. From Corollary 3.5, we can define a bandwidth allocation that represents the stationary behaviour of a MQNwS. We define a spinning allocation to be the stationary throughput of a closed multi-class queueing network. That is for each ∀ n ∈ Z I + , we define where e i is the i-th unit vector in R I + and B n is defined by (11). The stochastic flow level model defined by a spinning allocation policy is called a spinning network. Proutiere [9] notes that this bandwidth allocation is first defined by Laurent Massoulié. Insensitivity results on this bandwidth allocation are explored in Bonald and Proutiere [3].

Convergence of open queueing networks to spinning networks
We are now in a position to prove the main results of this paper. The stochastic flow level models of [7] are intended to represent the flow level dynamics of document transfer in a packet switched network. The aim of this section is to formally justify this interpretation for spinning networks. As a consequence of this analysis, we are able to formally prove insensitivity of spinning networks.
For exponential document sizes and processor sharing queues of fixed capacity, it has been demonstrated that a series of multi-class queueing networks converged weakly to the spinning network in the Skorohod topology [10]. In this section, we generalize theses argument to include general document size distributions and for the general queueing networks discussed in Section 3.2. Although our proof is applied to networks of quasi-reversible queues, the proof applied is phrased so that a more diverse range of queueing processes could be considered. In this sense we generalize Theorem 3.1 [10], whose proof is specific to the specific queueing and document sizes considered.

Limit and prelimit parameters
For our limit model, we consider the stochastic flow level model for the spinning network. We assume documents have a general positive distribution. As discussed in Section 4.1, we assume documents for transfer on route i ∈ I have a distribution given by positive random variable X (∞) i , with finite mean µ −1 i . We let process Y (∞) = (Y (∞) (t) ∈ Q : t ∈ R + ) give the flow level state of this generalised stochastic flow level model and we let N (∞) = (N (∞) (t) ∈ Z I + : t ∈ R + ) give the number of documents in transfer on each route of the spinning network. For our prelimit model, we consider a sequence of multi-class queueing networks with spinning indexed by c ∈ N. For this sequence, we assume that the parameters for queues J , routes I, route orders (j i 1 , ..., j i ki ) and Poisson arrival rates ν = (ν i : i ∈ I) are all fixed and coincide with the same parameters used to define our spinning network. In our sequence of multi-class queueing networks with spinning, we choose to vary the number of packets in each document and the rate at which packets are transferred through the network. For the c-th multi-class queueing network, we let route i document's size have a distribution X and we vary the queueing capacities so that φ For c ∈ N, we let process Q (c) = (Q (c) (t) ∈ Q : t ∈ R + ) give the explicit queueing description of the c-th multi-class queueing network with spinning and imsart-generic ver. 2011/01/24 file: insenscov.tex date: February 16, 2022 Queues process packets at rates given by cφ 1 (m 1 ) and cφ 2 (m 2 ). Thus the rate packets are transferred between queues is of order O(c). Documents on routes 0, 1 and 2 arrive as a Poisson processes of rates ν 0 , ν 1 , ν 2 . Therefore documents arrive at a rate of order O(1). In this example, documents on routes 0, 1 and 2 have a geometric distribution with parameters µ 0 /c, µ 1 /c and µ 2 /c, respectively. Now consider the rate documents depart the network. For route 0, for example, the rate documents depart is of the order of cφ 2 (m 2 ) × µ 0 /c = µ 0 φ 2 (m 2 ). Thus document departures occur at a rate of order O(1). This justifies a separation of timescales between packet transfer and document transfer. This separation of timescales will be required to form a limit process from a sequence of multi-class queueing networks with spinning.
imsart-generic ver. 2011/01/24 file: insenscov.tex date: February 16, 2022 we let process N (c) = (N (c) (t) ∈ Z I + : t ∈ R + ) give the number of documents in transfer on each route of the c-th network. Let Y (c) = (Y (c) (t) : t ∈ R + ) and Q (c) = (Q (c) (t) : t ∈ R + ) be the respective processes corresponding to the flow level state and packet level state of the c-th multi-class queueing network with spinning.
Associated with the multi-class queueing network with spinning Q (1) , we will considerQ n = (Q n (t) : t ∈ R + ) the closed multi-class queueing network with n ∈ Z I + packets on each route. We make the following assumption about each Q n Assumption 3. We assume Assumption 2 holds forQ n for all n ∈ Z I + . That isQ n is an irreducible Markov chain for all n ∈ Z I + . As noted in Section 3.3 this assumption excludes reducibility issues which can occur in closed queueing networks where a queue serves a single deterministically chosen packet.
We will also require an assumption on the spinning network Y (∞) .

Assumption 4.
We assume for Y (∞) that, almost surely, there are no simultaneous document arrival-departure events.
This assumption avoids complications associated with the definition of convergence in the Skorohod Topology. Later we will verify that if distribution X (∞) i is non-atomic ∀ i ∈ I then Assumption 4 holds. 2 Our main theorem, Theorem 5.1, considers weak convergence on bounded time intervals. Thus, we do not require assumptions on the networks long run behaviour, such as Assumption 1, however we will subsequently require some assumptions for results on insensitivity.

Theorem and proof
We now introduce and prove the main result.
Theorem 5.1. For c ∈ N, take an multi-class queueing network with spinning Q (c) , as described above. Assume Assumptions 3 and 4 hold for each c ∈ N. Let Y (∞) denote the flow level state of the spinning network, as described above. If the initial flow level state converges, then, for each T > 0, the stochastic processes converge in the Skorohod topology Proof of Theorem 5.1. We will prove this result using a coupling argument.
In between document arrival-departure events, a MQNwS behaves as a closed queueing network. We couple MQNwS so that in between arrival-departure events this closed queueing network behaviour is determined by a single closed queueing process. By doing this, Skorohod convergence results become a consequence of renewal theory results. We split the proof into four sections. In the first section, we couple the queueing network's initial states. In the second section, we state an induction hypothesis which we will use to prove weak convergence. In the third section, we form a coupling of our queueing networks. In the fourth section, we prove this coupling satisfies the induction hypothesis. Finally in the fifth section, we prove weak convergence in the Skorohod topology. Coupling the initial state: We start by coupling the initial state of our process. By (17) and the Skorohod Representation Theorem [2, Section 6] we may choose a sequence of coupled random variables {Y (c) (0)} c∈N∪{∞} such that, almost surely For c ∈ N and given our coupled sequence {Y (c) (0)} c∈N∪{∞} we know the required distribution of Q (c) (0). We may choose a sequence of functions f (c) : where here U is an independent uniform random variable on [0, 1]. Thus from a single uniform random variable and the coupled sequence {Y (c) (0)} c∈N , we may define the coupled initial state of each MQNwS by Induction Hypothesis: We now inductively construct our coupled process under the following induction hypothesis on κ ∈ Z + . For c ∈ N ∪ {∞}, let τ κ,(c) be the κ-th document arrivaldeparture event for the flow level state of our coupled process Y (c) , c ∈ N∪{∞}.
We will shortly define τ κ+1,(c) . We, also, define the flow level state of the c-th multi-class queueing network with spinning. We associate each packet in the closed queueing networkQ κ at time σqκ,(c) with a packet in the c-th MQNwS at time τ κ,(c) . For each route, let k index the packets associated with each document at time τ κ,(c) . We retain this same index until time τ κ+1,(c) . LetĀ κ ik (t) denote the number of transitions where the k-th packet on route i in the c-th MQNwS has traversed route i in closed queueing networkQ κ (t) by time t. We define the components of the flow level process of the c-th MQNwS by of all possible arrivals that could occur at time τ κ+1,(c) . In the second inequality we apply the triangle inequality to (39) by using the two facts, for t ∈ (τ κ,(c) , τ κ+1,(c) ∧ T ]. The first expression in equation (40) Thus, almost surely, we have convergence in the Skorohod topology on [0, T ], Since, the Skorohod convergence occurs almost surely in this coupling, for all continuous bounded functions f : Thus by the Bounded Convergence Theorem, imsart-generic ver. 2011/01/24 file: insenscov.tex date: February 16,2022 or, in other words, Y (c) c converges weakly to Y (∞) in the Skorohod topology. This completes the proof of Theorem 5.1.

Insensitivity of spinning networks
The insensitivity of the spinning network is a consequence of Theorem 5.1. To prove this we will first require two technical lemmas.
Proof. P(X (∞) ≥ z) can only have countably many points of discontinuity. Thus by integration by substitution and the Bounded Convergence Theorem, we have that, for all y ∈ R + , To prove Theorem 5.1, we assumed no simultaneous arrival-departure events occurred. We now demonstrate that these assumptions hold for the case of a spinning network, with non-atomic document sizes. Lemma 6.2. Suppose the initial distribution Y (0) conditional of N (0) consists of independent non-atomic random variables Y ik k = 1, ..., N i (0), i ∈ I. Given documents size distributions X i , i ∈ I are non-atomic, then, almost surely, a) There are no simultaneous document arrival-departure events, i.e. Assumption 4 holds. b) For all t ∈ R + , almost surely, no document arrival-departure event occurs at time t.
Proof. Let Y = (Y t : t ∈ R + ) be the spinning networks flow level process description. Since arrivals A 1 , A 2 , ... form a Poisson process almost surely no two arrivals occur at the same time and for each t ∈ R + almost surely no arrival occurs at time t. Since exponential random variable A k − A k−1 is independent of (Y t : t ≤ A k−1 ), there is zero probability that an arrival A k coincides with departures. Therefore, an arrival cannot coincide with a departure.
It remains to show that no two document departures may occur simultaneously. Let D k be the departure of some document k of initial size X k (or initial residual size Y ik at time 0). Let A k be that document's arrival time (take A k = 0 if the document is present at time zero). Let Y ′ be the process derived from Y in which document k never departs the SFLM (i.e. behaving as if X k = ∞). Note that Y ′ (t) = Y (t) for all t < D k and D k coincides with a document departure in Y iff D k coincides with a departure in Y ′ . Note that as that X k is conditionally independent of (Y ′ (t) : t > A k ) conditional on (Y ′ (t) : t ≤ A k ) and D k is non-atomic as it is a strictly increasing function of non-atomic independent random variable X k . Thus, conditional on (Y ′ (t) : t ≤ A k ) the probability that non-atomic random variable D k coincides with the countable set of departure events in (Y ′ (t) : t > A k ) or at a specific time t ∈ R + is zero. Thus, the probability two departure events coincide is zero and the probability that departure occurs at a specific time t is zero.
We can now prove one of the main results of this chapter: the insensitivity of the spinning network. Corollary 6.1. Given Assumption 1, the spinning network has a stationary distribution which is insensitive to all non-atomic document size distributions.
Proof. We can take document sizes X . As in Theorem 5.1, we consider a sequence of multi-class queueing networks with spinning associated with these document size distributions and with queue service rates cφ j (·). From Corollary 3.2 and Corollary 3.3, the prelimit stationary distribution of Y (c) and N (c) , c ∈ N is P(Y (c) (0) = y) = B n B i∈I n i n iy : y ∈ N ni k=1 ν i P(X i ≥ y ik ) , ∀ y ∈ Y, t ∈ R + , (43) We can construct Y (c) (0), by taking a vector N (0) according to distribution (44) then, for each i ∈ I and k = 1, ..., N i (0), Y     where Y (∞) (0) has density and also bP (N (∞) (0) = n) = B n B i∈I ρ ni i , ∀ n ∈ Z I + and for x ik ≤ x ik+1 k = 1, ..., n i − 1, i ∈ I. Note the above expression for N (∞) depends on X (∞) i only through its mean. Hence if (46) is the stationary distribution for our limit process then this distribution must be insensitive.
We now show that (46) provides a stationary distribution. It is known that if a sequence of processes weakly converge in the Skorohod topology and if, almost surely, there is not jump at time t then the marginal distribution at time t must weakly converge, see [2,Theorem 12.5]. By Lemma 6.2, almost surely no jump occurs at time t for N (∞) and, by Theorem 5.1, N (c) ⇒ N (∞) as c → ∞ in the Skorohod topology. Thus, N (c) (t) ⇒ N (∞) (t) i.e. the marginal distributions converge at time t. Thus, when processes Y (c) , c ∈ N are stationary, by for any continuous bounded function f : The first equality holds by (45); the second holds by the stationarity of Y (c) ; and the third holds by the weak convergence of the marginal distributions. This proves (46) gives a stationary distribution of the spinning network, and consequently, from (44) and (47), we see that the spinning network is insensitive.