A General Class of Throughput Optimal Routing Policies in Multi-hop Wireless Networks
This paper considers the problem of throughput optimal routing/scheduling in a multi-hop constrained queueing network with random connectivity whose special case includes opportunistic multi-hop wireless networks and input-queued switch fabrics. The main challenge in the design of throughput optimal routing policies is closely related to identifying appropriate and universal Lyapunov functions with negative expected drift. The few well-known throughput optimal policies in the literature are constructed using simple quadratic or exponential Lyapunov functions of the queue backlogs and as such they seek to balance the queue backlogs across network independent of the topology.
By considering a class of continuous, differentiable, and piece-wise quadratic Lyapunov functions, this paper provides a large class of throughput optimal routing policies. The proposed class of Lyapunov functions allow for the routing policy to control the traffic along short paths for a large portion of state-space while ensuring a negative expected drift. This structure enables the design of a large class of routing policies. In particular, and in addition to recovering the throughput optimality of the well known backpressure routing policy, an opportunistic routing policy with congestion diversity is proved to be throughput optimal.
This paper considers the problem of throughput optimal routing/scheduling in a general constrained queueing network with random connectivity whose special case includes opportunistic routing in multi-hop wireless network and input-queued switch scheduling. While it is often possible to intuitively design and propose various routing/scheduling policies, providing theoretical guarantees for the corresponding controlled Markov chains is far from straight forwards with the exception of the throughput optimality of backpressure routing  and maximum weight scheduling . These guarantees are obtained using Foster-Lyapunov Theorem which ensures the stability of a controlled Markov chain if a Lyapunov function with negative expected drift is shown to exist. More specifically, the throughput optimal backpressure-based policies [1, 3, 4, 5, 6] as well as maximum weight schedules [7, 2] are reverse-engineered to be the very rule under which the known quadratic Lyapunov function is ensured a negative expected drift.
While reverse engineering routing/scheduling in this function has the advantage of obtaining theoretical guarantees, it may result in schemes with undesirable structure. In particular, under the strict Schur-convexity of quadratic Lyapunov function [1, 2] (as well as the exponential Lyapunov functions ) with respect to the (weighted) backlog vector, the negativity of the expected drift is only achieved when nodes with large queues are prioritized in favor of those with small number of buffered packets (e.g. a node with small backlog must refrain from routing packets to a neighbor with large backlog). This very need to ensure a negative drift of the Lyapunov function (equivalently to balance the queues in a network), goes against the intuition behind many promising routing/scheduling schemes. For instance, consider the wired network in Fig. 1 where packets are to be routed from node 1 to node 8. It is intuitively desirable for the routing decisions in this network to be such that the bottle-neck link (7,8) is maximally utilized. Indeed, in Subsection V-B we discuss an opportunistic routing policy (ORCD) which attempts to achieve this goal. However, these very intuitive properties cause a positive expected drift in the quadratic Lyapunov function in an infinite number of states. This means that theoretical guarantee for this algorithm requires a significantly different approach (non-Schur-convex Lyapunov function).
In this paper, we provide a large class of throughput optimal policies by considering a class of piece-wise quadratic Lyapunov functions. The proposed class of Lyapunov functions are constructed by grouping the queues based on their relative size and the network topology and as such are not strictly Schur-convex. This allows for the Lyapunov function to have an expected negative drift even when packets are routed from a node with small backlog to one with large backlog so long as the queues are grouped together. We will see that the proposed class of Lyapunov functions establish the throughput optimality of a large class of routing/scheduling policies by indirectly incorporating the critical information about topology. In particular, we specialize our result to recover the throughput optimality of two known routing policies, backpressure (already known to be throughput optimal) and ORCD (discussed above and whose throughput optimality only was conjectured in ).
Before we close, we note that using the methodology in this paper, it is always possible to find uncountably many throughput optimal routing/scheduling policies among which most will suffer from an unreasonable complexity and overhead. In light of this observation, we believe that (even though beyond the scope of this paper) the contribution and utility of the proposed Lyapunov construction is of two folds: 1) The proposed class of Lyapunov functions can be used systematically to establish the throughput optimality of many intuitive solutions which do not locally balance the backlogs in the network. Furthermore, 2) the nature of the constructed Lyapunov function concretely establishes the intuition that queue stability in a network is only affected by the control applied at the boundaries of the state space where inevitable idling is likely to occur (these are the states in which one or more of the queues are near empty while others have extremely large backlogs). The consequence of the former is more flexibility when proving the throughput optimality of various routing/scheduling solutions such as the multi-hop schemes that would favor short paths, while the latter establishes a qualitative characterization of throughput optimality.
The remainder of this paper is organized as follows. In Section II, we formulate the problem for the general case of routing/scheduling in constrained queueing network with multiple destinations. For ease of exposition, the results are first presented in Section III for the multi-hop routing in a single destination network with orthogonal channels. The extensions of the results to the general constrained queueing is provided in Section IV where we also show that scheduling for input-queued switches is a special case of our framework. Section V discusses the structure of our proposed piece-wise quadratic Lyapunov function and provides a (alternative) proof of throughput optimality for some of the existing routing/scheduling policies. Finally, we conclude the paper and discuss future work in Section VI.
We close this section with a note on the notations used. Let . The indicator function takes the value whenever event occurs, and otherwise. For any set , denotes the cardinality of , while for any vector , denotes the euclidean norm of . For matrices and , let denote the inner product. For any set , is the set of all interior points of . When dealing with a sequence of sets , we define . Lastly, we use bold letters to discriminate vectors from scalar quantities as well as their components.
Ii Problem Formulation
We consider a time slotted system with slots indexed by where slot refers to the time interval . There are nodes in the network labeled by . We denote the set of all destinations by , .
Let random variable represent the amount of data (in units of packets) that exogenously arrives to node and destined for node , during time slot . Arrivals are assumed to be i.i.d. over time and bounded by a constant . All packets destined for node , are referred to as commodity packets. Let denote the exogenous arrival rate of commodity packets to node . We define to be the arrival rate matrix (of size ). We assume that each node maintains a separate buffer (with infinite queuing space) for each destination in which packets that arrive exogenously at that node as well as packets routed to that node from other nodes in the network are queued. Without loss of generality, we assume that after a packet is successfully received at its destination, the packet would be ejected from the network. Let denote the queue backlog of node corresponding to destination at time slot , i.e. denotes the number of commodity packets in node at time slot . Any data that is successfully delivered to its destination will exit the network and hence, for all and all time slots . We define to be the matrix (of size ) of all queue backlogs and to be the row of this matrix corresponding to commodity packets, i.e. .
We define a routing decision to be the (potential) number of commodity packets whose relaying responsibility is shifted from node to node during time slot . We assume each node transmits at most one packet during a single time slot which can be selected from any of the buffers maintained at that node. Note that forms the departure process from node , while it is an element of the endogenous arrival to node . Hence,
Here we assume a simple on-off channel model and we assume a perfect channel state information at every transmitter. More specifically, let represent the (random) set of nodes the channel to whom from node at time slot is in good state. We refer to as the set of potential forwarders for node and we assume that node has perfect knowledge of . Due to a perfect recall at any node , we assume for all time . We characterize the behavior of the wireless channel using the probabilistic model of local broadcast model . The local broadcast model is defined using a marginal probability mass functions , , . Note that, by definition, for all , successful reception at and are mutually exclusive and . We say node reaches node (we write ), if there exists a set of nodes such that and .
Under the simple on-off channel model considered here, the routing of packets can only occur over links in an on state. Often there might also be some constraints on the simultaneous activation of the links, i.e. certain links cannot provide service at the same time. Let an activation set be a set of links which can be activated in the same slot. We assume that at any time the collective routing decisions must be such that the set of links for which , , belong to an activation set. Letting denote the set of all such allowable routing decisions, the above constraints can be written as
The selection of routing decisions together with the exogenous arrivals impact the queue backlog of node corresponding to commodity in the following manner:
A routing policy is a collection of routing decisions where for all , , and , the decisions belong to the -field generated by .
A routing policy is said to stabilize the network if the time average queue backlog of each node remains finite when packets are routed according to . The stability region of the network is the set of all arrival rate matrices for which there exists a routing policy that stabilizes the network.
A routing policy is said to be throughput optimal if it stabilizes the network for all arrival rate matrices that belong to the interior of the stability region.
Fact 1 (Corollary 1 in ).
Let denote the stability region of the network. An arrival rate matrix is in the stability region if and only if there exists a stationary randomized routing policy that makes routing decisions solely based on the collection of potential forwarders at time , , and for which
Fact 1 provides a linear program whose solutions always stabilize the network, but requires a full knowledge of the arrivals statistics. In this paper, we are interested in a class of routing policies which are throughput optimal but do not require knowledge of the arrival rates.
Now we are ready to provide the main analytical results of the paper. For simplicity of exposition, we first consider the single destination scenario with no activation constraints. The generalization of the results to the multi-destination scenario with activation constraint is provided in Section IV.
Before we proceed, we introduce the following notations in the interest of simplicity: For a set of nodes, we define , , , and .
Iii Single Destination and Orthogonal Channel Scenario
In this section, we consider the single destination with orthogonal channel scenario and provide an overview of the results. The analysis of the results is provided in Subsection III-E while the generalization of the results to the multi-destination scenario with parallel transmission constraints is discussed in Section IV.
Without loss of generality, we consider node to be the destination, i.e. . Since each node maintains only one buffer (corresponding to destination ), we drop the commodity superscript when denoting various random variables such as routing decisions , . Furthermore, in this section we assume that all channels are orthogonal and there are no activation set constraints on simultaneous packet transmissions, i.e. .
Iii-a Priority-Based Routing
In this subsection, we introduce the class of priority-based routing policies. To define the priority-based routing policy, we need the following definitions.
A rank ordering is an ordered list of non-empty sets , referred to as ranking classes, that create a partition of , i.e. and , . We denote the set of all possible rank orderings of by . Note that when ’s are singleton, reduces to a simple permutation of the nodes . Given a rank ordering , we write to indicate that node has a lower rank than , . We write , if or for some .
A priority-based routing policy is a routing policy under which node , at time and among its set of potential forwarders , selects a node with the lowest rank according to . In other words, under , , only when and for all .
Next we introduce a class of priority-based routing policies under which is chosen as a time-invariant function of , i.e. there exists a function such that . In Subsection III-D, we proceed to establish the throughput optimality of this class of routing policies.
In this section, we introduce a class of priority-based routing policies each of which is associated with a bivariate function , hence referred to as an -policy. Each such policy partitions the space of queue backlogs, , into routing decision cones to each of which a unique rank ordering of nodes is assigned. In other words, it is possible to define the mapping such that at any time and for all in the cone associated with , . The specific shape of each cone (i.e. the set of its defining hyperplanes) is dictated by the corresponding function . In order to give the precise description of -policy, we need the following definitions which allow us to compare rank orderings and :
Let and . We define a mismatch as
For two rank orderings and , compares ranking classes of and from low to high and determines the index of the first ranking class in which they differ.
Given two rank orderings and , we say is a refinement of (and is a confinement of ) if implies that for any .
Given two rank orderings and , we say is a one-step refinement of (and is a one-step confinement of ) with regard to ranking class if
The union of the sets of all one-step refinements and one-step confinements of , denoted by and respectively, is referred to as adjacency of and is denoted by .
Given a bivariate function , a penalty function is defined on backlog vector , rank ordering , and natural number , :
Consider two rank orderings and and a bivariate function . We say penalizes less than and write if
Let , , be a subset of such that for all and all , , i.e.
Let and be two rank orderings and let be a constant. If then . In other words, is a cone in .
Due to the linearity of and finiteness of , the boundaries of the cone corresponding to rank ordering consists of finitely many hyperplanes of the form
Let bivariate function satisfy the following two conditions
(C1) For all and
(C2) For all and
Then for any , there exists a unique such that .
The proof is given in Appendix -B.
By Lemma 1, forms a partition of . Hence, it is meaningful to define a function such that .
Now we are ready to provide the precise definition of -policy as discussed earlier.
-policy is a priority-based routing policy where .
Consider a network of three nodes as given in Fig. 2(a). Let be the set of all rank orderings of , and (it is easy to show that function satisfies (C1) and (C2)). Since node is the destination and for all time slots , the space of queue backlogs can be reduced to . Furthermore, it suffices to restrict to the set of all rank orderings in which the first ranking class only consists of node , i.e. . Fig. 2(b) shows the structure of the cones .
Consider a network of four nodes as given in Fig. 3(a). Let be the set of all rank orderings of , and . Similar to Example 1 and since for all time slots , the space of queue backlogs can be reduced to . Furthermore, it suffices to restrict to the set of all rank orderings in which the first ranking class only consists of node , i.e. . Fig. 3(b) shows the structure of the cones .
By construction, -policy orders the nodes based only on their queue backlogs using a bivariate function independently of the topological characteristic of the network. In certain cases, this may cause packets to be routed away from the destination. In the next section, we introduce a modified version of -policy, referred to as path-connected -policy, which does not allow packets to be routed away from the destination. The main idea behind path-connected -policy is that the rank orderings are limited to those under which there exists a path from any node to the destination through the nodes with lower or the same rank as . The precise description of path-connected -policy is provided in the next section.
Iii-C Path-connected -policy
In order to give a detailed description of path-connected -policy, we have to define a path-connected rank ordering.
A rank ordering is referred to as path-connected if for each node there exist distinct nodes such that and for all .
The set of all path-connected rank orderings is denoted by , . Let be the union of the sets of all path-connected one-step refinements and one-step confinements of , denoted by and respectively. We define , , as
The network is said to be connected if for each node there exist nodes such that .
Next lemma renders the set of cones as a partition of .
Assume the network is connected.111If a node has no path to the destination, it cannot sustain any traffic and can be ignored without loss of generality. If bivariate function satisfies conditions (C1) and (C2), then for all , there exists a unique such that .
The proof is given in Appendix -B.
In other words, is the set of cones that partition and it is possible to define a function such that .
A priority-based routing policy is said to be a path-connected -policy if .
Next we state the main results of this paper.
Iii-D Overview of the Results
Let be a bivariate function that satisfies conditions (C1) and (C2). Then the associated -policy (path-connected -policy) is throughput optimal.
Let and be two priority-based routing policies. We say respects if is a refinement of for all time slots .
Suppose is a priority-based routing policy that is throughput optimal. Any priority-based routing policy that respects is also throughput optimal.
Note that Theorem 2 enables the proof of throughput optimality of specific routing policies. For example, in Subsection V-B, Theorems 1 and 2 are used to prove the throughput optimality of two known routing policies, backpressure  and ORCD . The proof of Theorem 2 is fairly straight forward and is given in Appendix -F.
Iii-E Throughput Optimality of -policy
In this section, we assume that routing decisions , are made under an -policy for which is a bivariate function satisfying conditions (C1) and (C2). In this setting, we prove that -policy is throughput optimal. The proof is based on the following corollary to Foster-Lyapunov Theorem.
Fact 2 (Lemma 4.1 in ).
Let be a Lyapunov function. If there exist constants , , such that for all time slots we have:
then the network is stable, i.e. the time average queue backlog of each node remains finite.
To prove Theorem 1, we identify a class of Lyapunov functions that under the corresponding -policy satisfy the conditions of Fact 2 for all arrival rate vectors . In particular, we construct a piece-wise Lyapunov function, , by assigning to each cone , , a quadratic function of the queue backlogs:
Since the collection of cones form a partition of , we can combine the above quadratic functions to arrive at a piece-wise quadratic function
is continuous and differentiable.
Note that the continuity and differentiability of follow 1) the continuity and differentiability of the construction of inside the cone corresponding to , as well as 2) the construction of penalty function on the separating hyperplanes at the boundary of . The details are given in Appendix -C.
Next we provide the main steps in showing has a negative expected drift.
Let us consider the Lyapunov drift when for some . By Lemma 3, is continuous and differentiable. Thus, we can write in terms of its first-order Taylor expansion around and we obtain
Let and . We have
where is a constant bounded real number.
The proof is given in Appendix -D.
Let , , and let represent routing decisions made under an -policy. For any collection of routing decisions , we have
The proof is given in Appendix -D.
Since , there exists a positive vector (vector of length with all elements equal to , ) such that . Thus, from Fact 1
Since is bounded, there exists a constant, say , such that for all time slots . Moreover, property (C2) of function implies that
Therefore, we can rewrite (9) as
Note that the proof of throughput optimality for path-connected -policy follows similar lines above and is provided in Appendix -E.
Iv Generalization: Multi-Destination Constrained -policy
In this section, we introduce multi-destination constrained -policy as a generalization of -policy in a multi-destination scenario with parallel transmission constraints. Next we provide a precise definition of multi-destination -policy.
Suppose , , , and let . Multi-destination constrained -policy is defined as to select routing decisions such that for any global channel state , they maximize
Let be a bivariate function that satisfies conditions (C1) and (C2). Then the associated multi-destination constrained -policy is throughput optimal.
Let represent routing decisions made under a multi-destination constrained -policy. Let us consider the Lyapunov drift when , , . Following similar steps as that of the proof of Theorem 1, we obtain
where is a constant bounded real number. However, the term
is maximized by the multi-destination constrained -policy for any global channel state . Hence, the negative drift term in (IV) is bounded by the negative drift under any other set of routing decisions, including the stabilizing randomized rule. Now from Facts 1 and 2, the proof of Theorem 3 is complete.
As a special case of routing in constrained queueing networks, scheduling for single-hop networks (e.g. wireless uplinks and downlinks) and input-queued switches have also been of great interest [7, 12, 13, 2, 14, 15]. Next, we show that throughput optimal scheduling for input-queued switches is a special case of our framework and discuss the -scheduling. In particular, we specialize our result to derive a class of throughput optimal scheduling policies for input-queued switches which will be compared with some of the existing scheduling policies.
Iv-a Input-Queued Switches
Consider the input-queued switch studied in  and as depicted in Fig. 5. A scheduling decision is defined to be the (potential) number of packets sent from input , to output , during time slot . In a crossbar switch, each input can send to at most one output and each output can receive from at most one input and hence,
This is nothing but a single-hop example of the setup introduced in Section II where the nodes in the network are partitioned into inputs labeled by to outputs (destinations) labeled by , with for all and a deterministic and fully connected local broadcast model for all . In this setup, choice of routing decisions are equivalent to scheduling decisions , while the set of allowable routing decisions is the space of all permutation matrices. This means that, in the input-queued switch problem, (1) and (2) reduce to (13). Now for any bivariate function , we introduce a class of scheduling policies each of which is constructed using our proposed framework, hence referred to as an -scheduling. Let denote the set of all possible rank orderings of . -scheduling partitions the space of queue backlogs corresponding to each destination, , into scheduling decision cones to each of which a unique rank ordering is assigned. Similar to (4) in Subsection III-B, we can define , , such that forms a partition of . Suppose and for all . Then -scheduling selects scheduling decisions in order to maximize