Towards a queueing-based framework for in-network function computation

Abstract

We seek to develop network algorithms for function computation in sensor networks. Specifically, we want dynamic joint aggregation, routing, and scheduling algorithms that have analytically provable performance benefits due to in-network computation as compared to simple data forwarding. To this end, we define a class of functions, the Fully-Multiplexible functions, which includes several functions such as parity, MAX, and kth-order statistics. For such functions we characterize the maximum achievable refresh rate of the network in terms of an underlying graph primitive, the min-mincut. In acyclic wireline networks we show that the maximum refresh rate is achievable by a simple algorithm that is dynamic, distributed, and only dependent on local information. In the case of wireless networks we provide a MaxWeight-like algorithm with dynamic flow-splitting, which is shown to be throughput-optimal.

Appendix: Scheduling with random packet forwarding: detailed proofs

We now present the complete proof for the throughput-optimality of Algorithm 1 in directed acyclic graphs. Since the proof closely follows the proof of Massoulié et al.[28], we do not go into complete details, but try mainly to highlight the modifications we make in order to perform aggregation rather than broadcast.

First we need a lemma that ensures that under the useful packet transmission rule, each round of packets follows a spanning tree. Recall that the footprint of a round of packets is defined as the set of nodes in which the packets of that round is present. Further, recall that a set S is said to be a valid footprint-set if each node in S has a path to a in the subgraph induced by S; the collection of such sets is denoted by $\mathcal{S}$. We assume throughout that $\mathcal{N}\in\mathcal{S}$, for otherwise the min-mincut is 0. Note that since we operate in continuous time, only one packet transmission occurs at a given time with probability 1; further, we require that the local state information is available at the time of making routing decision. Now we have the following lemma:

Lemma 3

For a round of packets with footprint $S\in\mathcal{S}$, the transmission of a useful packet results in a new footprint S′ which is also a valid footprint-set.

Proof

Since the underlying graph is directed acyclic, we re-label the nodes as {0,1,2,…,N−1} according to their topological ordering, where node 0 is the aggregator a, and all edges are from a higher numbered node to a lower numbered node. Further, given a round of packets on a valid footprint-set S, we have that each node k∈S has at least one route to a using only nodes in S; for short, we refer to such a route as a path from k to a in S.

Since we are operating in continuous time, with probability 1 only one packet transmission occurs at a given time. Now suppose a useful packet is transmitted on edge (j,i), where i<j, resulting in a new footprint-set S′=S∖{j}. For S′ to be a valid footprint, we need that even after the transmission, each node k∈S′ has a path to a in S′. To do this, we need to consider a partition of the nodes in S′ into 3 classes:

• Node k∈S′ such that k<j in the topological order: due to the topological ordering property, a path from k to a in S is clearly unaffected by the packet transmission from j to i.

• Node k∈S′,k>j such that there exists a path from k to a in S which does not include j: such a path is also unaffected by the packet transmission from j to i and hence is still present in S′.

• Node k∈S′, k>j such that all paths from k to a in S pass through j: we show by contradiction that this case is impossible under the rules of useful packet transmission. For any path from k to a in S, let k′≤k be the node immediately before j (i.e., the path is k→⋯→k′→j→⋯→a). Then k′ has no path to a in S that does not pass through j, for otherwise we have a path from k to k′, and then to a, which does not pass through j. This means that k′ becomes isolated upon transmission of packet from j to i, which violates the non-isolation condition of useful packet forwarding.

Thus we have that S′ is a valid footprint-set. □

The main idea behind the proof in [28] was to define the ‘footprint-counter’ variables to represent the state of the system, and considering an appropriate function of these that allowed translating the local decisions of the nodes in terms of global graph parameters. In order to modify the proof for broadcast, we defined a similar collection of counter variables in Sect. 4, and now define their associated dynamics as follows.

• Arrival of new round: $X_{\mathcal{N}}\rightarrow X_{\mathcal {N}}+1$ (this corresponds to adding a packet to the queue with footprint-set $\mathcal{N}$, as a packet of the new round is simultaneously generated at all the nodes).

• Completion of packet transfer: This is only for active packets, i.e., those currently under transmission. For active packet $r\in \mathcal{A}$ with corresponding (FP _r ,E _r ) and (u,v)∈E _r , we have:

  

  (The first equation corresponds to removing the edge over which packet transmission was completed, and also updating the footprint of the packet to include the new node. The second updates the list of idle packets in case there is no other instance of this packet being transmitted.)

• Initiation of a new transfer at an idle link. The new packet is selected uniformly at random among the set of useful packets at the node. If $(u,v)\notin E_{r}\ \forall r\in\mathcal{A}$, then a new packet transfer is formally described as follows:

  • Select a useful packet of an idle round with footprint-set $S\in \mathcal{S}, v\in S$, u∉S, with probability

    $$ p_{S}=\frac{X_{S+u}}{X_{+u-v}+X^a_{+u-v}}. $$

    Select a useful packet of an active round $r\in\mathcal{A}$ with $(FP_{r},E_{r})\in\mathcal{A}$ with probability

    $$ p_r=\frac{1}{X_{+u-v}+X^a_{+u-v}}. $$

  • If idle packet with footprint S is selected, then $X_{S}\rightarrow X_{S}-1, \mathcal{A}\rightarrow\mathcal{A}\cup\{r\}$, with (FP _r ,E _r )=(S,(u,v)). If packet of active round r is selected, then E _r →E _r ∪{(u,v)}.

We note here that the node itself does not need to know these global counters to perform packet selection; rather, this emerges from the use of the random useful packet forwarding rule. The idea of relating the local packet selection rule to the global counters is crucial in proving the optimality of the algorithm. The local rules for checking whether a packet is useful or not corresponds to selecting packets whose global footprint obeys certain properties; picking a useful packet uniformly at random therefore corresponds to picking a packet from such a useful global footprint with a probability proportional to the corresponding counter variable.

Observe that in order to determine the flow into a footprint-set S, we need to consider the collection of sets which include S and have one extra node. We now define the fluid limits of the system. This is similar in spirit to the fluid limit of the system in [28], so we try to use similar notation. The existence of the limit also follows immediately from their convergence results, so we omit it due to lack of space and refer interested readers to [28] for technical details.

The fluid limits of the system

The fluid trajectories $t\rightarrow x_{S}(t), S\in\mathcal{S}$ corresponding to the system are defined as follows:

• $\forall (u,v)\in\mathcal{L},\ \forall S$ s.t. v∈S, u∉S, ∃t→ϕ _S+u,(u,v)(t) s.t.

  

• Work Conservation: At almost every t, ϕ _S,(u,v)(t) is differentiable and if x _+u−v (t)>0 (where x _+u−v (t) is the fluid trajectory associated with X _+u−v ), then we have

  

• ϕ _S,(u,v)(t) are non-decreasing, Lipschitz continuous, with Lipschitz constant c _uv , and $\sum_{S\in\mathcal {S}:v\in S, u\notin S}\phi_{S+u,(u,v)}(t)$ is c _uv -Lipschitz.

For any $y\in\mathbb{R}^{|\mathcal{S}|}_{+}, S(y)\triangleq$ set of all fluid trajectories with initial condition $\in\nobreak C([0,\infty),\mathbb {R}_{+}^{|\mathcal{S}|})$, and further, we define $\{X_{S}^{N}(t)\}_{S\in \mathcal{S}}$ as the state of the MC with initial conditions (X ^N(0),A ^N(0)), $Y_{S}^{N}(t)=\frac{X^{N}_{S}(z_{N}t)}{z_{N}}$. Now, as in [28], for a sequence of initial conditions (X ^N(0),A ^N(0)),N>0 s.t. for a sequence of positive numbers (z _N ) _N>0,lim _N→∞ z _N =∞ and the limit

$$ \lim_{N\rightarrow\infty}\frac{X^N(0)}{z_N}\triangleq x(0) $$

exists in $\mathbb{R}^{|\mathcal{S}|}_{+}$, we have ∀T>0,ϵ>0:

$$ \lim_{N\rightarrow\infty}\mathbb{P}\Bigl[\inf_{f\in S(x(0))}\sup_{t\in [0,T]}\bigl\|Y^N(t)-f(t)\bigr\| \geq\epsilon\Bigr]=0. $$

The fluid Lyapunov function

Next, we define the candidate Lyapunov function that we use to analyze the stability of the system. In [28] the function was defined in terms of queues (or counters) that counted all the packets whose footprint was contained inside a set S. The advantage of these queues for studying broadcast was that their rate of increase was controlled by external arrivals to the system, while they were drained due to transfers across the cut defined by the set S.

For the purpose of studying aggregation, we need to identify an equivalent set of queues to reflect the unique dynamics of the system. In particular, we consider for each set S a queue of all rounds whose footprints are not entirely contained within S. These queues (counters) exhibit similar properties to the ones considered for broadcast in that every incoming round is counted by all these queues (as every node in the network generates a packet), while the drain of these queues is controlled by flow across the cut defined by the set S. Formally, we have the following theorem:

Theorem 5

Let $\{x_{S}\}_{S\in\mathcal{S}}$ denote the fluid trajectories. $\forall S\in\mathcal{S}$, define:

$$ x_{\nsubseteq S}=\sum_{S'\in\mathcal{S}, S'\nsubseteq S}x_{S'}. $$

Then (given λ,c _uv ) ∃β ₁,β ₂,…,β _K−1>0,ϵ>0 such that the Lyapunov function

$$ L\bigl(\{x_S\}_{S\in\mathcal{S}}\bigr)\triangleq \max_{S\in\mathcal{S}}\beta_{|S|}x_{\nsubseteq S} $$

(1)

verifies

$$ L\bigl(x(t)\bigr)\leq\max\bigl(0,L\bigl (x(0)\bigr)- \epsilon t\bigr). $$

(2)

As in [28], before proving this theorem we first we need a combinatorial lemma. This lemma and its proof parallels a corresponding lemma in [28], with modifications to deal with aggregation and the x _⊈S counter variables we have defined above.

Lemma 4

Let α>0 be fixed (but arbitrary). We define:

$$ \beta_{i}= \biggl(1+\frac{1}{\alpha} \biggr)^{i-1}, \quad i=1, \ldots,K. $$

Then $\forall \{x_{S}\}_{S\in\mathcal{S}}\in\mathbb{R}_{+}^{|\mathcal {S}|}$, the following conditions hold:

• 1. $\forall S\in\mathcal{S}$, v∈S, u∉S, we have

  $$ x_{+u-v}<(1+\alpha)^{-1}x_{\nsubseteq S} \quad \Rightarrow \quad \beta_{|S|+1}x_{\nsubseteq S+u}>\beta_{|S|}x_{\nsubseteq S}. $$

• 2. $\forall S\in\mathcal{S}$ such that ∀v∈S, u∉S, x _+u−v ≥(1+α)⁻¹ x _⊈S , if ∃v∈S, u∉S and some S′⊈S,v∈S′,u∉S′ such that x _S′+u >αx _+u−v , then

  $$ \beta_{|S\cup S'|}x_{\nsubseteq S\cup S'}>\beta_{|S|}x_{\nsubseteq S}. $$

Note that Lemma 4 does not depend on the algorithm, or the fluid model in any way. It is a pure combinatorial property of the way that the quantities are defined. In other words, any function mapping the sets $S\in\mathcal{S}$ to ℝ₊ obeys the lemma for any α>0. Later we use the ability to control α to obtain uniform bounds on the Lyapunov drift.

Proof

For the first condition, consider $S\in\mathcal{S}$, v∈S, u∉S such that

$$ x_{+u-v}<(1+\alpha)^{-1}x_{\nsubseteq S}. $$

Then we have

and thus

$$ x_{\nsubseteq S}<\frac{1+\alpha}{\alpha}x_{\nsubseteq S+u}. $$

However, from the definition of the β _i , we have $\beta _{i}\frac{1+\alpha}{\alpha}=\beta_{i+1}$ for all i=1,2,…,N−1. Hence we have

$$ \beta_{|S|+1}x_{\nsubseteq S+u}>\beta_{|S|}x_{\nsubseteq S}. $$

For the second condition, consider $S\in\mathcal{S}$ such that ∀v∈S, u∉S, x _+u−v ≥(1+α)⁻¹ x _⊈S . Further, consider set S′ such that S′⊈S, v∈S′, u∉S′ and satisfying

$$ x_{S'+u}>\alpha x_{+u-v}. $$

Then we have

Thus for our condition, we need

$$ \beta_{|S\cup S'|}\alpha(1+\alpha)^{-1}\geq\beta_{|S|}, $$

and noting the fact that β _i are increasing with i, it is sufficient to ensure

$$ \frac{\beta_{i+1}}{\beta_i}\geq\frac{1+\alpha}{\alpha}, \quad \forall i=1,2,\ldots,K-1. $$

This in fact holds with equality because of our choice of β _i . Thus, given any α>0, we can construct β _i such that the two conditions hold. □

Now, we use Lemma 4 to prove Theorem 5. The steps of this proof closely follow the corresponding proof in [28].

Proof of Theorem 5

Given α>0, we define β _i as in Lemma 4. Then, or any $y\in\mathbb{R}_{+}^{|\mathcal{S}|}$, if S ^∗ is a set which belongs to arg-max of $\max_{S\in\mathcal{S}}\beta _{|S|}x_{\nsubseteq S}$, then $x_{S^{*}_{+}}>0$ (unless all the fluid sample paths are identically 0).

Next, we use the optimality of S ^∗ to obtain some relations between $x_{S^{*}_{+}}$ and the weight across its cut-edges. ∀v∈S ^∗, u∉S ^∗ such that $(u,v)\in\mathcal{L}$, we have from the contrapositive of the first condition of Lemma 4 (as S ^∗ is in the arg-max) that

$$ x_{+u-v}\geq(1+\alpha)^{-1}x_{\nsubseteq S^*}. $$

Similarly from condition 2, ∀v∈S ^∗, u∉S ^∗, S′⊈S ^∗ such that v∈S′, u∉S′, we have

$$ x_{S'+u}\leq\alpha x_{+u-v}. $$

Now, we have

If we choose α and ϵ as follows:

then we get that, for all $S^{*}\in\arg\max_{S\in\mathcal{S}}\beta _{|S|}x_{\nsubseteq S}$,

To argue that this implies negative drift of the Lyapunov function, i.e. $L(x(t))=\max_{S\in\mathcal{S}}\beta_{|S|}x_{\nsubseteq S}\leq\max (0,L(x(0))-\epsilon t)$, we observe that by definition β _|S|≥1 $\forall S\in\mathcal{S}$. Finally, using the Lipschitz continuity of the trajectories, it is sufficient to show this property holds for the sets $S^{*}\in\arg\max_{S\in\mathcal{S}}\beta_{|S|}x_{\nsubseteq S}$. □

Finally we can prove Theorem 3 using the stability of the fluid limit process along with standard techniques from literature; for technical details, see [28].