Scheduling with pairwise XORing of packets under statistical overhearing information and feedback

Abstract

We study the problem of scheduling packets from several flows traversing a given node which can mix packets belonging to different flows. Practical wireless network coding solutions depend on knowledge of overhearing events which is obtained either by acknowledgments or statistically. In the latter case, the knowledge about each packet improves progressively with feedback from the transmissions. We propose a virtual network mechanism in order to characterize the throughput region of such a system for the case where we allow only pairwise XORing. We also provide the policy which achieves the stability region and compare it to simple heuristics. The derived policy is a modification of the standard backpressure policy, designed to take into account the fact that in the proposed virtual network the destination of a transmitted packet is known only probabilistically. We demonstrate simulation results according to which scheduling with statistical information can provide significant throughput benefits even for overhearing probabilities as small as 0.6.

Appendices

Appendix A

In this Appendix we give the proof for the necessity of the conditions of Theorem 1. The sufficiency of the the conditions follows directly from the proof of Theorem 2 which shows that Algorithm 2 stabilizes the system for any arrival rate vector that is in the interior of the throughput region.

The proof of this section is based on the technique used in [8]. We will need the following lemmas.

Lemma 1

If a random vector X(t) converges to X on a set A then for any ϵ>0

$$\lim_{t\rightarrow\infty}\mathrm{P}\bigl( \bigl\{ \big\vert\mathbf {X} (t )-\mathbf{X} \big\vert\leq\epsilon \bigr\} \cap A \bigr)=\mathrm{P}(A ). $$

Lemma 2

Consider two sequences of real numbers a _t ,b _t t=1,2,…. Then

$$\limsup_{t\rightarrow\infty} (a_{t}+b_{t} )\geq \limsup_{t\rightarrow\infty}a_{t}+\liminf_{t\rightarrow\infty}b_{t}. $$

Lemma 3

If a network is stable then

$$\lim_{V\rightarrow\infty}\liminf_{t\rightarrow\infty}\mathrm {P}\biggl(\sum _{j\in\mathcal{N}}X_{j} (t )>V \biggr)=0. $$

Lemmas 1 and 2 can be found in standard textbooks while a proof of Lemma 3 can be found in [19].

For the proof we consider a somewhat more general network than the one presented in Sect. 6. Specifically, we consider a network consisting of $\mathcal{N} \cup{d}$ nodes and $\mathcal{E}$ links, where the special node d represents the destination of traffic originated at the other nodes in $\mathcal{N}$. Let $\mathcal {E}_{o}^{i}$ represent the set of outgoing links from node i and $\mathcal{E}_{in}^{i}$ set of incoming links. A finite set of controls $\mathcal{I}$ is available. For each control $I\in\mathcal {I}$, “transmission” takes place over the sets of outgoing links of node $i\in\mathcal{N}$, $\mathcal{E}_{o}^{i}$, as follows.

• If at a given slot control $I\in\mathcal{I}$ is applied, then for any node $i\in\mathcal{N}$ at most $\hat{\mu}_{i}(I)\geq0$ packets may be transmitted over the set $\mathcal{E}_{o}^{i}$ in the following random manner.

• For each $i\in\mathcal{N}$ and $I\in\mathcal{I}$ there is a random sequence $\{ R_{n}^{i} (I ) \}_{n=1}^{\infty}$, where each $R_{n}^{i} (I )$ takes values in the set $\mathcal{E}_{o}^{i}$. The nth packet transmitted over the set $\mathcal{E}^{i}$ when control I is applied, is received only by the recipient of the link $R_{n}^{i} (I )$.

For a given n and I, the random variables $\{ R_{n}^{i} (I ) \}_{i\in\mathcal{N}}$ may be arbitrarily correlated. Moreover the random sequences $\{ R_{n}^{i} (I ) \}_{n=1}^{\infty}$ are obeying the strong law of large numbers, i.e., for any $I\in \mathcal{I}$, $i\in\mathcal{N}$,

(3)

(4)

Packets may arrive at each node $i\in\mathcal{N}$ and must be delivered to destination node d. We denote by A _i (t) the number of arrivals up to time t. We assume that

$$\lim_{t\rightarrow\infty}\frac{A_{i}(t)}{t}=\lambda_{i}, \quad i\in \mathcal{N}. $$

If in the virtual network developed in Sect. 6 all destination nodes $n_{d}^{i}$ are replaced by a single node d, maintaining the links of the destination nodes with the rest of the nodes in $\mathcal{N}$, the resulting network is a special case of the network model defined here. We mention that the model presented here can be easily generalized to include channel states and multi commodity flows but we opted for the current description since it suffices for our purposes and avoids further complicated notation.

In the following, to avoid trivial cases we assume that there is a path from every node $i\in\mathcal{N}$ to d (this is needed in the sufficient conditions to justify the existence of an interior point).

Assume that a policy π stabilizes the network for a rate vector $\{ \lambda_{i} \}_{i\in\mathcal{N}}$. Let the system operate under this policy and define the following random variables

• $\mathcal{W} (I,t )$: subset of time slots in {1,…,t} when control I is applied. Let $W (I,t )=\vert \mathcal{W} (I,t )\vert$.

• F ⁱ(I,t): the number of packets transmitted up to time t over set $\mathcal{E}_{o}^{i}$, when control I is applied.

• F _e (t): total number of packets transmitted over link e up to time t.

According to the definitions above, it follows that

$$ \sum_{I\in\mathcal{I}}W (I,t )=t $$

(5)

and, for any $e=(i,k)\in\mathcal{E}^{i}$,

(6)

where we define $\sum_{m=1}^{0}X_{i}=0$.

Let X _i (t) be the number of packets at node i at time t. Assuming that the queues are initially empty, the following holds for any node i other than the destination node.

$$ A_{i}(t)+\sum_{e\in\mathcal{E}_{in}^{i}}F_{e}(t)- \sum_{e\in \mathcal{E}_{o}^{i}}F_{e}(t)=X_{i}(t). $$

(7)

The rest of the proof involves several technical details. To clarify the steps, we provide first an outline of the proof making some simplifying assumptions. Specifically, assume that under policy π, for any $I\in\mathcal{I}$,

• 1. The (long term) proportion of time when control I is applied is well defined and positive (hence lim _t→∞ W(I,t)=∞),

  $$\lim_{t\rightarrow\infty}\frac{W(I,t)}{t}=\phi_{I}>0. $$

• 2. The average number of packets transmitted when control I is applied is well defined,

  $$\lim_{t\rightarrow\infty}\frac{F^{i}(I,t)}{W(I,t)}=\mu_{i}(I). $$

We note that these assumptions do not hold for all policies and thus some ill-behaving policies are left outside this consideration. The technical details that complicate the proof aim at removing these assumptions. We have from (5),

$$\sum_{I\in\mathcal{I}}\phi_{I}=1 $$

and from (3), (4), (6), for e=(i,k)

(8)

Since at most $\hat{\mu}_{i}(I)$ packets can be transmitted in each slot when control I is applied, it follows that

$$0\leq\mu_{i}(I)\leq\hat{\mu}_{i}(I). $$

From this and (8) we see that the vector $\{ f_{e} \}_{e\in\mathcal{E}}$ belongs to the region $\mathcal{C}$. Moreover, dividing both sides of (7) by t, taking limit as t→∞and using the fact that for a stabilizing policy lim _t→∞ X _i (t)/t=0, we have from (7),

$$\lambda_{i}+\sum_{e\mathcal{\in E}_{in}^{i}}f_{e}= \sum_{e\mathcal {\in E}_{0}^{i}}f_{e}, \quad\mathrm{\ for\ } i \in \mathcal{N} $$

which implies that the arrival rate vector belongs to the Stability region as claimed. We now proceed with the detailed proof.

In the following Ω denotes the underlying probability space. Let $\mathcal{P}_{\mathcal{T}}= (\mathcal{T}_{a},\mathcal {T}_{f} )$ denote a partition of a set $\mathcal{T}$, i.e., $\mathcal {T}=\mathcal{T}_{a}\cup\mathcal{T}_{f}, \mathcal{T}_{a}\cap \mathcal{T}_{f}=\emptyset$, and denote by $\mathbb{P}_{\mathcal{T}}$ the set of these partitions. We need the following preliminary lemmas.

Lemma 4

Let the stabilizing policy π be applied. Then there is a partition $\mathcal{P}_{\mathcal{I}}^{o}= (\mathcal {I}_{a}^{o},\mathcal{I}_{f}^{o} )$ of the control set $\mathcal {I}$, with $\mathcal{I}_{a}^{o}\neq\emptyset$, such that the set

$$\varOmega_{0}= \Bigl\{ \omega:\;\lim_{t\rightarrow\infty }W(I,t)=\infty, I \in\mathcal{I}_{a}^{o},\lim_{t\rightarrow\infty }W(I,t)<\infty, I\in\mathcal{I}_{f}^{o} \Bigr\} $$

has positive probability.

Moreover, there are partitions of the set $\mathcal{I}_{a}^{o}$, $\{ \mathcal{P}_{\mathcal{I}_{a}^{o}}^{i} \}_{i\in \mathcal{N}}= \{ (\mathcal{I}_{a}^{o,i},\mathcal {I}_{f}^{o,i} ) \}_{i\in\mathcal{N}}$ such that the subset of Ω ₀,

$$\varOmega_{1}= \Bigl\{ \omega\in\varOmega_{o}: \lim_{t\rightarrow \infty}F^{i}(I,t)=\infty, I\in\mathcal{I}_{a}^{o,i}, \lim_{t\rightarrow\infty}F^{i}(I,t)<\infty, I\in\mathcal {I}_{f}^{o,i}, i\in\mathcal{N} \Bigr\} $$

has positive probability.

Proof

Since the sequence W(I,t) is nondecreasing in t it converges either to a finite number or to infinity. Hence defining

$$\varOmega_{T}= \Bigl\{ \omega:\;\lim_{t\rightarrow\infty}W(I,t) \mathrm{\ exists} \Bigr\}, $$

we have that P(Ω _T )=1. Define now

$$\varOmega_{\mathcal{P}_{\mathcal{I}}}= \Bigl\{ \omega:\;\lim_{t\rightarrow\infty}W(I,t)=\infty, I\in\mathcal{I}_{a},\lim_{t\rightarrow\infty}W(I,t)<\infty, I\in \mathcal{I}_{f} \Bigr\} . $$

Then,

$$\varOmega_{T}=\bigcup_{\mathcal{P}_{\mathcal{I}}\in\mathbb {P}_{\mathcal {I}}}\varOmega_{\mathcal{P}_{\mathcal{I}}}. $$

Since P(Ω _T )>0, one of the sets on the right of the last equality must have nonzero probability and this establishes the existence of Ω ₀. Moreover, if $\mathcal {I}_{a}=\emptyset$ then (7) on Ω ₀ we would have

$$\lim_{t\rightarrow\infty}\sum_{I\in\mathcal{I}}W(I,t)<\infty, $$

which contradicts (5).

Decomposing Ω ₀ in a similar fashion based on the existence of limits of the sequences F ⁱ(I,t), establishes the second part of the lemma. □

Lemma 5

Let the stabilizing policy π be applied. Then there is a partition $\mathcal{P}_{\mathcal{I}}^{o}= (\mathcal {I}_{a}^{o},\mathcal{I}_{f}^{o} )$ of the control set $\mathcal{I}$, with $\mathcal{I}_{a}^{o}\neq \emptyset$ and partitions of the set $\mathcal{I}_{a}^{o}$, $\{ \mathcal{P}_{\mathcal{I}_{a}^{o}}^{i} \}_{i\in\mathcal {N}}= \{ (\mathcal{I}_{a}^{o,i},\mathcal {I}_{f}^{o,i} ) \}_{i\in\mathcal{N}}$, such that: for any ϵ>0 there is

• (a) a realization ω∈Ω,

• (b) a t large enough,

for which that the following hold

(9)

(10)

(11)

(12)

Proof

Consider the set Ω ₁ of Lemma 4. For any realization belonging to this set, since lim _t→∞ F ⁱ(I,t)=∞ for $I\in\mathcal{I}_{a}^{o,i}, i\in\mathcal{N}$ we have

(13)

Also, since lim _t→∞ F ⁱ(I,t)<∞ and lim _t→∞ W(I,t)=∞ for $I\in \mathcal{I}_{f}^{o,i}$, it follows that

$$ \lim_{t\rightarrow\infty}\frac{F^{i}(I,t)}{W(I,t)}=0,\quad I\in \mathcal{I}_{f}^{o,i}, i\in\mathcal{N} $$

(14)

and similarly,

$$ \lim_{t\rightarrow\infty}\frac{W(I,t)}{t}=0,\quad I\in\mathcal{I}_{f}^{o}. $$

(15)

By the ergodicity of the arrival process,

$$ \lim_{t\rightarrow\infty}\frac{A_{i}(t)}{t}=\lambda_{i},\quad i\in \mathcal{N} $$

(16)

Defining the set $\mbox{$\varOmega_{\epsilon}$(t)}$ as the subset where (9)–(12) hold simultaneously, it follows from (13)–(16) and Lemma 1 that

$$\liminf_{t\rightarrow\infty}\mathrm{P}\bigl(\varOmega_{\epsilon }(t)\cap \varOmega_{1} \bigr)= \mathrm{P}(\varOmega_{1} )>0. $$

Hence for any fixed V,

(17)

(18)

where the equality follows from Lemma 2.

From Lemma 3 it follows that

$$\lim_{V\rightarrow\infty}\limsup_{t\rightarrow\infty}\mathrm {P}\biggl(\sum _{j\in\mathcal{N}}X_{j} (t )\leq V \biggr)=1. $$

Hence we can pick V large enough so that

$$\limsup_{t\rightarrow\infty}\mathrm{P}\biggl(\sum_{j\in\mathcal {N}}X_{j} (t )\leq V \biggr)\geq1-\mathrm{P}(\varOmega_{1} )/2. $$

For this choice of V it follows from (18) that

$$\limsup_{t\rightarrow\infty}\mathrm{P}\biggl( \biggl\{ \sum _{j\in \mathcal {N}}X_{j} (t )\leq V \biggr\} \cap \varOmega_{\epsilon }(t)\cap\varOmega_{1} \biggr) \geq\mathrm{P}( \varOmega_{1} )/2>0. $$

Therefore there is a sequence of times t _m , with lim _m→∞ t _m =∞ such that

$$ \mathrm{P}\biggl( \biggl\{ \sum_{j\in\mathcal{N}}X_{j} (t_{m} )\leq V \biggr\} \cap\varOmega_{\epsilon}(t_{m}) \cap\varOmega_{1} \biggr)>0. $$

(19)

Picking a t _m large enough so that V/t _m ≤ϵ we conclude from (19) that there is a realization satisfying all the conditions of the lemma. □

Proof of Theorem 1 (Necessity)

For any ϵ>0 and for the corresponding realization of Lemma 5 define

These quantities are all bounded, specifically,

Moreover, by (5) it follows that

$$ \sum_{I\in\mathcal{I}}\phi^{\epsilon}(I)=1. $$

(20)

It then follows from (6) that for $e=(i,k)\in\mathcal{E}$ it follows that

Since , using the definitions and the inequalities in Lemma 5 we have

(21)

where $M=\max_{I\in\mathcal{I}}\max_{i\in\mathcal{N}}\hat{\mu}_{i}(I)$. Similarly,

$$ f_{e}^{\epsilon}\geq\sum_{I\in\mathcal{I}_{a}^{o,i}} \phi^{\epsilon}(I)\mu_{i}^{\epsilon}(I) \bigl(w_{e}^{i}(I)- \epsilon \bigr) $$

(22)

and taking into account (7),

$$ \lambda_{i}-\epsilon+\sum_{e\in\mathcal {E}_{in}^{i}}f_{e}^{\epsilon}- \sum_{e\in\mathcal {E}_{o}^{i}}f_{e}^{\epsilon}\leq \epsilon. $$

(23)

Consider now the sequence of vectors

$$\boldsymbol{V}_{n}= \bigl\{ f_{e}^{1/n}, e\in \mathcal{E}, \phi^{1/n}(I), I\in\mathcal{I}, \mu_{i}^{1/n}, i\in\mathcal{N} \bigr\} ,\quad n=1,\dots. $$

This sequence is bounded and hence it contains a convergent subsequence $\boldsymbol{V}_{n_{k}}, k=1,\dots$ . Let $\{ f_{e}, e\in \mathcal{E}, \phi(I), I\in\mathcal{I}, \mu_{i} \}$ be the limit of this subsequence. Taking limits in (20), (21)–(23) we see that using this limit sequence shows that $\{ \lambda_{i} \}_{i\in\mathcal{N}}\in \mathcal{C}$. □

Appendix B

In this Appendix we give the proof of Theorem 2.

Prof of Theorem 2

Define the following quadratic Lyapunov function

$$L(\mathbf{X})=\sum_{i\in\mathcal{N}}X_i^2 $$

and the following conditional expectation, called Lyapunov drift

$$\Delta\mathbf{X}(\tau)\doteq\mathrm{E}\bigl\{L \bigl(\mathbf{X}(\tau +1) \bigr)-L \bigl( \mathbf{X}(\tau) \bigr)|\mathbf{X}(\tau) \bigr\}. $$

We would like to show the system is stable under Algorithm 2 whenever the arrival vector $\pmb{\lambda}$ lies inside the region of Theorem 1. For this, it is enough to show that the Lyapunov drift is negative whenever the backlogs are large enough, i.e. that there exist positive constants B,ξ such that for all τ

$$\Delta\mathbf{X}(\tau)\leq B-\xi\sum_{i\in\mathcal{N}}X_i( \tau), $$

see [9] and in particular Lemma 4.1. In the same Sect. of [9], the Lyapunov drift for the queues of an arbitrary network is bounded above by (4.13)

(24)

where F _i,k (τ) is the actual service rate of link (i,k) at time slot τ when Algorithm 2 is in use.

For each control $I\in\mathcal{I}$, consider the quantity

where $Y_{i}(\tau,I)=X_{i}(\tau)-\sum_{k\in\mathcal {E}^{i}}w_{(ik)}(I)X_{k}(\tau)$. Note now that for a given control, the above maximization is attained by

$$\mu_i^{\star}(I)=\left \{ \begin{array}{l@{\quad}l} \hat{\mu}_i(I) & \mathrm{if\ } Y_i(\tau,I)>0\\ 0 & \mathrm{if\ } Y_i(\tau,I)\leq0, \end{array} \right . $$

i.e. transmitting with full rate from nodes with with positive Y _i (τ,I) and transmitting zero from nodes with negative Y _i (τ,I). This explains why the algorithm makes use of the function (.)⁺≡max[.,0]. Finally, note that Algorithm 2 solves a second maximization problem at each slot by selecting the control

$$I_{\tau}^*=\arg\max_{I\in\mathcal{I}}C(I). $$

Consider now any point in the interior of the throughput region, denoted by $\breve{\pmb{\lambda}}$. There exist flow variables $\mathbf{f}\in\mathcal{C}$ for which we will have for each node $\breve{\lambda}_{i}+\epsilon\leq\sum_{k\in\mathcal {E}_{o}^{i}}f_{ik}-\sum_{k:i\in\mathcal{E}_{o}^{k}}f_{ki}$. Also, we have for any τ

where $I^{*}_{\tau}$ is the control selected by Algorithm 2 at time slot τ and the equality below the second inequality follows from the fact that Algorithm 2 is designed exactly to maximize this term at each time slot. Now we can elaborate (24):

where the second inequality comes from the aforementioned derivation and the third inequality comes from the fact that the point $\breve {\pmb{\lambda}}$ is selected in the interior of the throughput region. Thus, we have shown that whenever the backlogs are large enough, Algorithm 1 guarantees a negative drift which brings the system to stability as long as the vector $\pmb{\lambda}$ is in the interior of the throughput region. □

Appendix C: More controls

In this subsection we examine closer the control that sends encoded packets (results of XOR operations of packet pairs belonging to two queues). Under this control, if queues a and b are selected, m packet pairs from the queues are transmitted in a slot by using “dummy” packets if necessary to form these pairs. In this case an inefficiency seems to arise as indicated by the following example. Suppose that the control selects to perform XOR operation of 10 packets, queue a has 20 queued packets and queue b 3 packets. Then according to the control specified, 10 packet pairs will be XORed by using 7 “dummy” packets from queue b. However, it would have been more efficient to send 3 pairs of XORed packets, without using any dummy packets, and then transmit 7 uncoded packets from queue a. Since this type of control is not included in the controls specified in the previous sections, the question arises whether one can do better by introducing more detailed controls. Below we show that this is not the case.

Let us extend the available controls by adding the following ones. If it is decided to transmit packets belonging to both of the queues a and b, then a control I(a,b,l _o ,l _a ,l _b ) may be selected, where l ₀,l _a ,l _b are nonnegative integers with the following interpretation. Let r _a ≥r _b . Then at most l ₀ XORed packets may be transmitted and at most l _x ,x=aorb uncoded packets may be transmitted from each of the queues. The l ₀+l _b packets must be seen by user b and hence they need to be transmitted at rate r _b . On the other hand the l _a packets need to be seen only by user a and hence they can be transmitted at the higher rate r _a . Since all these packets must be transmitted within a time slot, it must follow that

$$ \frac{l_{0}+l_{b}}{r_{b}}+\frac{l_{a}}{r_{a}}\leq1 $$

(25)

This type of controls covers the case described in the example above. The controls corresponding to transmission from one of the queues remain the same.

The model used in the previous sections can be extended to cover the case when this extended set of controls is chosen. The resulting stabilizing policy in this case is similar to the one described in Sect. 6. Specifically, at time t a reward C(I) is specified for each control, the reward depending on queue sizes at time t, and then the control whose reward is maximized is selected for slot (t,t+1]. The reward for a control I(a,b,l _o ,l _a ,l _b ) is given by

$$ C(I)= \biggl(\sum_{i\in\{ a,b \} } \biggl(X_{i}(t)- \sum_{k\in\mathcal{E}_{o}^{i}}X_{k}(t)w_{(i,k)}^{l}(I) \biggr)^{+} \biggr)l_{0}+X_{a}(t)l_{a}+X_{b}(t)l_{b}. $$

(26)

Let now c ₀,c _a ,c _b be the coefficients multiplying l ₀,l _a ,l _b in (26). Consider all controls I(a,b,l _o ,l _a ,l _b ) where a,b are fixed. If c ₀=max{c ₀,c _a ,c _b } then among all these controls those that set l _b =0 dominate. For the latter class of controls the reward becomes

$$C(I)=c_{0}l_{0}+c_{a}l_{a}. $$

Taking into account (25) we have

$$C(I)\leq \biggl(c_{0}-c_{a}\frac{r_{a}}{r_{b}} \biggr)l_{0}+c_{a}r_{a}. $$

Hence, if

$$c_{0}>c_{a}\frac{r_{a}}{r_{b}}, $$

then the control with l ₀=r _b ,l _a =l _b =0 dominates. If on the other hand c ₀≤c _a r _a /r _b then the control l ₀=0,l _a =r _a ,l _b =0 dominates. In either case we obtain one of the admissible controls of the policy defined in Sect. 6. In a similar fashion it can be shown that the cases c _b =max{c ₀,c _a ,c _b } or c _a =max{c ₀,c _a ,c _b } result in one of the admissible controls of the policy defined in Sect. 6.

We see from the discussion above, that under the new extended set of controls, the policy specified in Sect. 6 will still be optimal.

Appendix D: Beyond pairwise XOR

To provide some intuition as to why extensions are needed for the case where $|\mathcal{P}|>2$, we briefly explain here an example with three combined packets. If only partial state feedback is given, the virtual network that captures all state transitions might contain infinite number of nodes. Consider three packets from different flows which are combined together. Then assume that the scheduler receives a NACK from destination 1 and 2 and an ACK from 3. This means that both destination nodes 1, (2) did not correctly decode the packet, but this could be either because packet 2 (1) was not correctly overheard or packet 3 was not correctly overheard. To capture both cases, none of the packets can be characterized as bad; instead, the relay can now estimate new overhearing probabilities. A new state is required to capture this new partial knowledge of the scheduler thus obtained. Repeating this process, we see that in order to capture all the partial knowledge that the relay may have in the construction of the virtual network, we need to introduce infinite number of nodes. This introduces new technical challenges and extensions to the approach used in this paper.