On distributed scheduling with heterogeneously delayed network-state information

Abstract

We study the problem of distributed scheduling in wireless networks, where each node makes individual scheduling decisions based on heterogeneously delayed network state information (NSI). This leads to inconsistency in the views of the network across nodes, which, coupled with interference, makes it challenging to schedule for high throughputs.

We characterize the network throughput region for this setup, and develop optimal scheduling policies to achieve the same. Our scheduling policies have a threshold-based structure and, moreover, require the nodes to use only the “smallest critical subset” of the available delayed NSI to make decisions. In addition, using Markov chain mixing techniques, we quantify the impact of delayed NSI on the throughput region. This not only highlights the value of extra NSI for scheduling, but also characterizes the loss in throughput incurred by lower complexity scheduling policies which use homogeneously delayed NSI.

Appendix

Proof of Lemma 4.1

First, assume that the arrival rates E[A[t]] are such that $\tilde{\lambda} := (1 + \epsilon)E[\mathbf {A}[t]] \in \varLambda$ for some ϵ>0. Then, by the definition of the region Λ, it follows that we can construct a set of channel state dependent policies (i.e., f _l ’s) and “time-share” over those policies to get a long-term service rate of $\tilde{\lambda}$ (analogous to the proof of Theorem 1 in [3]). This, in turn, ensures that the network is stochastically stable.

Now for the other direction, given A[t] is supportable, by definition, there exists a scheduling algorithm $\mathcal{F}$ which makes the network stable. Since the system state Markov chain $\mathbf{Y}^{\mathcal{F}}[t]$ is positive recurrent, it exhibits a stationary distribution. Let us denote the scheduling decision under policy $\mathcal{F}$ as $S^{\mathcal{F}}(\mathbf{Y}[t])$. We will now construct a time-sharing scheduling policy $\mathcal{F}_{s}$ that depends on the steady state distribution of queue lengths and channel states (denoted as π(y), y={q(0:τ _max),c(0:τ _max)}) under policy $\mathcal{F}$. Let r(y)=Pr(q|c), computed using π(y).

At each time, when delayed channel state information C[t](0:τ _max)=c, the policy $\mathcal{F}_{s}$ probabilistically selects the scheduling decision $S^{\mathcal{F}}(\mathbf{q},\mathbf{c})$ with probability r(y=(q,c)). We observe that the time-sharing policy $\mathcal{F}_{s}$ allocates the same amount of service to each link as $\mathcal{F}$. Since A[t] can be supported by the time sharing policy, we have that E[A[t]]∈Λ. □

Proof of Theorem 4.2

The proof is split into two parts. Part one proves the threshold property of optimal solution and part two shows that optimal solution depends only up on the critical set of NSI. In other words, part two shows that the optimizing solution is independent of extra channel state NSI available at each node other than the critical NSI. (Proof: Part 1) We first show the following threshold property for the optimal solution to the optimization problem defined in Eq. (3),

Let us assume that we partly know the optimal solution. In particular, we assume that we are given the entire $\{F_{l}^{*}(\mathcal{P}_{l}(\mathbf{C}[t](0:\tau_{\max})))\}_{l\in \mathcal{L}}$ except $F_{k}^{*}(\mathcal{P}_{k}(\mathbf{C}[t](0:\tau_{\max})))$ at two different values of NSI ($\mathcal{P}_{k}(\mathbf{C}[t](0:\tau_{\max}))$ = {(C _k [t]=c _i ,r),(C _k [t]=c _j ,r)}) available at transmitter k.

To find $F_{k}^{*}(C_{k}[t]=c_{i},\mathbf {r}), F_{k}^{*}(C_{k}[t]=c_{j},\mathbf {r})$, we can solve the optimization (3) with other variables being fixed to the optimal solution. Consider the function that needs to be optimized:

Expanding this, we can write this as

Note that $\mathbf {z} \in \mathcal{ C}^{L\tau_{\max}}$ corresponds to one particular realization of channel states of the network for the past τ _max slots. Since the variables in the above optimization are only F _k (C _k [t]=c _i ,r) and F _k (C _k [t]=c _j ,r), we ignore the terms in the summation that do not involve these variables (as they are constant and do not affect the arg max). Let A _i denote the set $\{\mathbf {z} : \mathbf {z} \in \mathcal{ C}^{L\tau_{\max}}, \mathcal{P}_{k}(\mathbf {z}) = (c_{i},\mathbf {r})\}$. The new function we now have is

From the above expression, we observe that the above optimization for finding two variables F _k (c _i ,r),F _k (c _j ,r) splits into two independent optimization problems. First, let us consider the function that needs to be optimized to get F _k (c _i ,r):

From the above equation, we observe that the optimization function is linear in the variable F _k (c _i ,r). Using the fact that channels are independent across links, we have the above function of the form Pr(C[t]=c _i |r)(ac _i F _k (c _i ,r)+b(1−F _k (c _i ,r))), where parameters a and b are independent of value of c _i . Similarly, we can show that the function that needs to be optimized for variable F _k (c _j ,r) is of form ac _j F _k (c _j ,r)+b(1−F _k (c _i ,r)). Thus, the optimal solution is of the form

The above solution implies that if c _j ≥c _i and $F_{k}^{*}(c_{i},\mathbf {r}) = 1$, then $F_{k}^{*}(c_{j},\mathbf {r}) = 1$. This proves the threshold nature of optimal solution.

(Proof: Part 2) Let us consider the original function that needs to be optimized (3)

Expanding the above expression, we have

First, observe that each variable in the above expression has a unique notation. In particular, a variable that is associated with link l and a particular value of channel state $\mathbf {z} \in \mathcal{ C}^{L\tau_{\max}}$ is denoted by F _l (z) and more specifically $F_{l}(\mathcal{P}_{k}(\mathbf {z}))$. Consider a τ(≠τ ₁(l)∀l) and let the set $B(\tau) = \{ \mathbf {z} \in \mathcal{ C}^{L\tau_{\max}} : C_{1}[\tau] = c1 \, \mathrm{or} \, C_{1}[\tau] = c2 \}$ denote the set of variables whose optimal values are not known. In other words, assume that the optimal values of all the variables are known to us except those in set B.

We define the sets $B1 = \{ \mathbf {z} \in \mathcal{ C}^{L\tau_{\max}} : C_{1}[\tau] = c1 \}$ and $B2 = \{ \mathbf {z} \in \mathcal{ C}^{L\tau_{\max}} : C_{1}[\tau] = c2 \}$. The sets B1 and B2 satisfy B=B1∪B2. We now observe that the optimization functions that depend on variables in sets B1 and B2 are exactly identical up to a scaling factor. Therefore, the optimal solutions are also equal, and thus we have that optimal solution is independent of channel state information that is not critical NSI. □

Proof of Lemma 4.3

Consider the following Lyapunov function V[t], of the system state $\mathbf{Y}^{\mathcal{F}}[t]$, as follows:

We thus have

where ΔQ _l [t] is the difference Q _l [t+1]−Q _l [t]. Using the fact that arrivals and services are bounded in each time slot, we have

Using the queue update equation, we have

(6)

Since $(1+\epsilon) \mathbf {\lambda} \in \varLambda$, there exists $\{\bar{\eta}(\mathbf{c})\}_{\mathbf{c}}$ such that

From the scheduling algorithm optimization, we also have that

Taking the expectation on both sides of inequality (6) over C[t−τ _max], we have that

It now follows from the standard Foster–Lyapunov drift criterion [30] that the network is stochastically stable. □

Proof of Lemma 5.3

From Eq. (5), we have

Let us denote the summation $\sum_{i = 1}^{M} c_{i} P_{.i}^{\tau_{1}} 1_{c_{i} \geq T_{2,l}^{*}}$ by f _l (τ ₁) and the summation $\sum_{i = 1}^{M} P_{.i}^{\tau_{1}} 1_{c_{m} \geq T_{2,l}^{*}}$ by g _m (τ ₁). Thus, we have

Expanding the terms with γ _l and (1−γ _l ), we have

Using the triangle inequality, we have the following inequality,

By adding and subtracting the term $f_{l}(\tau_{2}) \prod_{m \in I_{l}} g_{m}(\tau_{1})$, we have

Using the triangle inequality results in

Let the set I _l be expressed as {m ₁,m ₂,m ₃,…,m _l }. By iterating the above idea of adding and subtracting terms on the second component of the above expression and using the triangle inequality, we have

Using the following upper bounds, |f _l (τ ₁)−f _l (τ ₂)|≤∑c _i β(τ ₁,τ ₂), |g _l (τ ₁)−g _l (τ ₂)|≤Mβ(τ ₁,τ ₂) and |f _l (τ ₁)|≤∑c _i , we have

where the last inequality follows from definition of γ=minγ _l . □

Proof of Corollary 5.4

From Eq. (4), we have

It is sufficient to prove that $\beta(\tau_{1}, \infty) \leq (1-M\delta)^{\tau_{1}}$ and $\beta(\tau_{1}, \tau_{2}) \leq 2(1-M\delta)^{\tau_{1}}$ ∀τ ₂≥τ ₁. Consider the following difference:

Let us denote ${\min}_{u} P_{uj}^{\tau}$ by $m_{j}^{\tau}$ and $\max_{u} P_{uj}^{\tau}$ by $M_{j}^{\tau}$. We now bound the above difference using $m_{j}^{\tau}$ and $M_{j}^{\tau}$, we have

By noticing that $\sum_{u : P_{iu} < P_{ku}} (P_{iu} - P_{ku}) + \sum_{u : P_{iu} \geq P_{ku}} (P_{iu} - P_{ku}) = 0$, we have

where the last inequality follows from the definition of δ.

Using the definition of $M_{j}^{\tau}$ and $m_{j}^{\tau}$, we have that

Using the fact that $m_{j}^{\tau}$ monotonically increases with τ, $M_{j}^{\tau}$ monotonically decreases with τ, and both have a common limit π _j , we have

(7)

Consider the following difference:

Using (7) in the above inequality, we have the desired corollary. □