AoI Minimization in Stationary Environments

In this section, we first describe the stochastic system model and then formulate the optimal scheduling problem. In the rest of the paper, the abbreviation UE will refer to any generic user equipment, and the term BS will refer to a Base Station. The area covered by a BS will be referred to as a Cell.

Channel model

We consider a cellular system where a set of $`N`$ UEs travel around in an area having $`M`$ BSs. Time is slotted, and at every slot, each BS can beam-form and schedule a packet transmission to one of the UEs in its coverage area. The wireless link to $`\textrm{UE}_i`$ from the BS in its current cell is assumed to be a stationary erasure channel with the probability of successful reception of a transmitted packet being $`p_i, 1\leq i \leq N`$. Hence, when a BS schedules a downlink packet transmission to $`\textrm{UE}_i`$ in its cell, the packet is either successfully received with probability $`p_i`$ or lost otherwise.

Mobility model

We assume that the UE mobility is modelled by a stationary ergodic process. Formally, let the random variable $`C_i(t) \in \{1,2,\ldots, M\}`$ denote the index of the cell to which $`\textrm{UE}_i`$ is associated with at time $`t`$ ¹. Then, according to our assumption, the stochastic process $`\{C_i(t)\}_{t \geq 1}`$ is a stationary ergodic process with the probability that $`\textrm{UE}_i`$ is associated with $`\textrm{BS}_j`$ at any time $`t`$ given by $`\mathbb{P}(C_i(t)=j)= \psi_{ij}, \forall i,j, t.`$ The probability measure $`\bm{\psi}`$ denotes the stationary occupancy distribution of the cells by the UEs. The mobility of different UEs is assumed to be independent of each other. Many different mobility models proposed in the literature fall under the above general scheme, including the i.i.d. mobility model, random walk model, and the random waypoint model . See Figure [AoI_mobility_fig] in the Appendix 13.1 for a schematic.

Packet arrival model to BS

We consider a saturated traffic model, where at the beginning of any slot, each BS receives a fresh update packet from a common external source (e.g., a high-speed optical backbone network). Since the UEs are interested in the latest updates only, the BS then deletes any old packet from its buffer and schedules the fresh packet for transmission to some UE following a scheduling policy. The saturated traffic model is standard in applications relying on continuous status updates , such as monitoring and surveillance with sensor networks , velocity and position updates for autonomous vehicles , command and control information exchange in mission-critical systems, disseminating stock-index updates and live game scores.

System states

For slot $`t`$, let $`t_i(t) < t`$ denote the last time before time $`t`$ at which $`\textrm{UE}_i`$ received a packet successfully from any BS. The Age-of-Information $`h_i(t)`$ of $`\textrm{UE}_i`$ at time $`t`$ is defined as

h_i(t) \equiv t-t_i(t).

In other words, the random variable $`h_i(t)`$ denotes the length of time elapsed since $`\textrm{UE}_i`$ received its last update before time $`t`$. Hence, the r.v. $`h_i(t)`$ quantifies the staleness of information available to $`\textrm{UE}_i`$. See Figure [AoI_fig] in the Appendix for a typical evolution of $`h_i(t)`$. The state of the UEs at time $`t`$ is completely specified by the Age-of-Information of all UEs, given by the random vector $`\bm{h}(t)\equiv \big(h_1(t), h_2(t), \ldots, h_N(t)\big)`$, and the association of the UEs with the cells, represented by the cell-occupancy vector $`\bm{C}(t)`$.

Policy space and performance metric

A scheduling policy $`\pi`$ first selects a UE in each cell (if the cell contains any UE), and then schedules the transmission of the latest packet from the BSs to the UEs over the wireless erasure channel described earlier. The scheduling decisions are required to be causal for it to be implementable in real-time. The set of all admissible scheduling policies is denoted by $`\Pi`$. Our goal in this paper is to design a distributed scheduling policy which minimizes the long-term average AoI of all users. In view of this, we consider the following average-cost problem:

\begin{eqnarray}
 \label{objective}
\textsf{AoI}^*=\inf_{\bm{\pi} \in \Pi } \limsup_{T \to \infty} \frac{1}{T} \sum_{t=1}^{T}\frac{1}{N}\bigg(\sum_{i=1}^{N} \mathbb{E}^\pi(h_i(t))\bigg).
\end{eqnarray}

Converse and Achievability

The AoI minimization problem given by [objective] is an example of an average-cost MDP with countably infinite state-space . Excepting a few cases with special structures (cf. ), such problems are notoriously difficult to solve exactly. Moreover, the standard numerical approximation schemes for infinite-state MDPs typically do not provide theoretical performance guarantees. In this paper, we take a different approach to approximately solve the problem [objective]. In the following Theorem, we obtain a fundamental lower bound to the optimal AoI. Finally, in Theorem [achievability_thm], we show that a simple online scheduling policy $`\pi^{\textsf{MMW}}`$ achieves the lower bound within a factor of $`2`$.

\begin{eqnarray}
 \label{lb_expr}
 \textsf{AoI}^* \geq    \frac{1}{2N g(\bm{\psi})}   \bigg(\sum_{i=1}^{N} \sqrt{\frac{1}{p_i}}\bigg)^2+ \frac{1}{2},
\end{eqnarray}

\begin{eqnarray*}
 \textsf{AoI}^* \geq    \frac{1}{2N \min \{M, N\}}  \bigg(\sum_{i=1}^N \sqrt{\frac{1}{p_i}}\bigg)^2 + \frac{1}{2}.
\end{eqnarray*}

Please refer to Appendix 13.1 for a proof of this theorem.

Discussion

Theorem [lb] gives a universal lower bound for the minimum AoI achievable by any admissible scheduling policy $`\pi \in \Pi`$. Interestingly, it reveals that the lower bound depends on the mobility of the UEs only through their stationary cell-occupancy distribution $`\bm{\psi}`$. Hence, given the stationary distribution $`\bm \psi`$, the lower bound [lb_expr] is agnostic of the details of the mobility model. The appearance of the quantity $`g(\bm \psi)`$ in the lower bound should not be surprising as it denotes the typical number of non-empty cells at a slot in the long run. Since a BS can transmit a packet only if at least one UE is present in its coverage area, the quantity $`g(\bm \psi)`$, in some sense, represents the multi-user diversity of the system.

Expression for $`g(\bm \psi)`$

To get a sense of the lower bound [lb_expr], we now work out a closed-form expression for $`g(\bm \psi)`$ for the uniform UE mobility pattern. Using linearity of expectation,

\begin{eqnarray}
\label{g_psi_eq}
    g(\bm{\psi})&=& \mathbb{E}_{\bm \psi} \sum_{j=1}^{M} \mathds{1}(\textrm{BS}_j \textrm{ contains at least one UE}\big)\nonumber \\ 
    &=&\sum_{j=1}^{M} \mathbb{P}_{\bm \psi} \big( \textrm{BS}_j \textrm{ contains at least one UE}\big).
\end{eqnarray}

Since the cells are disjoint, we readily conclude from [g_psi_eq] that $`g(\bm{\psi}) \leq \min\{M, N\}`$. Recall that $`\psi_{ij}`$ denotes the marginal probability that the $`\textrm{UE}_i`$ is in $`\textrm{BS}_j`$. Since the mobility of the UEs are independent of each other, the expected number of non-empty cells $`g(\bm \psi)`$ in Eqn. [g_psi_eq] simplifies to:

\begin{eqnarray}
 \label{g_eqn3}
g(\bm{\psi})= \sum_{j=1}^{M} \big(1-\prod_{i=1}^N(1-\psi_{ij})\big).
\end{eqnarray}

We now evaluate the above expression for the case when the limiting occupancy distribution of each UE is uniform across all BSs, i.e., $`\psi_{ij}=\frac{1}{M}, \forall i,j`$. The uniform stationary distribution arises, for example, when the UE mobility can be modelled as a random walk on a regular graph . In this case, Eqn. [g_eqn3] simplifies to

\begin{eqnarray}
 \label{g_spl}
g(\bm{\psi^{\textsf{unif}}}) = M \bigg(1-\big(1-\frac{1}{M}\big)^N\bigg).
\end{eqnarray}

For $`M=1`$, we have $`g(\bm{\psi})=1`$. For $`M\geq 2`$, we have the following bounds which are easier to work with

\begin{eqnarray}
 \label{g_unif}
 M\bigg(1-e^{-\frac{N}{M}}\bigg)\leq g(\bm{\psi^{\textsf{unif}}}) \leq M\bigg(1-e^{-1.387 \frac{N}{M}}\bigg).
\end{eqnarray}

For a derivation of the bounds in [g_unif], please refer to Appendix 13.2.

Conclusion and Future Work

This paper investigates the fundamental limits of Age-of-Information in stationary and non-stationary environments from an online scheduling point-of-view. In the stochastic setting, a $`2`$-optimal scheduling policy has been proposed for mobile UEs. For the non-stationary regime, a new adversarial channel model has been introduced. Upper and lower bounds for the competitive ratio have been derived for the adversarial model. As an immediate extension of this work, the effect of mobility in the non-stationary environment may be considered. The gap between the upper and lower bounds of the competitive ratio may be tightened. Also, it will be interesting to obtain the competitive ratio for $`w`$-step lookahead policies as a function of the prediction-window $`w`$.

Numerical Simulations

In this Section, we perform numerical simulations to compare the performance of the RHC and MA policies in the adversarial setting. Figure 1 shows the variation of time-averaged AoI with different number of UEs for $`T=500`$. A Monte-Carlo simulation with $`k=50`$ iterations was performed with randomly generated channels, and we plotted the worst-case AoI in Figure 1(a). For each of these iterations, at every time step, the number of Good Channels is selected uniformly at random between $`1`$ and $`N-1`$. From the plots, we see that RHC outperforms MA by a large margin even with just a small prediction window of $`w=3`$.

Figure 1(b) shows the variation of the AoI with the window size $`(w)`$ for the RHC policy. The number of UEs is $`N=5`$ and the simulation is performed for $`T=500`$ slots. The window-size is varied from $`1`$ to $`10`$. Each simulation is repeated for $`50`$ times and we plotted the maximum AoI value at the end of these iterations. We see that increasing the prediction window does not significantly decrease the average AoI.

A tight universal lower bound

In this section, we derive an improved universal lower bound for a single BS in which only one user may be scheduled among $`N`$ users at a slot. For simplicity, we assume that the UEs have statistically identical channels, i.e., $`p_i=p, \forall 1\leq i \leq N`$.
Let $`\mathcal{F}_{t}\equiv \sigma(\vec{h}(k), \vec{\mu}(k), 1\leq k \leq t)`$ be the sigma-algebra generated by the random age $`\vec{h}(t)`$ and scheduling decisions $`\vec{\mu}(t)`$ up to time $`t`$. For any online policy, the scheduling decision $`\vec{\mu}(t+1)`$ at the beginning of the slot $`t+1`$ must be measurable in the sigma-algebra $`\mathcal{F}_{t}`$. Let the random variable $`H_{\textrm{sum}}(t) \equiv (\sum_i h_i(t))`$ denote the sum of ages of all UEs at time $`t`$. Let $`B_i(t) \in \mathcal{F}_t`$ be the event under which the policy $`\pi`$ schedules the $`\textrm{UE}_i, \forall i`$. Clearly, $`B_i(t) \cap B_j(t)=\phi, \forall i\neq j`$ and $`\sqcup_i B_{i}(t)=\Omega.`$ Hence, we can write

\begin{eqnarray*}
%\mathbb{E}\big(h_i(t+1)| \mathcal{F}_{t}\big) = \begin{cases}
% 1+ h_i(t), ~~\textrm{if} ~i\neq j\\
% 1+ \bar{p}h_i(t), ~~\textrm{if} ~ i=j,    
% \end{cases}
\mathbb{E}\big(h_i(t+1)| \mathcal{F}_{t}\big) &=& \mathds{1}(B_i(t))\big(p + (1-p)(1+h_i(t)\big)\\
 &&+ \mathds{1}(B_i^c(t))\big(1+h_i(t)\big)\\
 &=& 1+ h_i(t) - p h_i(t) \mathds{1}(B_i(t)).
\end{eqnarray*}

Hence,

\begin{eqnarray*}
\mathbb{E}(H_{\textrm{sum}}(t+1)| \mathcal{F}_t)= N+ H_{\textrm{sum}}(t) - p \sum_{i}h_i(t)\mathds{1}(B_i(t)).
\end{eqnarray*}

Next, we define the set of exhaustive and mutually exclusive events $`\{B_i'(t)\}_{i}`$ as follows:

\begin{eqnarray*}
    B_i'(t) = \{ \omega \in \Omega: h_i(t, \omega) \geq h_j(t, \omega), \forall j \neq i\},
\end{eqnarray*}

where, in the above, we break ties uniformly at random. Then, it follows that

\begin{eqnarray*}
\mathbb{E}(H_{\textrm{sum}}(t+1)| \mathcal{F}_t) &\geq& N+ H_{\textrm{sum}}(t) - p \sum_{i}h_i(t)\mathds{1}(B_i'(t))\\
    &=& N+ H_{\textrm{sum}}(t) - p h_{\max}(t).
\end{eqnarray*}

It is clear that,

Introduction and Related work

T he Quality-of-Service (QoS) offered by any wireless network has traditionally been measured along three dimensions, namely, throughput, packet delay, and energy efficiency. There exists an extensive body of literature addressed to optimizing the cross-layer resource allocations to improve the QoS along these axes . However, it has been argued that the standard QoS metrics are primarily geared towards quantifying the degree of utilization of the system resources, and less towards measuring the actual user experience . With the explosive growth of hand-held mobile devices, Internet of Things (IoT), real-time AR and VR systems powered by the emerging 5G technology, the Quality of Experience (QoE) for the users plays a major role in today’s network design . In order to integrate QoE with the design criteria, a new metric, called Age-of-Information (AoI), has been proposed recently for measuring the freshness of information available to the end-users .

Designing efficient schedulers to minimize the AoI is currently an active area of research. The papers and study the average AoI minimization problem for static User Equipments (UEs) associated with a single Base Station (BS). In these papers, the authors propose a $`4`$-optimal Max-Weight-type scheduling policy (Theorem 12 of ). The paper proposes an optimal scheduling policy for the same setup, where the objective is to minimize the maximum AoI of all UEs. All of these papers consider a single-hop network model with static UEs only. The problem of AoI minimization in a multi-hop network with static UEs has been studied in . The paper considers the problem of designing an AoI-optimal trajectory for a mobile agent which facilitates information dissemination from a central station to a set of ground terminals. The effect of mobility on the capacity of wireless networks has been investigated in the classic work of . It has been shown that mobility, in general, increases the capacity of ad hoc networks. However, to the best of our knowledge, the effect of UE-mobility on the Age-of-Information has not been studied before. One of the main objectives of this paper is to study the AoI-optimal scheduling with mobile UEs.

Most of the existing works on wireless networks assume a stationary channel model for analytical tractability. In rapidly varying environments, such as high-speed trains and vehicle-to-vehicle communication, the standard stationary channel model assumption no longer holds in practice. This is particularly true for the 5G mmWave regime ($`\geq 28`$ GHz), which suffers from severe attenuation loss . On the other hand, designing an accurate and analytically tractable non-stationary wireless channel model remains an overarching challenge to the research community . To overcome this difficulty, in the second part of this paper, we propose a simple adversarial channel model for non-stationary environments and study the scheduling problem in this model. In addition to the emerging 5G technology, the adversarial channel model is also useful for ensuring reliable communication in the presence of tactical jammers, where the interferers, in reality, behave adversarially .

Our contributions:

We make the following contributions in this paper.

• We study the multi-user scheduling problem in stationary and non-stationary environments. The stationary environment is modelled stochastically, and the non-stationary environment is modelled using an adversarial framework. To the best of our knowledge, this is the first paper that considers the AoI-optimal scheduling problem in an adversarial setting.

• In the stationary setting described in Section 8.1, we design a $`2`$-optimal scheduling policy for mobile UEs. Our result improves upon the $`4`$-optimality bound known for static UEs .

• Our analytical result enables us to precisely characterize the effect of mobility on the overall AoI as a function of the long-term user mobility statistics. The results may also be effectively used for small-cell network planning .

• In the non-stationary setting of Section 9, we show that a simple online scheduling policy achieves $`O(N^2)`$ competitive ratio. Using Yao’s minimax principle, we show that no online policy can have a competitive ratio better than $`\Omega(N)`$.

• We propose a heuristic scheduling policy in Section 9.2 for the scenario where the future channel states can be accurately estimated for the next $`w`$ slots. We validate the efficacy of the proposed policy through numerical simulations.

The rest of the paper is organized as follows. In Section 8, we describe the stochastic model and formulate the problem in the stationary regime. Section 8.1 and 9 study the problem in the Stationary and Non-stationary environments respectively. In Section 10, we compare the performance of the proposed scheduling policies via numerical simulations. Section 11 concludes the paper with some pointers to open problems.

AoI Minimization in Non-Stationary Environments

In this Section, we consider the problem of AoI-optimal scheduling with $`N`$ static users in a non-stationary environment. Since non-stationary channels are difficult to model and analyze, we propose a new adversarial channel model in this setting. Besides being analytically tractable, all positive results in this model (e.g., Theorem [comp_ratio_ub]) carry over to less adversarial environments.

Channel model

A set of $`N`$ UEs are under the coverage of a single BS (i.e., $`M=1`$). The BS can transmit to any one UE at a slot. The channel state $`\textsf{Ch}_i(t)`$ of any $`\textrm{UE}_i`$ at any time slot $`t`$ could be either Good ($`1`$) or Bad ($`0`$). If the BS schedules a packet to a UE having a Good channel at that slot, the UE decodes the packet successfully. Otherwise, the packet is lost. We assume that, the states of the $`N`$ channels (corresponding to $`N`$ different UEs) are selected by an omniscient adversary from the set of all possible $`2^N`$ states at every slot. The scheduling policy is online and has no information on the channel states for the current or future slots. We will partially relax this assumption in Section 9.2, by considering a more general class of adversarial channel models with future channel estimations. The cost function over a horizon of $`T`$ slots is given by:

\begin{eqnarray}
\label{cost_fn}
\textsf{AoI}(T) = \sum_{t=1}^{T}\bigg(\sum_{i=1}^N h_i(t)\bigg).
\end{eqnarray}

The packet arrival model to the BS remains the same as in the stationary environment in Section 8.

Performance Metric

As standard in the literature on online algorithms , we gauge the performance of an online scheduling policy $`\mathcal A`$ using competitive ratio ($`\eta^{\mathcal A}`$), which compares the cost of $`\mathcal A`$ with that of an optimal offline policy OPT equipped with hindsight knowledge. More precisely, let $`\bm{\sigma} \in \{\{0,1\}^N\}^T`$ be a sequence of length $`T`$ representing the vector of channel states chosen by the adversary for the entire horizon. Then, the competitive ratio of the policy $`\mathcal A`$ is defined as :

\begin{eqnarray}
\label{comp_rat_def}
\eta^{\mathcal{A}} = \sup_{\bm \sigma}\bigg(\frac{\textrm{Cost of the online policy } \mathcal A \textrm{ on } \bm{\sigma}}{\textrm{Cost of OPT on } \bm{\sigma}}\bigg),
\end{eqnarray}

where the supremum is taken over all finite-length input sequences $`\bm \sigma`$, and the cost function is given by [cost_fn]. In the definition [comp_rat_def], while the online policy $`\mathcal A`$ has only causal information, the policy OPT is assumed to be equipped with full knowledge on the entire channel-state sequence $`\bm \sigma.`$

Characterization of the optimal offline (OPT) policy

For a given sequence of channel states $`\bm \sigma`$ of length $`T`$, the optimal offline policy OPT may be obtained by using Dynamic Programming. Let the variable $`C_t^*(h_1(t), h_2(t), \ldots, h_N(t))`$ denote the optimal cost-to-go from time $`t`$ when the AoIs of the the $`N`$ UEs are given by the vector $`\bm{h}(t)\equiv (h_1(t), h_2(t), \ldots, h_N(t)).`$ Using standard notations, we have the following backward DP recursion

\begin{eqnarray}
\label{opt_dp}
C^*_{t}(\bm{h}(t))&=& \underbrace{\sum_{i=1}^N h_i(t)}_{\textrm{cost for slot } t} + \underbrace{\min_{i: \textsf{Ch}_i(t+1)=1} C^*_{t+1}(\bm{h}_{-i}(t)+\bm 1, 1)}_{\textrm{optimal future cost}}, \nonumber \\
C^*_{T+1}(\bm{h})&=&0 \hspace{10pt} \forall \bm{h},
\end{eqnarray}

where the minimization in Eqn. [opt_dp] is over all UEs $`i`$ having a Good channel at slot $`t+1`$. When there is no UE with a Good channel at slot $`t+1`$ (i.e., $`\textsf{Ch}_i(t+1)=0, \forall i`$), the second term denoting the future cost is replaced with $`C^*_{t+1}(\bm{h}(t)+\bm{1})`$.

Comparison with the throughput maximization problem

It is interesting to note that the competitive ratio for the sum-throughput maximization problem in this adversarial model can be arbitrarily bad (i.e., unbounded). It can be understood from the following. Consider a system with two users. If an online scheduler $`\mathcal{A}`$ schedules $`\textrm{UE}_1`$ at any slot, the adversary can set the channel corresponding to $`\textrm{UE}_1`$ to Bad and set $`\textrm{UE}_2`$’s channel to Good and vice versa. At any slot, the optimal policy schedules the user with the Good channel state. Hence, any online scheduler $`\mathcal{A}`$ receives zero throughput, but OPT achieves the full throughput of unity.
Surprisingly enough, Theorem [comp_ratio_ub] shows that the Max Age (MA) scheduling policy, which schedules a user having the highest age (i.e., Scheduled UE at time $`t`$ $`\in \arg\max_i h_i(t)`$), is $`O(N^2)`$-competitive for minimizing the AoI.

For a proof of Theorem [comp_ratio_ub], please refer to Appendix 13.4. On a related note, in our recent work , we showed that the MA policy is exactly optimal for minimizing the maximum AoI of all UEs in the stochastic setting.

A Lower bound to the competitive ratio

In this section, we use Yao’s minimax principle for obtaining a universal lower bound to the competitive ratio [comp_rat_def] in the adversarial setting. In connection with online problems, Yao’s minimax principle may be stated as follows:

Using the above principle, it is clear that a lower bound to the competitive ratio of all deterministic online algorithms under any input channel state distribution $`\bm p`$ yields a lower bound to the competitive ratio in the adversarial setting, i.e.,

\begin{eqnarray}
\label{Yao_lb}
\eta \geq \frac{\mathbb{E}_{\bm{\sigma} \sim \bm{p}}(\textrm{Cost of the Best Deterministic Online Policy})}{\mathbb{E}_{\bm \sigma \sim \bm p}\textrm{(Cost of OPT)}}.
\end{eqnarray}

To apply Yao’s principle in our setting, we construct the following distribution $`\bm{p}`$ of the channel states: at every slot $`t`$, a UE is chosen independently and uniformly at random, and assigned a Good channel. The rest of the UEs are assigned Bad channels. The rationale behind the above choice of the channel state distributions will become clear when we compute OPT’s expected cost in Appendix 13.5. In general, the cost of the optimal offline policy is obtained by solving the Dynamic Program [opt_dp], which is difficult to analyze. However, with our chosen channel distribution $`\bm{p}`$, we see that only one UE’s channel is in Good state at any slot. This greatly simplifies the evaluation of OPT’s expected cost. The following Theorem gives the universal lower bound:

Please refer to Appendix 13.5 for a proof of this Theorem.

AoI minimization with Channel Predictions

The converse result in Theorem [comp_ratio_lb] states that under the adversarial channel model, any online scheduling policy has a worst-case competitive ratio $`\eta`$ which grows at least linearly with the number of UEs ($`N`$). This is quite a disappointing result when the number of UEs is large. On the flip side, the fully adversarial channel model may also be too restrictive in practice. To circumvent this situation, we now exploit the physical fact that wireless channels with block-fading may often be estimated quite accurately for a few subsequent future slots . We consider a relaxed adversarial model, where at any slot $`t`$, the BS can estimate the channels perfectly for a window of the next $`w \geq 0`$ slots. Here, $`w`$ is an adjustable system parameter that can be adaptively tuned by the policy in accordance with the scale of time-variation of the channels (e.g., fading block length). Similar to the adversarial model in Section 9, we continue to assume that the channel states are binary-valued and chosen by an omniscient adversary. Thus, the adversarial model discussed in Section 9 is a special case of this model with the window-size $`w=0`$. We now propose the following policy which exploits the $`w`$-step look-ahead information:
Receding Horizon Control (RHC:) The UE scheduled at each time $`t`$ is chosen by minimizing the total cost for the next $`w`$ time-steps. Hence, the scheduling decision at time $`t`$ is obtained by solving the DP [opt_dp] with the boundary condition $`C^*_{t+w+1}(\bm{h})=0, \forall \bm h`$.
The RHC policy was considered in in the context of load-balancing in data centers. It was shown that the RHC policy has a competitive ratio of $`1+O(\frac{1}{w})`$- approaching $`1`$ as the prediction window size $`w`$ is increased. Since the result of is not directly applicable to our problem, we examine the gain for AoI due to channel prediction capabilities via numerical simulations in the next section. Unsurprisingly, RHC reduces to the MA policy when the prediction window $`w=0.`$