I Introduction
[]K eeping information fresh is an essential requirement for a variety of control and communication tasks. Stale feedback information in networked control systems may reduce the gain or phase margin, which may, in turn, push the system towards the verge of instability [2],[3]. Realtime status updates are necessary for a plethora of communication tasks including effective traffic monitoring [4], online gaming [5], intrusion detection [6], environment sensing using IoT devices [7] etc. While designing routing and scheduling policies for maximizing the throughput region is wellunderstood [8], [9], designing optimal policies for maximizing the information freshness is currently an active area of research [10].
With the advent of the 5G technology, it is becoming increasingly common for the Base Stations (BS) to serve the following two different types of UEs at the same time  TypeI: Delayconstrained UEs (e.g., UEs with URLLC type of traffic [11], such as control information updates for autonomous vehicles), and TypeII: Throughputconstrained UEs (e.g., UEs with eMBB type of traffic, such as HD multimedia streaming). Moreover, in NetworkControl applications where the delayconstrained UEs often perform a global task jointly (e.g., by sensing different parts of a sizeable cyberphysical system), it is critical to uniformly maximize the information freshness across all TypeI UEs to avoid information bottlenecks. In this paper, we study the problem of optimal joint scheduling of TypeI and TypeII UEs over wireless erasure channels.
We characterize the freshness of information at a UE by a metric called the Age of Information (AoI) [12], [13]. In our context, the AoI for a UE at a time indicates the time elapsed since the UE received a new packet from the BS prior to time . The larger the value of AoI for a UE at a time, the more outdated the UE is at that time. In this short paper, we consider two related problems on minimizing the AoI  Problem (1): In the presence of only TypeI UEs, our goal is to design a scheduling policy which minimizes the longterm peakAoI uniformly across all UEs, and Problem (2): When the TypeII UEs are to be scheduled simultaneously along with the TypeI UEs by the same BS, we consider the problem of minimizing the longterm peakAoI, subject to throughput constraint to the TypeII UEs.
Related Work
In a recent paper [14] Bedewy et al., consider the problem of optimal scheduling of status updates over an errorfree delay channel. They showed that, in that setting, the greedy MaxAge First scheduling policy is an optimal policy for both peakage and total age metrics. In the paper [15]
, He et al. consider the problem of link scheduling to transmit a fixed number of packets over a common interferenceconstrained channel such that the overall age is minimized. They proved the problem to be NPhard and proposed an Integer Linear Program and a fast heuristic. The authors continued studying the previous problem in
[16] for minimizing the peakage, and obtained similar results. See the monograph [13] for a detailed survey of the recent literature on Age of Information.Closer to our work is the paper [1], which studies a similar problem with singlehop wireless erasure channel. However, contrary to this paper, the objective of [1] is to design a policy to minimize the longterm average AoI. Using Lyapunovdrift based methodology, the paper [1] designs an approximately optimal policy for this problem. Designing an optimal policy in this setting still remains elusive. As we argued before, with distributed sensing applications, where all sensors need to stay updated uniformly, a more suitable objective is to minimize the longterm peakAoI across all users. In this paper, we design an exactly optimal policy for the problem (1) using MDP techniques. We also show that the proposed policy achieves the optimal large deviation exponent among all scheduling policies. Moreover, inspired by the analysis for the problem (1), we propose a heuristic policy for the problem (2), where we incorporate an additional throughput constraint for the eMBB UEs. Operating performances of these proposed policies have been compared extensively with other wellknown scheduling policies through numerical simulations.
The rest of the paper is organized as follows. Section II outlines the system model. In Section III, we consider the problem of minimizing the longterm peakAoI across all UEs when only TypeI UEs are present. In Section IV, we consider the problem of optimal joint scheduling in the presence of both TypeI and TypeII UEs. Section V presents numerical simulation results comparing the proposed policies with other wellknown scheduling policies, such as Proportional Fair and Randomized Policies. Finally, we conclude the paper in Section VI.
Ii System Model
We consider the downlink UE scheduling problem where a Base Station (BS) serves wireless users, each with fullbuffer traffic, meaning, each user is infinitely backlogged. The channel from the BS to the ^{th}
UE is modelled by a binary erasure channel with erasure probability
, where . Time is slotted, and the BS can transmit to only one user per slot. If the BS transmits a packet to the ^{th} UE at slot , the packet is either successfully decoded by the UE with probability , or, the packet is permanently lost with probability , independently of everything else. Refer to Figure 1 for a schematic diagram of the model. The objective is to design suitable downlink UE scheduling policy optimizing a given metric. In this paper, we consider two related problems  (1) designing peakAoIoptimal scheduler without any throughput constraints (Section III) and (2) Designing peakAoIoptimal scheduler with throughput constraint for a UE (Section IV).Iii Minimizing the PeakAgeofInformation
In this section, we consider the problem of minimizing the longterm peakAoI metric, which denotes the maximum Age of Information among all receivers associated with a basestation in a cellular network. With hard deadline constraints for each user in the case of URLLC traffic in 5G, minimizing the peakAoI metric is more practically meaningful than the minimizing the average AoI [1], [17].
At a given slot , define to be the peak instantaneous Age of Information among all users. Our objective is to design a scheduling policy , which minimizes the timeaveraged expected peakAoI. More formally, we consider the following stochastic control problem :
(1) 
subject to the constraint that at most one user may be scheduled at any slot. Define a greedy scheduling policy MA (MaxAge) which, at any given slot , schedules the user having the highest instantaneous age. More formally, at a slot , a user is scheduled which maximizes the metric (ties are broken arbitrarily). We establish the following theorem for MA:
Theorem 1 (Optimal Policy).
The greedy policy MA is an optimal policy for the problem (1). Moreover, the optimal longterm peakAoI is given by .
Theorem 1 states that the greedy policy MA
is optimal for the peakAoI metric. Interestingly, the optimal policy is independent of the channel statistics (the probability vector
). This should be contrasted with the approximately optimal policy MW for minimizing the averageAoI metric proposed in [1].We prove this theorem by proposing a closedform candidate solution of the Bellman’s equation of the associated averagecost MDP and then verifying that the candidate solution indeed satisfies the Bellman’s equation.
Proof:
The stochastic control problem under investigation is an instance of a countablestate averagecost MDP with a finite action space. The state of the system at a slot given by the current AoI vector of all users, i.e., . The perstage cost at time is , which is unbounded, in general. Finally, the finite action space corresponds to the index of the user scheduled at a given slot.
Let the optimal cost for the problem be denoted by and the differential costtogo from the state be denoted by . Then, following the standard theory of average cost countable state MDP (Proposition 4.6.1 of [18]), we consider the following Bellman Eqn.
where the vector denotes the dimensional vector of all coordinates excepting the ^{th} coordinate and is a allone vector.
Explanation
The Bellman Equation (III) may be explained as follows. Suppose that the current AoI state is given by . If the scheduler schedules a transmission to the ^{th} user, the transmission is successful with probability and is unsuccessful with probability . If the transmission is successful, the AoI of all users, excepting the ^{th} user, is incremented by , and the AoI of the ^{th} UE is reduced to . This explains the first term. On the other hand, if the transmission to the ^{th} UE is unsuccessful, the AoI of all users are incremented by . This explains the second term within the bracket. Finally, the term denotes the current stage cost.
Solution to the Bellman Equation (Iii)
We verify that the following constitutes a solution to the Bellman Equation (III):
(3) 
To verify the solution, we start with the of (III). Upon substitution from Eqn. (3), the expression corresponding to the ^{th} user inside the operator of Eqn. (III) is simplified to:
(4)  
Hence,
The optimality result now follows from [18]. ∎ The following interesting features of the optimal scheduling policy MA should be noted:

Unlike the approximately optimal policy for the averageAoI metric proposed in [1], the optimal policy for the peakAoI metric is completely agnostic of the channel statistics parameter . Hence, the policy MA
is simple to implement in practice as it requires no complex channel estimation procedures.

The proof of optimality of the MA policy gives an explicit expression for the associated costtogo function and the optimal cost . This is one of the rare cases where the associated Bellman Equation of an MDP has an analytic solution.
Iiia Large Deviation Rate Optimality and Stability
Although Theorem 1 establishes that the MA scheduling policy is optimal in terms of minimizing the longterm expected peakAoI, for missioncritical URLLC applications, we need to additionally ensure that the peakAoI metric stays within a bounded limit with high probability. The following Proposition 2 shows that the peakAge process has an exponentially light tail under the action of the MA policy. This ensures highprobability delay guarantees to URLLC traffic having a strict latency requirement.
Proposition 2.
Under the action of the MA policy, there exists a constant such that, for any fixed time and any ,
(5) 
Our objective in the rest of this subsection is to show that the large deviation bound (5) is asymptotically optimal in the sense that no other scheduling policy has lower probability exceeding a given sufficiently large AoIthreshold . Towards this end, in the next proposition, we establish a fundamental performance bound of the peakAoI tail probability under the action of any scheduling policy.
Proposition 3.
Under the action of any arbitrary scheduling policy , at any slot and for all , we have
(6) 
Proof.
Let . Now,
where the inequality (a) follows from the fact that consecutive erasures just prior to time for UE_{i∗} (which takes place with probability ) ensures that the age of UE_{i∗} at time is at least . ∎
Combining Propositions 2 and 3, we conclude that the MA policy achieves the optimal largedeviation exponent for the peakAoI metric.
Theorem 4.
The MA policy achieves the optimal largedeviation exponent for the maxage metric and the value of the optimal exponent is given by
Proof.
From Proposition 2, since the inequality in Eqn. (5) holds for any time , for the MA policy, we can write
Hence, taking limit as , we obtain
(7) 
On the other hand, for any fixed and at any time slot , Eqn. (6) of Proposition 3 states that for any scheduling policy , we have
(8) 
Since the bound (8) is valid for any , for any fixed , we can let to obtain
(9) 
Finally, since the bound (9) is valid for any , we can now let to obtain
(10) 
Discussion
Interestingly enough, although the timeaverage optimal performance depends on the statistical parameters of all UEs (Theorem 1), the optimal largedeviation exponent depends only on the parameter of the worst UE.
We conclude this Section with the following stability result of the Ageprocess under the MA policy.
Theorem 5.
See Appendix VIIIB for the proof.
Iv Minimizing the PeakAgeofInformation with Throughput Constraints
In this Section, we consider a generalization of the above system model, where, in addition to maintaining a small peakAoI, there is also a TypeII UE (denoted by UE_{1}), which is interested in maximizing its throughput. This problem can be motivated by considering a residential subscriber who is running one highthroughput application (with eMBBtype traffic), such as, downloading an HD movie, while also using several smart home automation IoTs, which have URLLCtype traffic, and hence, require lowlatency. In a smallcell residential network, all of these devices are served by a single BS typically located within the house [19].
Objective
Define a sequence of random variables
such that if the UE_{1} did not successfully receive a packet at the end of slot and otherwise. Let be a nonnegative tuning parameter. We are interested in finding a scheduling policy which solves the following problem :(11) 
The singlestage cost described above may be understood as follows: the first term denotes the usual maximum AoI across all UEs as in the previous Section. The second term imposes a penalty of if the UE_{1} does not receive a packet at the current slot. By suitably controlling the value of , a tradeoff between the peakAoI and achievable throughput to UE_{1} (serving the eMBB traffic) may be obtained [20].
Similar to the problem of Eqn. (1), the problem is also an instance of an infinitestate averagecost MDP with an additional actiondependent additive perstage cost term (). Arguing as before, and introducing an additional cost term arising due to the throughput constraint, the Bellman Equation for this problem may be written down as follows:
(12)  
where
(13) 
Here denotes the expected cost when the UE_{1}, which receives eMBB traffic, does not successfully receive a packet at slot . The above Bellman Equation (12) may be explained along similar line as the equation (III).
Inspired by its similarity to the problem , we try the same differential costtogo function as before. Using Eqn. (4), the RHS of Eqn. (12) can be evaluated to be
This yields the MATP (MaxAge with Throughput) scheduling policy, which minimizes the RHS of the Bellman Equation (12):
MATP: At any slot , schedule the user having the highest value of , where is given in Eqn. (13).
The MATP policy strikes a balance between minimizing the peakAoI (through the first term ) while also ensuring sufficient throughput to the eMBB user (through the second term ). As is increased, it gradually dominates the AoI term, which, in turn, facilitates scheduling the eMBB user.
Analysis of Matp
Note that,
(14) 
Furthermore, since , we have
(15) 
Hence, by taking , we see that under the action of the MATP policy, the supnorm of the difference between the RHS and LHS of the Bellman Equation (12) is bounded by the constant . In other words, upon denoting the RHS of the Bellman operator of (12) by (see [18] for this operator notation), we have
(16) 
Hence, we conclude that the policy MATP approximately solves the Bellman Equation (12). The efficacy of this policy is studied extensively in the Simulation Section V.
It should be noted that unlike the MA policy, the MATP policy takes into account the channel statistics (the value of ) for the eMBB user. It is oblivious to the channel statistics of other URLLC UEs, however.
Following a similar line of argument as in the proof of Theorem 5, we can establish the result below.
Theorem 6.
The Markov Chain is Positive Recurrent under the action of the MATP policy.
V Numerical Simulation
Simulated Policies
In this section, we simulate the following five scheduling policies for the downlink wireless system described in Section II  1) MaxAge Policy (MA) 2) Randomized Policy (RP) 3) MaxWeight Policy (MW) 4) Proportional Fair (PF) 5) MaxAge Policy with Throughput Constraints (MATP). The policy MA is described in Section III. The second policy RP chooses a UE randomly in each time slot. The third policy MW was proposed in [1] for approximately minimizing the longterm averageAoI metric. In every time slot , the policy MW chooses the UE which maximizes the metric amongst all UEs. The fourth policy is the wellknown Proportional Fair policy [21], [22] which at every slot selects the UE maximizing the metric . Here is the exponentiallysmoothed average rate, which is updated at every time slot as: where is the instantaneous throughput to the UE_{i} at slot . The fifth policy (MATP) is described in Section IV of this paper.
Simulation SetUp
We simulate a downlink wireless network with nodes, each with a binary erasure channel. The probability of successful transmission for the ^{th} channel
is sampled i.i.d. from a uniform distribution in
. Each simulation is run for slots, and an average of simulations is taken for the plots. For the PF algorithm, the value of is set to .Discussion
In Figure 2, we have compared the performance of five different scheduling policies on the basis of longterm MaxAge in the setup described in Section III. The number of TypeI UEs associated with the BS has been varied from to . For reference, we have also included the Theoretical Optimal value of AoI, given in Theorem 1. As expected, we see that the performance of the MaxAge (MA) policy matches with the optimal value. The MaxWeight policy performs slightly worse than the optimal MA policy. However, we find that the randomized and the PF policy performs very poorly in terms of the longterm peakage metric. The bottom line is that a utilitymaximizing policy (such as PF, which maximizes the summation of logarithmic rates of the UEs) may be far from optimality when maximizing freshness of information on the UE side.
Figures 3 and 4 pertain to the problem discussed in Section IV, where the objective is to minimize the longterm peakAoI, while providing a certain throughput guarantee to UE_{1}, which serves eMBB type of traffic. Consistent with the observation that the parameter amplifies the cost for throughputloss to UE_{1} (viz. Eqn. 11), Figure 3 shows that, under the action of the MATP policy, UE_{1} receives more throughput as the parameter is increased. For very large value of the (), the throughput to UE_{1} saturates to ( in the Figure 3), which is the maximumthroughput obtainable for UE_{1} if UE_{1} is scheduled exclusively. Figure 4 compares the performance of various scheduling policies in terms of the metric given in Eqn. (11). Similar to Figure 2, we see that the MW and the Randomized Policies perform poorly in this case. However, the proposed approximately optimal policy MATP performs close to the theoretical bound and the performance of the MW policy is also not very far from that of the MATP policy.
Vi Conclusion
In this paper, we have derived an optimal downlink scheduling policy for minimizing the longterm peakAgeofInformation for UEs with URLLC type of traffic. We have also proposed a heuristic scheduling policy in the case when one of the UEs is throughputconstrained. Extensive numerical simulations have been carried out comparing the efficacy of different scheduling policies. Deriving an optimal scheduling policy for minimizing peakAoI with throughput constraint is an interesting research direction which will be pursued in the future.
Vii Acknowledgement
The second author would like to thank Prof. Eytan Modiano and Igor Kadota from MIT, for the useful discussions that led to this paper.
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Viii Appendix
Viiia Proof of Proposition 2
Using Union bound, we have
(17) 
Next, for any UE , the event occurs iff at time , it has been at least slots since UE_{i} received a packet successfully. Let and . Since the MA policy successfully serves other UEs exactly once between two consecutive successful service of UE_{i}, it follows that, during the last slots prior to time , at most UEs have successfully received a packet. Thus,
where we have used the bound and defined ^{1}^{1}1In the case , we take .. The final result now follows from Eqn. (17).
ViiiB Proof of Theorem 5
It is clear that forms a countablestate Markov Chain under the action of the MA policy. To show the positive recurrence of the chain , we analyze the stochastic dynamics of the random variable , and choose it as our Lyapunov function for the subsequent drift analysis.
Let , where we break ties arbitrarily. Then, the MA policy transmits a fresh packet to the ^{th} user at time . Over the binary erasure channel that we consider, this packet transmission is successful with probability and is unsuccessful w.p. . In case the packet transmission is unsuccessful, the age of all users increase by . Thus,
(18) 
On the other hand, in the case when the packet transmission
is successful, the age of the ^{th} user drops to , and the age of all other users increases by . Hence, we can write
Finally, note that, under the action of the MA policy, we have . Hence, from the above equation, we conclude that
(19) 
Let be the sigmafield generated by the random variables , i.e., Using Equations (18) and (19), we upperbound the oneslot conditional drift as follows:
(20)  
where . The drift upperbound (20) shows that if for any , we have . Thus, the oneslot conditional drift of the chosen Lyapunov function is strictly negative whenever the state lies outside the bounded dimensional box . Finally, using the FosterLyapunov Theorem for stability of Markov Chains (Proposition 6.13 (b) of [23]), we conclude that the Markov Chain is Positive Recurrent.
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