I Introduction
In the times of COVID19 pandemic, realtime networked applications such as autonomous vehicles, immersive gaming, telemedicine and telesurgery played a vital role in enabling people to maintain socialdistancing while minimizing the loss in lives and productivity [21]. However, such applications need to maintain datafreshness at the nodes, which requires minimizing the age of latest datapacket (formally called the age of information (AoI) [12]) at the nodes.
In practice, AoI minimization is a complicated task. Factors such as limited energy, unavailability of fresh datapackets, etc., often restrict the control options for minimizing AoI. For example, to maximize the operational life, devices with limited energy need to restrict the number of datapackets that they transmit, as well as their transmission rate (speed), thus leading to large AoI. Hence, there is an inherent AoIenergy tradeoff that any transmission policy must consider.
In prior work, AoIenergy tradeoff has been considered primarily for energy harvesting model, and energy conservation model. In energy harvesting model, energy arrives at nodes intermittently, and at any time, depending on the energy level, the expected future energy arrivals, and packet availability, nodes need to make transmission decisions to minimize the AoI [23, 1, 2, 3]. In the energy conservation model, there is no limit on the energy that a policy may consume, and the AoIenergy tradeoff is formulated as an optimization problem that seeks to minimize either AoI (energy consumption) subject to energy (AoI) constraint [11, 19], or a linear combination of AoI and energy consumption [15, 20, 16, 24].
In this paper, we consider an energy conservation model, and the objective is to minimize the energy consumption subject to a peak AoI constraint, at all times over a fixed horizon of time. Essentially, we consider a node where datapackets (in short, packets) of fixed size (say, bits) are generated at arbitrary time instants, with bounded intergeneration time, and the node requires that by transmitting these packets, the peak AoI at the monitor is maintained below a threshold at all times, while consuming minimum energy. Thus, for every generated packet, the node needs to decide whether to transmit it or not. If the node chooses to transmit a packet, the AoI drops at the instant the node finishes transmitting the entire bits of the packet.
For the considered problem, at any time, if a new packet is generated, the node may discard (or, preempt) a packet already being transmitted, and switch to transmitting the new packet, thus seeking larger reduction in instantaneous AoI at the monitor, at the cost of wasting energy already consumed for partially transmitting the preempted packet. Note that the peak AoI constraint only requires that at any time , the generation time of the latest packet that has been entirely transmitted (all bits) by the node (until time ) is less than the peak AoI constraint. Moreover, once the node transmits a newly generated packet, it (node) does not need to transmit any old packet that it has not transmitted completely.
Additionally, we consider a tuneable/variable speed model (henceforth, called the speed scaling model), where at each time, the node can choose its transmission speed (in bits/sec) and correspondingly pay a price in terms of power/energy consumption. A large transmission speed reduces the AoI, however, at the cost of increased power consumption. We assume that the power consumed when transmitting at speed is that is convex and increasing. Two popular examples are and . Thus, there are two decisions to make at each time, whether to transmit an available packet, and at what speed.
Ia Prior Work
In past, in the context of AoI, the speed scaling model has been considered in [8, 1]. In particular, in [8], a node controls the number of bits of packet delivered to the monitor in a fixed time slot by controlling transmission power. However, it cannot transmit a single packet over multiple time slots. Therefore, to decrease the AoI, a node should deliver a certain number of bits in a single time slot. In [1], a node can control the transmission delay of packets by controlling the transmission power at the instant the packets are transmitted. However, [1] assumes all the packet generation times to be known in advance, which greatly simplifies the problem.
The speed scaling model has also been considered in context of AoIdistortion tradeoff [5, 7, 15], where a node controls the number of bits it transmits for each packet. Transmitting fewer bits take less time, and is assumed sufficient for minimizing AoI, but this results in distorted information at the monitor, where the level of distortion increases with decrease (increase) in the number of bits transmitted (transmission time).
The closest work to the considered problem in this paper, is on job scheduling with the speed scaling model [22, 4]. In this model, jobs arrive at a server sequentially, and the server needs to process all the arriving jobs (with adaptive speed) before their corresponding deadlines, while consuming minimum energy. Note that the job scheduling problem allows policies to be preemptive, however, all jobs have to be completed by their deadlines. When power function is (), [22] proposed a causal preemptive policy for which the competitive ratio (i.e., the ratio of the energy consumed by the causal policy to the energy consumed by an optimal offline policy, maximized over all inputs) is at most [4]. In the special case when the power function is , and all the packets are of equal size, with a common deadline, [10, 14] gave a competitive policy for the job scheduling problem.
The problem considered in [22, 4] differs from what we consider in this paper in one key aspect. In [22, 4] every job that arrives at the server needs to be processed completely, while in the considered problem, there is only a peak AoI constraint which could be satisfied by transmitting only a subset of the generated packets. Since the objective is to minimize total energy, subject to peak AoI constraint, an additional challenge compared to [22, 4] (where only speed and the order of processing is to be decided) is to identify the subset of packets to transmit.
In this paper, we consider a very general setting, where the time horizon is finite, the packet generation times are arbitrary, and there is a peak AoI constraint. To put this into perspective, we briefly discuss them below.
Finite Time Horizon
Most of the prior work pertaining to the AoI minimization problem considers infinite time horizon [12, 23, 15, 16]. Infinite time horizon assumption makes it possible to utilize stochastic properties of the problem, such as distribution over the packet intergeneration times/energy arrival instants, etc. This also makes it possible to ignore the edge effects (such as initial AoI, cost/energy incurred at times close to or time horizon , etc.). However, in this paper, we consider finite time horizon (such as in [6]), and fully acknowledge the edge effects.
Arbitrary Packet Intergeneration Times
In prior work, packet intergeneration time patterns have been considered for following three models: generateatwill model — a fresh packet is available at the node at all times (e.g., [17]), stochastic arrival model — fresh packets are generated with intergeneration time following particular distribution (e.g., [16]), and arbitrary arrival model — packet intergeneration times are arbitrary, and hence, need not follow any particular distribution or pattern (e.g., [15]). Clearly, arbitrary arrival model is the most general one, and may even model adversarial inputs.
Peak AoI Constraint
Primarily in AoI minimization problems, the actual metric being minimized is either timeaverage AoI (where is the time horizon, while denotes instantaneous AoI at time ) [16, 12], or average of Peak AoI (where denotes the number of packets successful transmitted by the the node until time horizon , while denotes the AoI just before the completion of transmission) [9, 18]. However, in practice these metrics (that consider only average values), may not be sufficient for critical applications such as telesurgery, driverless car, etc. For such applications, peak AoI constraint is a more suitable choice as it guarantees that at any time the instantaneous AoI is below a given threshold.
IB Our Contributions
The main contributions of this paper are as follows.

When all the generated packets are of same size ( bits), we derive a lower bound on the competitive ratio of any causal policy that seeks to minimize energy consumption while satisfying a peak AoI constraint. We show that for convex power functions (where denotes the transmission speed at time ), the competitive ratio of any causal policy is , where denotes the ratio of packet size to the maximum allowed peak AoI , and , and are finite positive constants such that and . Thus for any causal policy , if , the competitive ratio increases exponentially with increase in , while if (), the competitive ratio increases exponentially with increase in .

We propose a simple nonpreemptive causal greedy policy , and show that for convex power function , when the size of all the generated packets is equal to bits, the competitive ratio of the proposed greedy policy is . Consequently, we get that for (), , while for , (where ). Note that the derived upper bound on has similar order of dependence on the system parameters as the lower bound discussed in point 1. For a nonpreemptive causal greedy policy , such a competitive ratio is significant, as it shows that even an optimal causal policy (that can be much more complicated than , and may even preempt packets) cannot perform arbitrarily better (consume arbitrarily less energy) than .

When the generated packets are of arbitrary sizes in the range bits, we show that for , and (where denotes the maximum allowed peak AoI) the competitive ratio of the proposed greedy policy is . Moreover, if is unbounded, the competitive ratio of any causal policy is unbounded.
Ii System Model
Consider a node where datapackets (in short, packets), each of size bits are generated intermittently. In particular, the packet at the node is generated at time , where is determined by external factors (possibly adversarial), with intergeneration time . A packet is said to be delivered (at the monitor) at time if the node finishes transmitting the bits of packet at time .
At any time , the age of information (AoI) at the monitor is equal to , where is the generation time of the latest packet that has been delivered to the monitor until time . Further, in an interval , peak AoI at the monitor is defined to be . We consider a peak AoI constraint, i.e., for any given time horizon , the node requires that the peak AoI at the monitor in the interval is less than , where is a known constant.
Remark 1
If intergeneration time for any , then the peak AoI constraint is infeasible. Therefore, for the problem to be meaningful, we need the intergeneration time of packets to be bounded, and in particular, less than . So, we assume that there exists some such that , . In fact, we also need the initial AoI to be less than .
Remark 2
At any time , the peak AoI constraint only requires that , i.e., the generation time of the latest packet delivered to the monitor until time is less than time units old. Therefore, to satisfy the peak AoI constraint, it is not necessary to transmit every generated packet. For example, at time , if packets and (with generation times and respectively) are available at the node (i.e., neither of the packets have been completely transmitted until time ), and , then for satisfying the peak AoI constraint, it is sufficient to transmit packet only, without transmitting packet . Moreover, after packet gets delivered to the monitor, transmitting packet is no longer useful (and hence, need not be transmitted) because the peak AoI will still be determined by , i.e., the generation time of the latest packet delivered at the monitor.
Definition 1
A policy that at any time , can preempt (discard) an undelivered packet (i.e., a packet that is either transmitted partially, or not transmitted at all), and begin transmitting a newly generated packet, is called a preemptive policy. Note that the class of preemptive policies, by definition, includes all nonpreemptive policies, that never preempts any packet.
We consider a speed scaling model, where at any time , the node can transmit a packet at speed (in bits/sec) adaptively, i.e., the node can choose using causal information available at time . Also, transmitting a packet at speed consumes power , which is an increasing and convex function of speed , e.g., (), or , motivated by Shannon’s rate function. Therefore, if the node transmits packets at high speed, it incurs low AoI, but consumes large amount of energy. Hence, finding an optimal transmission speed is a nontrivial task.
Remark 3
Throughout this paper, we assume that when speed , the power consumption is . This assumption does not affect the results derived in this paper, but allows us to ignore the terms containing which merely appear as an offset.
In this paper, we consider the problem of finding a causal policy that chooses the packets to transmit (where preemption is allowed), the time interval over which the packets are transmitted, and their instantaneous transmission speed, so that the peak AoI is maintained below at all times over a time horizon , while consuming minimum energy. Formally, the objective can be stated as follows.
(1a)  
(1b) 
where is the set of all causal policies^{1}^{1}1Although packet generation instants are not known in advance, we assume that the time horizon is known. for packet scheduling and the choice of speed , and is the sequence of packet generation times. Note that the choice of packet a policy transmits at any time is inherently captured by (1a).
Remark 4
In this paper, the set of all causal policies implicitly refer to the set of all preemptive causal policies that may preempt packets (Definition 1).
Definition 2
A policy is said to be offline optimal if it satisfies the peak AoI constraint (1b), and there exists no other policy that can simultaneously satisfy the peak AoI constraint (1b), and consume less energy than , even if knows the generation time of all the packets in advance. Optimal offline policies are useful as they provide a lower bound on the energy consumed by any causal policy .
From prior work [22, 10, 14], it is known that finding an optimal causal policy for energy minimization problems under hard constraints (such as individual/common deadline for packets) is a challenging task. Hence, a usual approach is to find a causal policy whose competitive ratio, defined as the ratio of the energy consumed by a causal policy (to satisfy the peak AoI constraint) and the energy consumed by an optimal offline policy (Definition 2), maximized over all possible sequence of packet generation times , is small. Mathematically, the competitive ratio of policy is
(2) 
By definiton, a policy with small competitive ratio is robust. In the rest of the paper, we will consider a particular nonpreemptive causal policy, and show that its competitive ratio is at most , where . We will also derive a lower bound on the competitive ratio of any causal policy , and show that for different power function of interest, the competitive ratio of the considered policy has similar characteristics (dependence on parameters) as the derived lower bound. Note that for a nonpreemptive causal policy which is much simpler to implement than a general preemptive causal policy [13], this is a significant result.
Definition 3
At any time , a packet is defined to be fresh if its generation time is greater than , i.e. . Otherwise, the packet is stale.
Iia An Equivalent Deadline Constraint Problem
The peak AoI constraint (1b) can also be interpreted as a deadline constraint, where a deadline is defined as follows.
Definition 4
At any time , if is the generation time of the latest packet that has been delivered to the monitor until time , then the deadline at time is defined as , which is the earliest time instant at which the peak AoI constraint (1b) will be violated if no packet is delivered to the monitor after time . Equivalently, .
Remark 5
Note that is a nondecreasing function of . In fact, deadline increases in steps whenever a fresh packet (; Definition 3) is delivered to the monitor. This happens because (i.e., the generation time of the latest packet delivered until time ) increases discontinuously to the generation time of packet , at the instant packet is delivered to the monitor.
At any time , since the deadline , and the peak AoI constraint (1b) requires , therefore the peak AoI constraint (1b) is equivalent to the following deadline constraint:
(3) 
In other words, the peak AoI constraint (1b) is equivalent to the constraint that at any time , the current deadline must be in future. Hence, hereafter we consider the deadline constraint (3) (instead of peak AoI constraint (1b)) while minimizing the objective function (1a). Also, we define a feasible policy as follows.
Definition 5
A policy is defined to be feasible if it satisfies the deadline constraint (3) at all times .
Proposition 1
At any time , if , then for any feasible policy , at least one fresh packet has to be delivered to the monitor in interval .
Proof:
If the deadline at time is , and no fresh packet is delivered to the monitor in interval , then the deadline constraint (3) will be violated at time (follows from the definition of deadline; Definition 4). Hence, a feasible policy must deliver at least one fresh packet to the monitor in interval , for all , if .
Remark 6
In order to satisfy the deadline constraint (3), note that only fresh packets are needed/useful. Therefore, if a policy transmits a stale packet, it wastes energy. Hence, in rest of the paper, we only consider packets that are fresh, and at any time, the term ‘packet’ implicitly means a fresh packet.
Definition 6
For each packet generated at time , we define .
IiB Property of Convex Power Function
Convexity of power function implies the following property.
Lemma 1
Energy consumed in transmitting bits in a fixed interval is minimum if the bits are transmitted at a constant speed . Also, the minimum energy consumed in interval is .
Proof:
See Appendix A.
Corollary 1
For fixed , decreases with increase in .
Proof:
See Appendix B.
Iii Limitations of a Causal Policy
Before we discuss a particular causal policy for minimizing the energy consumption (1a) (under the deadline constraint (3)), it is important to note the fundamental limitations of any causal policy . Towards that end, Theorem 1 provides a lower bound on the competitive ratio of any causal policy , and shows (in Corollary 2) that for certain power functions , the competitive ratio of any causal policy is an increasing function of .
Theorem 1
For any causal policy , its competitive ratio
(4) 
where , and are finite positive constants, , and .
Proof:
To prove Theorem 1, we consider a particular scenario where the AoI at the monitor at time is , the time horizon (for ), and the packets are generated according to one of the two instances of packet generation times : , and . Then, for indexing all causal policies, we consider different cases based on the packet(s) that any causal policy may transmit in interval , and for each case, we compute the competitive ratio (2), where the maximization is with respect to . Finally, we take the minimum over the competitive ratio obtained for different cases considered above, and obtain (4), where , and are finite positive constants, , and . For detailed proof, see Appendix C.
Corollary 2
For any causal policy , if (), the competitive ratio increases exponentially with increase in , while for , the competitive ratio increases exponentially with increase in .
In the next section, we propose a feasible nonpreemptive (causal) greedy policy , and show that the competitive ratio (2) of is upper bounded by (where , , and ). Thus, we show that the dependence of the competitive ratio of on the system parameters is similar to the policyindependent lower bound (4) in Theorem 1.
Iv A Greedy Policy
Consider a greedy policy (Algorithm 1) that at any time , if the node is idle (i.e., not transmitting any packet) and the deadline , transmits the latest available (fresh) packet with constant speed (5), starting at time , throughout until the bits of the packet are delivered to the monitor (and waits otherwise),
(5) 
Remark 7
Note that the greedy policy does not preempt any packet. However, this is not a constraint, and in general, a policy is allowed to preempt packets to solve (1a).
Although greedy policy appears obvious, the speed (5) has been chosen carefully such that is feasible (since , if begins to transmit a packet at time , the packet will be delivered to the monitor before deadline ), and speed cannot be arbitrarily large (in fact, cannot be greater than ), unless a particular event happens (defined in Proposition 2) with regard to the packet generation time and for which we can lower bound the energy consumed by the optimal offline policy (Lemma 4 in Appendix F).
Proposition 2
If begins to transmit a packet at time , then the speed only if no packet was generated in interval .
Proof:
Consider time , when begins to transmit a packet with speed . To prove Proposition 2, we need to show that no packet must have been generated in interval . For this, let at least one packet be generated in the interval . Then we must have one of the following two cases:
There exists a packet generated in interval that begins to transmit at time
Recall that is a nonpreemptive policy (Remark 7). Hence, must have delivered packet (to the monitor) until time . Also, by hypothesis, begins to transmit packet at time with speed . Thus, from (5), it follows that . This implies that , which is possible only if no packet (including packet ) that was generated in interval was delivered to the monitor until time . This contradicts the fact that packet was generated in interval , and delivered until time . Hence, there cannot be a packet generated in interval that is transmitted by starting at time .
No packet generated in interval is transmitted by starting at any time
Since never idles if a fresh packet is available, this case is possible only if remains busy in the entire interval , transmitting packets that were generated before time . However, such an event cannot happen. To show this, let packet be the latest packet generated before time (say, at time , where ). Because only transmits a fresh packet, in interval , may transmit at most two packets generated before — a packet that was being transmitted by when packet was generated at time , and packet itself. Since transmits a packet with speed at least (Eq.(5)), it takes at most time units to completely transmit (deliver) a packet. Therefore, would finish transmitting both packet and packet before , and hence, must begin to transmit a packet generated in interval in subinterval , thus proving that this case is not possible.
Thus, we conclude that none of the above two cases are possible. This contradicts the assumption that a packet was generated in interval .
Corollary 3
If transmits a packet with speed greater than , then it must have begun to transmit packet immediately after it was generated at time , and completed the transmission at time .
Proof:
See Appendix D.
We next provide some intuition for the choice of speed (5) used by the greedy policy , that is at least equal to . For a greedy policy such as Algorithm 1, it is critical to have , because as shown in Example 1 below, a smaller value of speed such as may require to transmit some packets at much higher speed (compared to an optimal offline policy ), thus consuming large amount of energy (compared to the offline optimal policy ).
Example 1
Let at , , , and , where . Also, let three packets be generated at time , and , respectively. Then, an optimal offline policy will only transmit the third packet, with constant speed for time units, whereas the greedy policy (Algorithm 1) with speed will transmit all three packets with constant speed (each for time units).
However, if the speed choice (5) for is replaced with (where is the time when transmission of packet begins), then will still transmit all three packets, but the third packet will be transmitted with a constant speed .
Iva Competitive Ratio of the Greedy Policy
Theorem 2
Let . Then, the competitive ratio () of greedy policy is bounded as follows.
(6) 
Remark 8
Recall Theorem 1 that states that for any causal policy , , where , and , and . Theorem 2 shows that the dependence of on the system parameters is similar to the lower bound on in Theorem 1. So, for any (convex) and (i.e., ), if there exists a causal policy with bounded competitive ratio , then is also bounded.
Remark 9
In (6), if (), , while if , .
To prove Theorem 2, we need some structural results for an optimal offline policy , which we derive as follows.
V Properties of an Optimal Offline Policy
Consider an optimal offline policy . In this section, for simplicity, we only consider the packets that are transmitted by in interval , and index them as in ascending order of their generation times (i.e., ). Therefore, between the generation time of packets and , many other packets might have been generated, however, they are not transmitted by .
Lemma 2
If chooses to transmit packet , and (where ), then in interval , transmits at least two packets completely (i.e., bits).
Proof:
Thus, Lemma 2 implies that for , the intervals are special, and hence, we call them periods, defined rigorously next.
Definition 7
With respect to , the interval (where ) is called a period .
By definition, a period starts at the generation time of a packet transmitted packet by , and in each period , transmits at least two packets (from Lemma 2). Thus, consecutive periods overlap as shown in Figure 1(b). Hence, it is difficult to generalize Lemma 2 directly to the whole interval . Therefore, we further define frames (that are nonoverlapping) with respect to as follows.
Definition 8
With respect to , the interval (where and ) is called frame .
Remark 10
As shown in Figure 1(b), consecutive frames partition the timeaxis between (assuming the first frame starts at time ). Therefore, , , and there exists consecutive frames , such that lies in frame , and . So, the properties of in a frame can be easily generalized to the entire interval .
In Definition 7 and Definition 8, note that periods and frames are defined with respect to the packets transmitted by an optimal offline policy . Also, note that frame (interval ) is a subset of period (interval ), with . Hence, length of a frame is always less than .
Figure 1(a) shows a typical relation between periods and frames, where at time , the deadline is , and packet is delivered to the monitor at time . Thus the deadline is updated at time to . Then, the interval is called frame . Similarly, within interval , packet (with generation time ) is delivered to the monitor at time . Therefore, at time , the deadline gets updated to , and the interval is called frame .
Remark 11
Although Figure 1(a) shows typical properties of frames and periods defined with respect to the packets transmitted by , the speed profile shown in Figure 1(a) is not necessarily the speed chosen by , since that () is unknown. We show in Proposition 3 that within a frame, speed of exhibits several structural properties.
Proposition 3
The optimal offline policy

transmits the bits of a packet with constant speed,

never preempts a packet,

delivers packet in frame (), and

never decreases the transmission speed within a frame.
Proof:
See Appendix E.
Remark 12
The third property in Proposition 3 follows due to optimality of , and not from definition of frames. To understand this, note that frame depends only on packet and transmitted by , and does not restrict packet from being transmitted in frame , in addition to packet .
Vi Proof of Theorem 2
We prove Theorem 2 in two steps. In step 1, we show that , while in step 2, we show that .
Step 1: Upper Bound on
Consider an arbitrary sequence of packet generation times . From Remark 10, we know that the time axis can be partitioned into frames defined with respect to (Definition 8). Therefore, consider the consecutive frames such that the total time horizon interval is a subset of the union of frames to , and time horizon lies in frame (as shown in Figure 1(b) for ). Note that if (i.e., is less than the initial deadline ), then neither , nor transmits any packet because the deadline constraint (3) is trivially satisfied in the interval . Hence, we only consider the case of .
Since length of a frame is always less than (the length of a period that is equal to the peak AoI constraint), the time horizon . Therefore, in interval , if transmits number of packets with speed consuming units of energy, then (product of power consumption and upper bound on the length of time interval ). So, if denotes the number of packets that transmits with speed greater than (recall that transmits an entire packet at a constant speed), and denotes the total energy consumed by in transmitting these packets, then the total energy consumed by in interval is^{2}^{2}2Note that at any time , either transmits packets at speed greater than or equal to , or remains idle. We assume that when is idle, power consumption is (see Remark 3).
(7) 
Remark 13
Since transmits each of the packets at a constant speed greater than , the energy consumed by in transmitting the number of packets is .
From Proposition 3 (Property 3), it follows that delivers exactly one packet in each frame . Thus, transmits packets in interval .^{3}^{3}3 should not transmit any packet in frame because by definition, lies in frame , and hence, at the start of frame , the deadline (end of frame ) is already greater than . Also, the length of each frame is less than (length of a period). Hence, the energy consumed by in transmitting each of these packets is at least . Next, we show in Lemma 3 that there exists a subset of number of packets such that consumes at least units of energy in transmitting the packets in .
Lemma 3
There exists a subset of packets with cardinality , such that transmits all the packets in , and consumes at least units of energy in transmitting the packets in .
Proof:
Note that for , the claim is trivially satisfied. So, for the rest of the proof, we assume . Recall that transmits number of packets at speed greater than , consuming units of energy. Without loss of generality, let the packets be indexed as . Also, let transmits a packet } over the time interval , and consumes energy (in transmitting packet ). Since transmits only one packet at a time, for . Therefore, to prove Lemma 3, it is sufficient to show that in interval (for ), transmits at least one packet completely (entire bits), consuming at least units of energy. This follows from Lemma 4 (in Appendix F), where we show that for each packet that transmits with speed greater than , transmits at least one packet (where packet and may be same) completely during the time interval when transmits packet , at a constant speed at least equal to the constant speed with which transmits packet .
Remark 14
transmits a total of packets in interval (exactly one packet in each of the frames ). Also, from Lemma 3, it follows that transmits at least number of packets. Therefore, .
Hence, the total energy consumed by in interval is
(8) 
Hence, from (7) and (8), we obtain an upper bound on the competitive ratio (2) for as follows.
(9)  
(10) 
where in , we have used the fact that (Remark 13), follows because (transmitting bits at speed consumes more energy than transmitting bits at speed ), and we get by maximizing the R.H.S. of (9) with respect to . With , we get the result.
Step 2: Lower Bound on
Let , and for a fixed , consider the following problem instance, where , initial AoI , and three packets are generated at time , and , respectively. Optimal offline policy transmits only the third packet, with speed over the interval . On the other hand, transmits all three packets with speed , over the intervals , and , respectively. Therefore, , where .
Vii Related Setup with Arbitrary Packet Size
In Section II, we assumed that each packet generated at the node is of size bits, where is a constant. However, this need not be true in practice. So, if the system model considered in Section II is relaxed to allow packets of arbitrary sizes, then we have the following results.
Theorem 3
Among all the packets generated in the interval , let the size of the smallest packet be bits, and the size of the largest packet be bits.

The competitive ratio of the greedy policy (Algorithm 1) is , where .

When , the competitive ratio of any causal policy will be unbounded.
Proof:
See Appendix G.
The second result of Theorem 3 implies that if the packet sizes can vary arbitrarily, then the competitive ratio of any causal policy is unbounded. This is primarily because unlike causal policies, an optimal offline policy knows the generation time of all the packets in advance, and hence, can avoid transmitting large size packets, while satisfying the peak AoI constraint (1b).
If in addition to packets having arbitrary sizes, it is required that each generated packet has to be transmitted, then the offline optimal policy no longer has this advantage (i.e., cannot avoid transmitting large sized packets). In fact, if the packets are required to be transmitted on FirstComeFirstServe (FCFS) basis^{4}^{4}4In context of AoI, transmitting packets on FCFS basis is meaningful because receiving an old packet out of order is generally not useful., then a causal transmission policy with bounded competitive ratio can be obtained for particular power functions as follows.
Theorem 4
Proof:
To prove Theorem 4, we model the optimization problem (1a) with deadline constraint (3) and FCFS constraint, as an equivalent job scheduling problem, and use the existing results [4] for job scheduling problems to conclude the proof.
When all the packets generated in interval are to be transmitted on FCFS basis, at the instant the node begins to transmit a packet (generated at time ), the latest packet delivered to the monitor is packet . Hence, the deadline for packet is always . So, each packet can be considered as a job that needs to be processed before the deadline , such that the overall energy consumption is minimized.
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