I Introduction
Consider a monitoring system that observes a physical process, and sends its observations to another location, over a wireless network. Such systems arise naturally in various contexts, and are set to become even more prominent in the context of the Internet of Things (IoT) [1]. These include health monitoring systems, security devices and safety monitors. In many of these applications, it is crucial that the data received at the destination is fresh. More recently generated data is considered fresher than data generated earlier. Age of Information (AoI) is a recently introduced metric, that quantifies the freshness (or staleness) of information in such a communication system. It has received considerable interest owing to the fact that there are a number of applications which require fresh information to be delivered from one point to another; the relevance of packets that are generated earlier decays over time.
The notion of AoI [2, 3] enabled a new understanding of freshness of information. Consider a source generating packets to be sent to a destination, across a network. Let the packets be generated at the source at times , and received at the destination at (the packets need not be received in the same order in which they were generated). Define,
(1) 
This is the index of that packet among all packets received at the destination, till time , which has been generated most recently, i.e., the freshest packet present at the destination. The age of information is defined as the age of this packet, i.e.,
(2) 
The evolution of the age function is given in Figure 1. Note that AoI can be defined for the source as well, seeing it as a point that receives the packets with zero delay.
Define the Average AoI as,
(3) 
We will refer to the (average) age at the destination node to be the (average) age of the flow. Between the source and destination, packets experience queueing delays and transmission delays. While queueing delay contributes to the age process, delay is not identical to age. The age process depends on both the queueing delay and the rate at which packetized updates are being generated at the source. One can reduce the packet generation rate, which may lead to lower buffer levels, and hence, lower delays. However, owing to fewer updates, the age process may not reduce. On the other hand, sending too many updates may lead to congestion in the network.
The earliest works in AoI literature model the source to destination transmission system as a single hop queue. In [2], the problem of minimizing the average AoI for , and queues, under the First Come First Served (FCFS) discipline, is studied. Analytical expressions were obtained for Average AoI in the first two cases, and it was seen that there was an optimal load factor at which Average AoI was minimized. However, obtaining explicit expressions for AoI may not be easy under other service disciplines or complex network assumptions. Later works looked at AoI for other single queue models, such as sharing of an FCFS queue by two traffic streams [4], an Last Come First Served (LCFS) queueing system with and without preemption [5], and an system [6]. In [7], the authors consider a single base station, with multiple nodes trying to communicate time sensitive data to it, and propose three policies to minimize average AoI subject to throughput requirements. They also show that the AoI obtained in their policies is a multiplicative factor away from the optimal value. In [8] the authors study the problem of giving preemptive priority to one flow over another, in a single queue system, and obtain closedform expressions for average age and peak age. In works such as [9, 10], the problem of optimal sampling in order to minimize age is addressed, in the context of single hop transmission. However, they do not take into account the effects on queueing owing to a higher sampling rate.
More generally, one may model the source to destination transmission system as a multihop network. This models transmission of observations across a network which could be local, or even the internet. In the case of multihop networks, there have been a number of studies of the AoI problem. In [11]
, the authors consider a multihop network with a single flow. Under the assumption that service times are exponentially distributed, they show that the (preemptive) Last Come First Served (LCFS) service discipline minimizes the age among all disciplines, in a stochastic ordering sense. In
[12], the authors study distributed stationary policies that are not dependent on the channel state. Using these policies, they obtain tractable expressions for Average and Peak AoI, which are then optimized over this class of policies. However, this class of policies may be a small subset of all possible policies, and therefore not very likely to contain the policy that minimizes age among all possible policies. In [13], the authors propose an age based maxweight type scheduling policy that is throughput optimal, and further provide heavy traffic approximations for its performance. A concise survey covering diverse aspects of AoI, and giving a number of available AoI results for different system models, is [14].In this work we look at the AoI problem for a multihop network with multiple flows. The contributions of this paper are summarized below.

We present the State Dependent Scheduling with Packet Dropping policy (SDSPD). The system state consists of the queue lengths at different nodes, channel gains and the age of each flow at its destination. SDSPD provides a scheduling rule and a service discipline. The service rule consists of dropping older packets at each queue.

The SDSPD policy results in an age at least as low as that achieved by LCFS. Due to reduction of number of packets in the system, we are actually able to perform better, as demonstrated by simulations. We also compare the ages obtained with a theoretical lower bound, and show that the system performs close to this.

Due to the packet dropping rule, we are ensured of stability at all arrival rates. The system can accommodate samples arriving at any rate, without congestion. From simulations, it can be seen that it outperforms policies that do not drop packets, as well as stationary policies which do not take into account the system state (queue lengths and channel gains) while making control decisions.

Packet dropping also enables us to increase arrival rates without leading to congestion. Age is seen to reduce with arrival rate. Thus, network capacity is not a constraint on the sampling rate, and can be optimized independently.

Using an optimization with dynamically varying weights, we provide average age close to the desired targets. We provide a distributed version of the algorithm as well.
The remainder of this paper is organized as follows. In Section II, we provide the system model, formulate the problem and propose a control policy. The performance of the policy is analysed by simulations and compared with existing policies. The results obtained are discussed in Section III. Subsequently we have the concluding remarks in Section IV.
Ii System Model and Problem Formulation
We consider a multihop wireless network (see Fig (2)), modelled as a graph , where is the set of nodes, and is the set of edges (links) on . Packets are generated at source nodes, to be sent to various destination nodes. Each such stream of packets, corresponding to a sourcedestination pair, is called a flow. The set of all flows in the network will be denoted by . For any flow , we use and to denote its source and destination nodes. For each flow, a path is a set of nodes connecting the source to its destination. We assume that paths are fixed and known a priori. This would imply that a routing algorithm was employed beforehand to create these routes (see [15] for a survey of common routing algorithms in wireless sensor networks).
We have a slotted system, with time index . Each slot is of unit length and time duration denotes slot . The arrival process for a flow with source node is denoted by . We assume that evolves as an independent and identically distributed (i.i.d.) sequence across time slots and independent of other flows. The wireless channel gain of a link at time will be denoted by . This is also i.i.d. across time for a link, and is independent across links. The overall channel state is denoted by . We transmit at a constant power and a fixed rate. If a channel gain is above a threshold and interference from other channels is limited then we assume that there is a successful transmission. At each node , there is a queue which consists of packets of flow present at node . Let denote the number of packets of flow sent over link in time slot . The queues evolve as,
(4) 
where . By and we denote the AoI and average AoI of flow at its destination (see (2), (3)).
We assume that the links fall into interference sets. An interference set is a subset of such that no two members of that set can transmit simultaneously. These define the interference constraints of the system. Subject to these constraints, only certain configurations of links can be activated at a time.
A schedule is a mapping . If , then flow is scheduled to be transmitted on link in that slot. Not all mappings from to are feasible schedules. The links that are active must obey the interference constraints. Further, two flows cannot be simultaneously scheduled on a link. The schedules that obey these constraints are called feasible schedules. Denote the set of all feasible schedules by . Corresponding to each feasible schedule and channel state
, there is a rate vector
. We are interested in obtaining control policies that can reduce the AoI at the destinations. To this end, we propose the following policy.Iia Control Policy
The control policy we propose is called State Dependent Scheduling with Packet Dropping (SDSPD). This policy consists of a service discipline and an optimization rule.
IiA1 Service Discipline
Under the SDSPD policy, at each queue, we keep only the latest packet of a flow, and all others are discarded. Thus, if a more recently generated packet of a flow is received at a queue, all packets generated prior to that packet of that flow at the queue are dropped. This is a local decision that can be implemented at the node level. There is no need for exchange of information between the nodes for this purpose. Consequently, at all nodes and for all flows , . Such a service discipline will result in a performance similar to (or better than) an LCFS discipline.
IiA2 Optimization Rule
The schedule at time is chosen to be , where,
(5) 
where is the weight for flow , which is a function of the age of flow at time at its destination node. Also,
(6) 
where is a desired average age for flow , and is a fixed positive quantity. This represents a weighted queue policy with dynamic weights. The weight function enables us to differentiate between the flows, and gives higher priority to some flows, if desired. A flow with a higher weight will be scheduled more often, and consequently its age should decrease. A lower gives higher priority to flow .
Note that the quantity being optimized is different from the traditional maxweight metric, which involves a backpressure term. Owing to the packet dropping in our system, the vector remains in a bounded set for all time , and consequently, the system is always stable. Hence, we do not use a maxweight formulation, which is used generally to guarantee stability (within the capacity region of the system).
We will see in Section III that this policy is seen to yield a good performance in terms of the average AoI metric. We compare it with multiple policies, and see the benefit of dropping packets, even compared to policies which do LCFS. In the following section, we describe how we may solve the optimization problem in a distributed manner.
IiB Distributed Implementation
While the optimization (5) may be nonconvex in general, in case of smaller state spaces, it can be computed by a brute force search. For larger state spaces, it can be approximated by a linear relaxation (relaxing the scheduling variables to belong to the interval rather than the set ). The relaxed set of feasible vectors will be denoted by
. The relaxed linear program can be written in the form,
(7)  
(8) 
where , and is the rate that is achievable for the link if it is transmitting at fixed power, and none of the links it interferes with is on. This is now a separable linear program, and can then be solved in a distributed fashion.
One algorithm that can be used to solve it in a distributed fashion is the Incremental Gradient Descent algorithm (IGD) [16]. Let denote the set of all linkflow pairs, i.e., all elements of the form where and . Then, IGD provides,
(9) 
with , is a small positive number, is a vector which is one at its th position and is zero elsewhere, and denotes projection onto the set . Due to the vector , the update of the vector can be performed in a component wise manner. One can perform the update in (9) in a cyclic manner, going from one element of to the next. At each node, we can do the increment step in (9) for all the links that originate at that node, and then move to a neighbour. This process then continues cyclically. Thus, we can peform the optimization (7)(8) in a distributed manner, with messages passed between neighbouring nodes.
Since the power of transmission is fixed, and we assume that the channel gains take values from a bounded set, it follows that the rates are bounded by some . Further assume that the weights are bounded by some . Let us define,
(10) 
Then, the following result from [16] holds.
Lemma 1.
Thus, we can choose small enough to come close to the optimal value. Note that the algorithm does not require that the age at the destination be available at every node having that flow for computing the optimization. It is only necessary that it be known whether the age exceeds a threshold or not. We can have mini slots at the beginning of each slot, during which the destination node can broadcast a signal at a fixed power, to indicate whether the age has exceeded a threshold. Absence of the signal would indicate that the age is below the threshold. Using this simple signalling scheme, the one bit information corresponding to each flow can be broadcast.
Iii Simulation Results and Discussion
We compare the proposed policy, SDSPD, with five other policies. First, we have Backpressure with Dropping (BPD), which is the same as SDSPD, except that the optimization (5), we replace by , and . This can be considered as a maxweight (backpressure) policy with dropping. There are two other variants of the SDSPD policy, which use the same scheduling rule as SDSPD, but they do not drop packets. The first of these is SDSPnDFCFS, which has the FCFS service discipline, and the second, SDSPnDLCFS, has LCFS service. We also compare with BPLCFS and BPFCFS. which are backpresssure policies without dropping packets, with LCFS and FCFS service respectively. Finally we have the randomized scheduling policy of [12]
, which is a randomized stationary policy. It solves an optimization to obtain activation probabilities for links. It does not use instantaneous state information. Comparing with all these schemes allows us to evaluate the performance of the SDSPD algorithm against common scheduling schemes, some of which have been shown to perform well in terms of age.
We consider two example networks. All simulations are run for time slots, and averaged over such trials. For a theoretical comparison, we use the following lower bound.
Iiia An Approximate Lower Bound for Age
Consider a discrete time queue, with a Bernoulli arrival process, so that in each slot, a packet arrives with probability , and with probability , no packet arrives. Let denote the time between two packet arrivals. Clearly,
(11) 
The average age of the arrival process will be,
(12) 
If we assume that the channel takes values or with probability and respectively, the mean time between two time slots in which the channel state is , is , and this adds to the average age. Across a system of such links, we can obtain a lower bound on average age as,
(13) 
Observe that this is a loose bound, because it assumes that there is only one flow in the system. In a system with multiple flows, we may be far away from this lower bound.
IiiB Example Network 1
The network considered in this example is given by Figure 3.
The channel gains take value or with probability , in each slot. We will assume that if channel gain equals , exactly one packet can be successfully transmitted. This models a situation where the channel is above a threshold with probability , and hence ensures succesful transmission. The flows are from node to (path ), from node to node (path ), from node to (path ), from node to (path ) and from node to node (path ). The interference model assumes that any two links that have a common node interfere, and therefore cannot be active simultaneously. All weights in the optimization (5) are identically set to one (by choosing for all ). The arrival process is i.i.d Bernoulli across slots, with packet arrival rate for all the flows.
Table I gives the value of average AoI obtained at the destination for each flow, for SDSPD, SDSPnDFCFS, SDSPnDLCFS, BPD, BPFCFS, BPLCFS and the stationary policy of [12], as well as the loose lower bound .
Flow 15  Flow 67  Flow 810  Flow 119  Flow 112  
Lower Bound  17.5  17.5  17.5  13.5  13.5 
SDSPD  22.2  20.1  19.2  14.6  17.4 
BPD  24.6  20.5  19.6  14.8  17.9 
SDSPnDLCFS  25.5  24.6  22.8  15.6  18.9 
BPLCFS  37.4  31.9  27.6  16.3  23.5 
SDSPnDFCFS  33.9  30.5  26.2  15.9  21.9 
BPFCFS  47.2  37.3  30.1  16.3  25.4 
Policy of [12]  190.2  242.8  149.5  61.65  112.75 
It is easy to see that SDSPD is the best performing, and improves over the LCFS policy as well. The FCFS policy performs decently, but the age performance of the FCFS policy will deteriorate as we increase the arrival rates. The stationary policy of [12] performs an order worse than the other three, because it does not take into account channel or buffer state information. For SDSPD, the flows also have ages close to the lower bound. Recall that the lower bound was assuming a single flow using up all the resources. Even with five flows in the network, SDSPD performs quite close to the lower bound. The BPD policy performs close to SDSPD. However, SDSPD offers a slight improvement over BPD, especially for the first flow.
We repeated the simulation for arrival rate for all the flows (see Table II). Here we see that the age performances of the nondropping policies begin to deteriorate, owing to congestion. The SDSPD and BPD policies perform well. The age of all the flows of the SDSPD system have reduced, when compared to Table I. The policy is able to utilize the higher rate of updates to reduce the overall age.
Flow 15  Flow 67  Flow 810  Flow 119  Flow 112  
Lower Bound  15.2  15.2  15.2  11.2  11.2 
SDSPD  21.2  18.4  17.3  12.5  16.2 
BPD  24.9  19.2  17.9  12.7  16.9 
SDSPnDLCFS  43.1  51.6  40.4  16.2  19.9 
BPLCFS  95.5  98.3  79.6  19.2  51.1 
SDSPnDFCFS  97.8  100.3  81.9  17.6  50.6 
BPFCFS  160.3  154.0  121.9  20.1  78.1 
Policy of [12]  186.5  250.5  163.2  62.6  111.2 
From the above two tables, it may seem that the policy of [12] has the worst performance. However, this is not true in general. As we increase the arrival rates further, the average AoI for the non dropping policies begin to blow up as expected, owing to congestion. Table III summarizes the average AoI values for the different algorithms when arrival rate is . The BPFCFS algorithm performs the worst.
Flow 15  Flow 67  Flow 810  Flow 119  Flow 112  
Lower Bound  14.6  14.6  14.6  10.6  10.6 
SDSPD  20.9  18.1  16.8  11.9  16.1 
BPD  25.1  18.9  17.5  12.2  16.9 
SDSPnDLCFS  184.7  195.8  181.2  17.5  21.3 
BPLCFS  251.1  259.6  234.3  21.6  132.6 
SDSPnDFCFS  388.7  396.1  371.9  19.7  200.8 
BPFCFS  408.9  409.2  368.3  23.5  247.8 
Policy of [12]  199.1  264.4  163.8  65.8  102.2 
The above results demonstrate that the SDSPD policy can give low average AoI, close to the lower bound. Next, we demonstrate how we can use the weights to reduce the average AoI even further. This is done by fixing the values in (6). The results are given in Table IV, for the network in figure 3, with arrival rates of all flows fixed at .
Target average age for each flow  Flow 15  Flow 67  Flow 810  Flow 119  Flow 112 
*****  20.9  18.1  16.8  11.9  16.9 
18****  17.3  19.6  17.7  12.0  16.4 
15****  16.7  20.3  18.0  12.0  16.6 
15***11  16.7  21.6  18.3  12.7  12.3 
*16**12  22.2  16.6  17.8  12.9  12.8 
The first row gives the values of average AoI without targets. In the second row, we fix a target of for the first flow, and obtain an average AoI of 17.3. In the next row, we set the target to be , and obtain an average AoI of . Recall from Table III that the loose lower bound for AoI assuming that only one flow is present was , and therefore is a good value for average AoI. The AoI of other flows is only marginally increased. In the next row, we set targets of and for the first and last flows (with lower bounds and respectively), and obtain average AoI values of and . In the last row we set targets of and for the second and last flows, respectively, and obtain and respectively. Thus, the algorithm can provide close to optimal performance, and can prioritize some flows over others if necessary.
IiiC Example Network 2
The network considered in this example is given in Figure 4.
The channel, arrival and interference models are the same as in the previous example. The flows are , , and . Table V and Table VI depict values of Average AoI for the four flows, under the different policies considered, at arrival rates and , respectively. In this set of simulations too, we see that the patterns observed in the previous example hold.
Flow 19  Flow 38  Flow 410  Flow 411  
Lower Bound  19.5  15.5  17.5  19.5 
SDSPD  25.9  17.5  20.5  20.6 
BPD  29.8  17.7  21.2  21.1 
SDSPnDLCFS  28.0  19.2  26.5  25.9 
BPLCFS  42.7  21.7  27.8  26.2 
SDSPnDFCFS  37.2  20.7  27.9  27.4 
BPFCFS  59.0  22.8  28.9  26.9 
Policy of [12]  238.2  104.7  185.7  209.7 
Flow 19  Flow 38  Flow 410  Flow 411  
Lower Bound  17.2  13.2  15.2  17.2 
SDSPD  25.9  15.6  18.9  18.6 
BPD  32.1  15.9  20.1  19.3 
SDSPnDLCFS  28.8  20.3  48.5  47.2 
BPLCFS  83.1  35.4  55.4  56.1 
SDSPnDFCFS  79.9  32.4  55.9  58.5 
BPFCFS  179.6  49.5  66.0  68.4 
Policy of [12]  231.9  101.7  178.7  204.7 
IiiD Discussion
These experiments seem to suggest that dropping of packets locally at queues can help reduce age. Moreover, we get a policy that is robust to arrival rate variation. Now it may be that in certain applications, it is imperative to get all the packets from the source to the destination, without losing any information. In such cases one may use the SDSPnDLCFS scheme, which performs the best among all policies without packet dropping. The disadvantage of nondropping policies, however, is that in case of large arrival rates, the queues will be large, and the time to move all the packets across, from source to destination, will be huge. If the arrival rates are outside the stability region of the policy, this time may very well be not finite. In such a case, it is not even feasible to get all the packets across. Moreover, as the queue lengths build up, the complexity of optimizations used for resource allocation, may also increase. Against all these, SDSPD offers a distinct advantage. Additionally, the dynamically varying weight function allows us to obtain targeted age.
Iv Conclusion and Future Directions
In this work, we have presented a control policy which reduces the average AoI in a multihop wireless network. The control policy involves dropping of older packets at each queue, in favour of the youngest packet, and using the queue lengths and channel gains at each link. This policy is seen to perform better than policies without dropping, including LCFS schemes. Indeed, in many cases the scheme of dropping packets offers a huge improvement over LCFS schemes. It also performs much better than policies which do not use state information. Further, the average age obtained by the proposed policy is quite close to a theoretical lower bound as well. We further show that we can come even closer to the lower bound by using the age information at the destination. For applications for which there is no need to get all packets across to the destination, dropping of packets in the manner presented can help improve the performance in terms of age. Not keeping a backlog of older packets reduces buffering requirements. Moreover, there is no need to spend energy in transmitting packets which are not fresh. The network capacity is not a bottleneck in the transmission of fresh information. With packet dropping, higher rates of arrivals of packets do not result in an increase in the age due to queueing. We see a monotone decrease in the average age of different flows, as arrival rate increases. This suggests that in systems with packet dropping, the network is no longer a constraint on the optimal sampling rate. Thus, we can fix the sampling rate independent of network considerations, and dependent only on the energy or other requirements of the sampler at the source node.
Obtaining better theoretical bounds on the age for multihop networks and characterizing age optimal policies would be relevant directions for future research.
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