3-Competitive Policy for Minimizing Age of Information in Multi-Source M/G/1 Queuing Model

I Introduction

Reliance on time-sensitive networked applications (such as remote monitoring, telehealth services, control, etc.) for critical roles, have necessitated the need for a general transmission policy that could ensure timely delivery of fresh updates of each source, at the corresponding destination. The policy should mitigate the effect of constraints on update generation, transmission delay, capacity of the shared channel, number of sources, etc., and must be simple to implement. In this paper, we propose a particular randomized policy, and show that it adheres to the above-mentioned requirements.

In particular, we consider a general multi-source M/G/1 queuing model, where at each source, the updates are generated with exponentially distributed inter-generation time. Also, at any time, at most one source can transmit, and transmission delay for each source follows some general distribution. Note that the mean update generation rate and transmission delay distribution may be different for each source, however they do not change with time.

The objective is to find an online transmission policy, such that at any time, the latest update of each source available at the destination is fresh. Formally, we quantify freshness using age of information (AoI) metric [kaul2012real, kaul2012status, saurav2021game, saurav2021online]. At any time, AoI of a source is equal to the time elapsed since the generation time of the latest update of the source, available at the monitor. Thus, for any policy, if the long-term average AoI (AAoI) is small (less than a certain threshold) for each source, then the policy is considered to have ensured timely delivery of fresh updates. In the considered model, we assume that each source $ℓ$ has its own recommended AAoI threshold $α_{ℓ}$ .

Ideally, for each source $ℓ$ , its AAoI should be less than the recommended threshold $α_{ℓ}$ . However, due to constraints on availability of updates, transmission delay, excessive number of sources, etc., this may not be possible. In fact, the set $C$

of vectors

α = (α_{1}, α_{2}, \dots)

, such that for some online policy

π^{⋆}

, the expected AAoI of each source

ℓ

is less than

α_{ℓ}

, is not known. Therefore, in this paper, we first derive a necessary condition to characterize this set

C

. Then, we propose a simple randomized policy

π_{R}

, that at any time, among all the sources, picks a source

ℓ

with a fixed probability

p_{ℓ}

, and transmits its updates (if it has an update). We show that for any vector

α

that satisfies the derived necessary condition, the expected AAoI of each source

ℓ

under

π_{R}

is at most

3 α_{ℓ}

In the later part of this paper, we also consider the setting, where instead of threshold vector $α$ , only relative weights $w = (w_{1}, w_{2}, \dots)$ are known for the sources, and the objective is to minimize the weighted sum of the expected AAoI (WSAAoI) of the sources, following an online policy. We show that for an appropriate choice of randomization parameter $p_{ℓ}$ ’s, the same randomized policy $π_{R}$ proposed for the earlier setting, guarantees WSAAoI that is at most three times the WSAAoI for an optimal online policy.

Although the theoretical guarantee for $π_{R}$ has a multiplicative gap of $3$ (relative to an optimal online policy), for the general multi-source M/G/1 queuing model, this is a significant result. In prior work, for multi-source setting with stochastic packet generation, the best known online policy [kadota2019minimizing] has a multiplicative gap of $4$ (the results in [kadota2019minimizing] are for slotted time M/M/1 queuing model, unlike the general continuous time M/G/1 model considered in this paper).

In most of the prior work on AoI (e.g., [kadota2018optimizing, kadota2018scheduling, kadota2019minimizing, saurav2021minimizing]), a major reason for the large gap in AAoI guarantee of an online policy, and an optimal online policy $π^{⋆}$ , is the use of a weak lower bound on the AAoI of $π^{⋆}$ . Generally (as in [kadota2018optimizing, kadota2018scheduling, kadota2019minimizing, saurav2021minimizing, bhat2020throughput]

), the lower bound disregards the effect of variance

σ_{ℓ}^{2}

of inter-generation time of updates on the AAoI of sources under

π^{⋆}

. In this paper, we specifically include the effect of

σ_{ℓ}^{2}

(for each source

ℓ

) in the lower bound on the AAoI of sources, and were able to improve the lower bound (and minimize the gap).

Currently, in this paper, the major limitation in improving the theoretical guarantee for $π_{R}$ (or, designing a better online policy), is the weak lower bound on the waiting time of updates under on optimal online policy $π^{⋆}$ . For any transmitted update, the waiting time is equal to the difference between the time when the update got generated, and time when a policy transmits the update. Thus, waiting times have a significant impact on the AAoI of any policy. However, in general multi-source setup with stochastic update generation, to the best of our knowledge, none of the prior work has been able to derive a lower bound on the waiting time of updates under $π^{⋆}$ , that is better than $0$ .

However, when the number of sources in the system is $1$ , the problem gets simplified, and under specific assumption [saurav2021minimizing, sun2017update] optimal online policies are known. Hence, for fair evaluation of the performance of $π_{R}$ , instead of comparing $π_{R}$ with a lower bound, we considered a particular setting of [sun2017update] (source can generate a new update at any time, and the transmission delay for each update follows some general distribution), for which an optimal online policy $π^{⋆}$ is known, and compared $π_{R}$ with $π^{⋆}$ , using numerical simulation. Despite the simplicity, and broader applicability of $π_{R}$ , we found that its AAoI is close to the AAoI of $π^{R}$ (for the two transmission delay distributions that we considered).

The rest of this paper is organised as follows. In Section II, we discuss the considered M/G/1 queuing model in detail, and formally define the objective. In Section III, we derive the necessary condition to characterize the set $C$ of AAoI vector $α$ that an optimal online policy may achieve. In Section IV, we propose the randomized policy $π_{R}$ , and derive an upper bound on the expected AAoI of each source under $π_{R}$ . We show that for any $α \in C$ , the expected AAoI of the sources under $π_{R}$ is at most $3 α$ . In Section V, we consider weighted sum expected AAoI (i.e., WSAAoI) minimization problem, and generalize $π_{R}$ (and the corresponding guarantee) for this setting. Finally, in Section VI, we discuss the numerical simulations.

Ii System Model

Consider a system consisting of $N$ sources and a monitor. At each source $ℓ \in {1, \dots, N}$ , updates (henceforth, packets) are generated with exponentially distributed inter-generation time $X_{ℓ}$ , with mean $μ_{ℓ} < \infty$ . The sources transmit their packets to the monitor, over a common channel, that at any time $t$ , allows at most one source to transmit (one packet). Each packet transmitted by source $ℓ$ gets received at the monitor after random transmission delay $d_{ℓ} \sim D_{ℓ}$ , where $D_{ℓ}$ is some general distribution with mean $γ_{ℓ} < \infty$ .¹¹1For different sources, $D_{ℓ}$ ’s may belong to different family of distributions. The sources may choose whether to transmit a packet, or discard it, but only until the transmission of the packet is initiated. Once initiated, a transmission cannot be preempted.

Definition 1

While a packet is under transmission, the channel is said to be busy. Otherwise, the channel is free. A transmission can be initiated only when the channel is free.

At any time $t$ , the age of information (AoI) of a source $ℓ$ (denoted $Δ_{ℓ} (t)$ ) is equal to the time elapsed since the generation time of the latest packet of the source that has been received at the monitor. Thus, as shown in Figure 1, $Δ_{ℓ} (t) = t - λ_{ℓ} (t)$ , where $λ_{ℓ} (t)$ denotes the generation time of the latest update of source $ℓ$ that has been received at the monitor until time $t$ . Average AoI (in short, AAoI) of source $ℓ$ until time $t$ is defined as

{¯ ¯¯¯ ¯ Δ}_{ℓ} (t) = \frac{\int_{0}^{t} Δ_{ℓ} (i) d i}{} t .

(1)

Fig. 1: Sample AoI plot for source

ℓ

(assuming AoI at time

g_{ℓ 0}

to be

0

). Here,

g_{ℓ i}

and

r_{ℓ i}

respectively denote the generation time and transmission completion time of packet

i \geq 1

. Also,

T_{ℓ i}

’s denote the inter-generation time of transmitted packets.

Definition 2

A centralized online transmission policy (in short, an online policy) is an algorithm, that at each time $t$ (when the channel is free), using only the causal information available at all the sources at time $t$ , decides which source gets to transmit (at time $t$ ). In this paper, we only consider the set of online policies $Π_{O N}$ that never preempt any packet that is under transmission.

The objective in this paper is to find an online policy $π \in Π_{O N}$ (Definition 2), such that for any given vector $α = (α_{1}, \dots, α_{N})$ of recommended AAoI for the $N$ sources, the policy $π$ minimizes the ratio ${¯ ¯¯¯ ¯ Δ}_{ℓ} / α_{ℓ}$ for each source $ℓ$ . To make this formal, for any given $α$ , we define the cost for a policy $π$ as

Γ (π; α) = max ℓ {lim t \to \infty E_{π} [{¯ ¯¯¯ ¯ Δ}_{ℓ} (t)] / α_{ℓ}},

(2)

where $E_{π} [\cdot]$ denotes expectation with respect to policy $π$ , as well as the packet generation and transmission delay distribution of the sources. Thus, the objective is to find

π_{O N}^{⋆} = arg min π \in Π_{O N} Γ (π; α) .

(3)

Remark 1

Since $μ_{ℓ}$ and $γ_{ℓ}$ are finite for each source $ℓ$ , as shown in Appendix A, there exists an online policy $π \in Π_{O N}$ , such that the expected AAoI of each source under policy $π$ is finite (except when $N \to \infty$ , in which case, $Γ (π; α) \to \infty$ for all online policies $π \in Π_{O N}$ , and the objective (3) becomes meaningless). Hence, the expected AAoI of each source under $π_{O N}^{⋆}$ must also be finite. Therefore, without loss of generality, in the rest of this paper, we disregard the policies in $Π_{O N}$ , for which the expected AAoI of any of the source is infinity.

Note that $Γ (π; α) \leq 1$ implies that the AAoI of each source under policy $π$ , achieves the recommended value. However, for arbitrary $α$ , such policy $π$ may not exist (e.g., if $α = (0, \dots, 0)$ , $Γ (π, α) = \infty$ , $\forall π \in Π_{O N}$ ). Therefore, for problem (3) to be meaningful, we only consider $α \in C$ , where

C = {α \in R^{N} | \exists π \in Π_{O N}, s.t. Γ (π; α) \leq 1}

(4)

denotes the set of feasible $α$ (Definition 3), called the capacity region.

Definition 3

$α$ is said to be feasible under policy $π \in Π_{O N}$ , if $Γ (π, α) \leq 1$ , i.e., ${lim}_{t \to \infty} E_{π} [{¯ ¯¯¯ ¯ Δ}_{ℓ} (t)] \leq α_{ℓ}$ , $\forall ℓ$ .

Further, because the cost $Γ (π; α)$ for a policy $π$ depends on $α \in C$ (e.g., when $α \to \infty$ , $Γ (π; α) \to 0$ for all reasonable policies in $Π_{O N}$ ), we quantify the performance of policy $π$ using its competitive ratio $\textscCRπ$ (5), which is equal to the cost $Γ (π; α)$ (2) for the policy, maximized over all $α \in C$ .

\textscCRπ=maxα∈C  Γ(π,α).

(5)

To solve (3) and analyze (5), it is critical to first characterize the capacity region $C$ (4). Hence, in next section, we derive a necessary condition that any $α$ that lies in $C$ , must satisfy.

Iii Capacity Region $C$

Consider the following lower bound on the expected AAoI of source $ℓ$ under policy $π \in Π_{O N}$ .

Lemma 1

For any policy $π \in Π_{O N}$ , the expected AAoI of source $ℓ$ satisfies

lim t \to \infty E_{π} [{¯ ¯¯¯ ¯ Δ}_{ℓ} (t)] \geq \frac{1}{2} ⎛ ⎝ \frac{μ_{ℓ}^{2} / 2}{{¯ ¯¯ ¯ T}_{ℓ}^{π}} + {¯ ¯¯ ¯ T}_{ℓ}^{π} + 2 γ_{ℓ} ⎞ ⎠,

(6)

where ${¯ ¯¯ ¯ T}_{ℓ}^{π}$ denotes the average of the inter-generation time of packets of source $ℓ$ that are transmitted by policy $π$ , and satisfies

N \sum ℓ = 1 \frac{γ_{ℓ}}{{¯ ¯¯ ¯ T}_{ℓ}^{π}} \leq 1.

(7)

Proof:

See Appendix B.

Remark 2

Note that $μ_{ℓ}^{2}$ is the variance of the exponentially distributed inter-generation time of packets (with mean $μ_{ℓ}$ ) at source $ℓ$ .

Recall that $α$ lies in $C$ , only if it is feasible with respect to some policy $π \in Π_{O N}$ (Definition 3), i.e., for some $π \in Π_{O N}$ , $α_{ℓ} \geq {lim}_{t \to \infty} E_{π} [{¯ ¯¯¯ ¯ Δ}_{ℓ} (t)]$ , $\forall ℓ \in {1, \dots, N}$ . Hence, using (6), we get that for any $α \in C$ , $\exists π \in Π_{O N}$ such that for each source $ℓ$ ,

α_{ℓ} \geq \frac{1}{2} ⎛ ⎝ \frac{μ_{ℓ}^{2} / 2}{{¯ ¯¯ ¯ T}_{ℓ}^{π}} + {¯ ¯¯ ¯ T}_{ℓ}^{π} + 2 γ_{ℓ} ⎞ ⎠ .

(8)

But solving the quadratic inequality (8), we find that (8) can be true only if $\forall ℓ \in {1, \dots, N}$ , $α_{ℓ} \geq γ_{ℓ}$ , $(α_{ℓ} - γ_{ℓ})^{2} \geq μ_{ℓ}^{2} / 2$ , and $\exists π \in Π_{O N}$ , such that for each source $ℓ$ ,

{¯ ¯¯ ¯ T}_{ℓ}^{π} \leq (α_{ℓ} - γ_{ℓ}) + \sqrt{(α_{ℓ} - γ_{ℓ})^{2} - μ_{ℓ}^{2} / 2} .

(9)

Note that the conditions $α_{ℓ} > γ_{ℓ}$ and $(α_{ℓ} - γ_{ℓ})^{2} \geq μ_{ℓ}^{2} / 2$ are simultaneously true only if $α_{ℓ} \geq γ_{ℓ} + μ_{ℓ} / \sqrt{2}$ . Also, (7) and (9), together imply $\sum_{ℓ = 1}^{N} γ_{ℓ} / ((α_{ℓ} - γ_{ℓ}) + \sqrt{(α_{ℓ} - γ_{ℓ})^{2} - μ_{ℓ}^{2} / 2}) \leq 1$ . Hence, we get the following necessary condition for any $α$ that lies in $C$ .

Lemma 2

$α$ lies in $C$ , only if

$α_{ℓ} \geq γ_{ℓ} + μ_{ℓ} / \sqrt{2}$ , $\forall ℓ \in {1, \dots, N}$ , and
$\sum_{ℓ = 1}^{N} γ_{ℓ} / {¯ ¯¯ ¯ T}_{ℓ}^{max} \leq 1$ , where

${¯ ¯¯ ¯ T}_{ℓ}^{max} = (α_{ℓ} - γ_{ℓ}) + \sqrt{(α_{ℓ} - γ_{ℓ})^{2} - μ_{ℓ}^{2} / 2} .$ (10)

Remark 3

Note that for minimizing $Γ (π; α)$ (3), it is sufficient to consider only those sources for which $α_{ℓ}$ ’s are finite. Hence, without loss of generality, we assume that $α_{ℓ}$ is finite for each source $ℓ$ . Thus, by definition, ${¯ ¯¯ ¯ T}_{ℓ}^{max}$ (10) is also finite for each source $ℓ$ .

Corollary 1

For any $α \in C$ , ${¯ ¯¯ ¯ T}_{ℓ}^{max}$ (10) satisfies

\frac{1}{2} ⎛ ⎝ \frac{μ_{ℓ}^{2} / 2}{{¯ ¯¯ ¯ T}_{ℓ}^{max}} + {¯ ¯¯ ¯ T}_{ℓ}^{max} + 2 γ_{ℓ} ⎞ ⎠ = α_{ℓ}, \forall ℓ \in {1, \dots, N} .

(11)

Proof:

Substituting (10) in the L.H.S. of (11), we get $α_{ℓ}$ (R.H.S. of (11)).

Next, we propose a randomized policy $π_{R} \in Π_{O N}$ , and show that for any given $α$ that satisfies the two conditions in Lemma 2, has competitive ratio (5) at most $3$ .

Iv Randomized Policy $π_{R}$

Consider a randomized policy $π_{R}$ that at any time $t$ , if the channel is free (Definition 1), among all the sources, picks source $ℓ$ , with probability

p_{ℓ} = \frac{1 / {¯ ¯¯ ¯ T}_{ℓ}^{max}}{\sum_{i = 1}^{N} 1 / {¯ ¯¯ ¯ T}_{i}^{max}},

(12)

(where ${¯ ¯¯ ¯ T}_{ℓ}^{max}$ is defined in (10)), and transmits its latest generated packet (if it has a packet to transmit, otherwise, idles for $d_{ℓ} \sim D_{ℓ}$ time units). If the channel is busy, $π_{R}$ waits for the channel to become free.

At any time

t

if the channel is free then

among

N

sources, pick source

ℓ

with probability

p_{ℓ}

(12).

if the source has a fresh packet to transmit then

transmit its latest generated packet.

else

idle for

d_{ℓ} \sim D_{ℓ}

time units.

end if

else

wait for the channel to become free.

end if

Algorithm 1 Randomized Policy

π_{R}

The following lemma provides an upper bound on the expected AAoI of each source $ℓ$ under $π_{R}$ .

Lemma 3

Under $π_{R}$ (Algorithm 1), the expected AAoI for each source $ℓ$ satisfies

lim t \to \infty E_{R} [{¯ ¯¯¯ ¯ Δ}_{ℓ} (t)] \leq \frac{1}{2} ⎛ ⎝ \frac{μ_{ℓ}^{2}}{{¯ ¯¯ ¯ T}_{ℓ}^{max}} + 3 {¯ ¯¯ ¯ T}_{ℓ}^{max} + 2 γ_{ℓ} ⎞ ⎠ .

(13)

Proof:

See Appendix B.

The main result of this paper is as follows.

Theorem 1

The competitive ratio for $π_{R}$ (Algorithm 1) is $\textscCRπR≤3$ .

Proof:

From Corollary 1 and Lemma 3, we get that for any $α \in C$ , ${lim}_{t \to \infty} E_{R} [{¯ ¯¯¯ ¯ Δ}_{ℓ} (t)] / α_{ℓ} \leq 3$ , for each source $ℓ$ . Hence, $Γ (π_{R}; α) = {max}_{ℓ} {{lim}_{t \to \infty} E_{R} [{¯ ¯¯¯ ¯ Δ}_{ℓ} (t) / α_{ℓ}]} \leq 3$ . Thus, $\textscCRπR=maxα∈C Γ(πR;α)≤3$ .

Theorem 1 shows that if any policy can guarantee AAoI $α$ for the sources, then under $π_{R}$ , AAoI for the sources cannot be more than $3 α$ . Given the ease in implementing $π_{R}$ ( $π_{R}$ does not depend on the family of distribution that $D_{ℓ}$ ’s belong to), this is an interesting result. However, in its current form, $π_{R}$ needs to know $α$ . In next section, we generalize $π_{R}$ for systems where instead of $α$ , relative weights are known for the AAoI of the sources, and the objective is to minimize the weighted sum expected AAoI. We also derive an upper bound on the competitive ratio bound of $π_{R}$ with respect to an optimal policy that minimizes the weighted sum of the expected AAoI of all sources.

V Weighted Sum Expected AAoI Minimization

Let $w = {w_{1}, w_{2}, \dots, w_{N}}$ denote the relative weights for each source in the system, and define weighted sum expected AAoI to be

Γ (π; w) = lim t \to \infty N \sum ℓ = 1 w_{ℓ} E_{π} [{¯ ¯¯¯ ¯ Δ}_{ℓ} (t)] .

(14)

The objective is to find an online policy $π \in Π_{O N}$ that minimizes $Γ (π; w)$ for any given $w$ . Formally, the objective is to solve the following optimization problem:

π^{⋆} = arg min π \in Π_{O N} Γ (π; w) .

(15)

Although $π^{⋆}$ is not known, for analysis, let $α^{⋆} = {α_{1}^{⋆}, \dots, α_{N}^{⋆}}$ be the expected AAoI for the source under $π^{⋆}$ , as $t \to \infty$ . Then, $Γ (π^{⋆}; w) = \sum_{ℓ = 1}^{N} w_{ℓ} α_{ℓ}^{⋆}$ . Using Lemma 1, we get

N \sum ℓ = 1 w_{ℓ} α_{ℓ}^{⋆} \geq \frac{1}{2} N \sum ℓ = 1 w_{ℓ} ⎛ ⎝ \frac{μ_{ℓ}^{2} / 2}{{¯ ¯¯ ¯ T}_{ℓ}^{⋆}} + {¯ ¯¯ ¯ T}_{ℓ}^{⋆} + 2 γ_{ℓ} ⎞ ⎠,

(16)

where ${¯ ¯¯ ¯ T}_{ℓ}^{⋆}$ denotes the average of the inter-generation time of packets of source $ℓ$ that are transmitted under policy $π^{⋆}$ . Since $π^{⋆} \in Π_{O N}$ , ${¯ ¯¯ ¯ T}_{ℓ}^{⋆}$ ’s satisfy (7), and the R.H.S. of (16) is at least

		$min 1 / {¯ ¯ ¯ T}_{ℓ}, \forall ℓ \frac{1}{2} N \sum ℓ = 1 w_{ℓ} (\frac{μ_{ℓ}^{2} / 2}{{¯ ¯¯ ¯ T}_{ℓ}} + {¯ ¯¯ ¯ T}_{ℓ} + 2 γ_{ℓ}),$		(17)
		$s.t. N \sum ℓ = 1 (γ_{ℓ} / {¯ ¯¯ ¯ T}_{ℓ}) \leq 1.$

Therefore, denoting the minimizer of (17) by $1 / {¯ ¯¯ ¯ T}_{ℓ}^{o}$ , $\forall ℓ$ , we get $\sum_{ℓ = 1}^{N} w_{ℓ} α_{ℓ}^{⋆} \geq \sum_{ℓ = 1}^{N} w_{ℓ} α_{ℓ}^{o}$ , where

α_{ℓ}^{o} = \frac{1}{2} ⎛ ⎝ \frac{μ_{ℓ}^{2} / 2}{{¯ ¯¯ ¯ T}_{ℓ}^{o}} + {¯ ¯¯ ¯ T}_{ℓ}^{o} + 2 γ_{ℓ} ⎞ ⎠

(18)

is a lower bound on $α_{ℓ}^{⋆}$ .

Remark 4

Note that (17) is a convex optimization problem, and can be easily solved using standard optimization tools such as CVX in Matlab. Hence, in the rest of this paper, we assume that ${¯ ¯¯ ¯ T}_{ℓ}^{o}$ and $α_{ℓ}^{o}$ are known for each source $ℓ$ .

Now, for each source $ℓ$ , define

p_{ℓ}^{o} = \frac{1 / {¯ ¯¯ ¯ T}_{ℓ}^{o}}{\sum_{i = 1}^{N} (1 / {¯ ¯¯ ¯ T}_{ℓ}^{o})} .

(19)

Theorem 2

For the randomized policy $π_{R}$ (Algorithm 1) with $p_{ℓ} = p_{ℓ}^{o}$ (19) (for each source $ℓ$ ), the weighted sum expected AAoI $Γ (π_{R}; w) \leq 3 \cdot Γ (π^{⋆}; w)$ , for any relative weight vector $w$ .

Proof:

Replacing ${¯ ¯¯ ¯ T}_{ℓ}^{max}$ by ${¯ ¯¯ ¯ T}_{ℓ}^{o}$ in (11), (12) and (13), we get $p_{ℓ} = p_{ℓ}^{o}$ (19),

	$\frac{1}{2} ⎛ ⎝ \frac{μ_{ℓ}^{2} / 2}{{¯ ¯¯ ¯ T}_{ℓ}^{o}} + {¯ ¯¯ ¯ T}_{ℓ}^{o} + 2 γ_{ℓ} ⎞ ⎠ = α_{ℓ}^{o}, and$		(20)
	$lim t \to \infty E_{R} [{¯ ¯¯¯ ¯ Δ}_{ℓ} (t)] \leq \frac{1}{2} ⎛ ⎝ \frac{μ_{ℓ}^{2}}{{¯ ¯¯ ¯ T}_{ℓ}^{o}} + 3 ¯ T_{ℓ}^{o} + 2 γ_{ℓ} ⎞ ⎠,$		(21)

for each source $ℓ$ . Substituting (20) in (21), and taking the weighted sum of the resulting expression over all sources, we get ${lim}_{t \to \infty} E_{R} [{¯ ¯¯¯ ¯ Δ}_{ℓ} (t)] \leq \sum_{ℓ = 1}^{N} w_{ℓ} (3 α_{ℓ}^{o})$ . Since $\sum_{ℓ = 1}^{N} w_{ℓ} α_{ℓ}^{o} \leq \sum_{ℓ = 1}^{N} w_{ℓ} α_{ℓ}^{⋆}$ , we get

Γ(πR;w)=limt→∞ER[¯¯¯¯¯Δℓ(t)]≤3N∑ℓ=1wℓα⋆ℓ=3⋅Γ(π⋆;w).\IEEEQEDhereeqn

Theorem 2 shows that the weighted sum expected AAoI for the randomized policy $π_{R}$ is at most thrice compared to any other online policy. For M/G/1 queuing model, this is the best guarantee known for any online policy. Also, the proof of Theorem 2 provides a general recipe for generalizing the results for ‘per source AAoI minimization problem’ to the ‘weighted sum AAoI minimization problem’.

Vi Numerical Results

Theorems 1 and 2 provide some important analytical results regarding the performance of randomized policy $π_{R}$ (Algorithm 1). In this section, we use numerical simulations to verify these results, and derive new insights.

Remark 5

For all the simulations, we assume the initial AoI of the sources to be $0$ , and the time horizon $t = 10^{6}$ units.

First, to analyze the effect of number of sources on the AAoI of an individual source, we consider a system with $N$ identical sources (for each source $ℓ$ , $α_{ℓ} = 40$ , $μ_{ℓ} = 4$ , and $D_{ℓ} = D$ ), and simulate the system under policy $π_{R}$ for $N \in {1, \dots, 20}$ , assuming $D$ to be $(i)$ an exponential distribution with mean $γ \in {2, 8}$ , and $(i i)$ uniform distribution with mean $γ \in {2, 8}$ (the rationale is to consider memoryless distribution, as well as a non-memoryless distribution $D$ ). Then, we plot the AAoI of source $1$ for different choices of $N$ and $D$ in Figure 2.

As shown in Figure 2, as $N$ increases, AAoI of source 1 increases linearly, with slope proportional to $γ$ . This is because when sources are identical, the probability (12) of picking a source for transmission is $p_{ℓ} = 1 / N$ , which implies the expected time interval between two successive instants when source $ℓ$ gets to transmit is proportional to $N γ$ (since expected transmission delay for every transmitted packet is $γ$ ).

Fig. 2: Effect of number of sources on AAoI of an individual source.

Further, to analyze the effect of the recommended AAoI $α_{ℓ}$ of source $ℓ$ , we consider a system with $N = 5$ sources, with mean packet inter-generation time $[μ_{1}, \dots, μ_{5}] = [2, 4, 4, 8, 10]$ , mean transmission delay $[γ_{1}, \dots, γ_{5}] = [3, 3, 6, 2, 4]$ , and $α = [α_{1}, 10, 15, 20, 20]$ . Then, we simulate the system under policy $π_{R}$ for $α_{1} \in [9.2, 20]$ , and plot the corresponding AAoI values for two of the sources in Figure 3.

Remark 6

The choice of parameters $μ_{ℓ}$ ’s, $γ_{ℓ}$ ’s and $α_{ℓ}$ ’s are arbitrary, to avoid symmetry between sources. Further, we only consider $α_{1} \geq 9.2$ , because for the considered choice of other parameters, $α \in C$ , i.e., the conditions in Lemma 2 are satisfied, only when $α_{1} \geq 9.2$

Note that with increase in $α_{1}$ , ${¯ ¯¯ ¯ T}_{1}^{max}$ (10) increases, whereas ${¯ ¯¯ ¯ T}_{ℓ}^{max}$ (for $ℓ \neq 1$ ) decreases. Hence, $p_{1}$ (12) (proportional to $1 / {¯ ¯¯ ¯ T}_{1}^{max}$ decreases, while $p_{ℓ}$ (for $ℓ \neq 1$ ; inversely proportional to $1 / {¯ ¯¯ ¯ T}_{1}^{max}$ ) increases with increase in $α_{1}$ . This is also illustrated in Figure 3, where with increase in $α_{1}$ , AAoI of source $1$ increases, while AAoI of other source (source $3$ ) decreases. In addition, Figure 3 shows that when $α_{1} \geq 9.2$ (i.e., when $α \in C$ ), AAoI of source $ℓ \in {1, 3}$ is less than $3 α_{ℓ}$ , which is expected because of Theorem 1.

Fig. 3: Effect of parameter $α$ on AAoI of sources.

Next, to verify Theorem 2, we again consider a system with $N = 5$ sources, and parameters $[μ_{1}, \dots, μ_{5}] = μ \cdot [2, 4, 4, 8, 10]$ , mean transmission delay $[γ_{1}, \dots, γ_{5}] = γ \cdot [3, 3, 6, 2, 4]$ , and the relative weight vector $w = [0.8, 0.8, 0.2, 0.2, 0.4]$ . Then, we simulate the system under policy $π_{R}$ for different values of $μ$ , $γ$ and family of transmission delay distribution $D_{ℓ}$ (family of distribution is same for each source $ℓ$ ), and plot the output in Figure 4.

Remark 7

For simulating $π_{R}$ when the objective is to minimize the weighted Sum AAoI, we compute ${¯ ¯¯ ¯ T}_{ℓ}^{o}$ ( $\forall ℓ \in {1, \dots, N}$ ) by solving (17) (using CVX toolbox in Matlab), and use it to obtain $p_{ℓ}^{o}$ by substituting ${¯ ¯¯ ¯ T}_{ℓ}^{o}$ ’s in (19).

It is evident from Figure 4 that the weighted sum AAoI for policy $π_{R}$ is less than $3$ times the theoretical lower bound $\sum_{ℓ = 1}^{N} w_{ℓ} α_{ℓ}^{o}$ computed by solving the optimization problem (17). Also, it can be noted that the mean transmission delay has significant impact on the weighted sum AAoI of $π_{R}$ , whereas the family of transmission delay distribution $D_{ℓ}$ (i.e., exponential or uniform) has comparatively negligible effect. Note that for any given value of the mean transmission delay for the sources, the lower bound (17) is independent of the family of distribution of $D_{ℓ}$ .

Fig. 4: Weighted sum AAoI as a function of mean packet inter-generation time.

Finally, we consider a system with single source ( $N = 1$ ) that can generate fresh packets at any time (i.e., the mean packet inter-generation time $μ \to 0^{+}$ ), and each packet transmitted by the source suffers random transmission delay according to some general distribution $D$ , with mean $γ$ . Also, when $N = 1$ , since a policy does not need to choose among multiple sources, we consider a simplified version of $π_{R}$ , where the source generates and transmits a packet whenever the channel is free, and the previously transmitted update is at least ${¯ ¯¯ ¯ T}^{o} = γ$ (17) time units old.

Remark 8

${¯ ¯¯ ¯ T}^{o} = γ$ is the inter-generation time of successive transmitted packets in the lower bound $α^{o} (???)$ (on AAoI of the source), when $N = 1$ and $μ \to 0^{+}$ .

We simulate the system under $π_{R}$ , and an optimal online policy $π^{⋆}$ proposed in [sun2017update], for different choices of $D$ and $γ$ , and plot the AAoI for the policies in Figure 5. Interestingly, for exponential and uniform distribution $D$ , we find that the difference between the AAoI of the source under $π_{R}$ and $π^{⋆}$ is negligible. This is despite the fact that $π_{R}$ is independent of the family of distribution that $D$ belongs to, and only depends on $γ$ . Whereas, $π^{⋆}$ is a threshold-based policy, where the threshold needs to be computed for each distribution $D$ , which might be difficult for certain family of distributions.

Fig. 5: $π_{R}$ versus $π^{⋆}$ in a single source generate-at-will setup.

Vii Conclusion

In this paper, we considered M/G/1 queuing model with multiple sources, where the objective is to minimize the expected average age of information (AAoI) of each source. We proposed an online randomized policy for prioritizing sources (and their packets), and showed that if there exists any online policy that can minimize expected AAoI of the sources below $α$ , then the proposed policy can guarantee expected AAoI of the sources, less than $3 α$ . We further showed that in the setting where $α$ is not known, and the objective is to minimize the weighted sum expected AAoI (WSAAoI) of the sources, the proposed policy guarantees WSAAoI that is at most $3$ times the minimum possible value (under any online policy). Using numerical simulations, we also showed that in special cases of the problem, where an optimal online policy is known, the proposed randomized policy might still be preferable due to ease of implementation, and near-optimal performance.

References

Appendix A Existence of an online policy for which the expected AAoI of each source is finite

Recall that $\forall ℓ \in {1, \dots, N}$ , $μ_{ℓ}$ and $γ_{ℓ}$ are finite. Therefore, packet inter-generation times and transmission delays are finite with probability 1, for each source $ℓ$ . Hence, when $N$ is finite, for a round-robin policy, that picks a source, waits until a fresh packet is generated at the picked source, then transmits the generated packets, and then picks another source in cyclic order when channel becomes free, will have finite expected AAoI for each source.

Appendix B Proof of Lemma 1 and Lemma 3

Let $ℓ_{1}^{π}, ℓ_{2}^{π}, ℓ_{3}^{π}, \dots$ denote the sequence of packets of source $ℓ$ that get transmitted under policy $π$ . Also, let $g_{ℓ i}^{π}$ , $s_{ℓ i}^{π}$ and $r_{ℓ i}^{π}$ respectively denote the generation time of packet $ℓ_{i}^{π}$ , time when transmission of packet $ℓ_{i}^{π}$ begins, and the time when transmission of packet $ℓ_{i}^{π}$ completes. Now, define $T_{ℓ i}^{π} = g_{ℓ i}^{π} - g_{ℓ (i - 1)}^{π}$ , $W_{ℓ i}^{π} = s_{ℓ i}^{π} - g_{ℓ i}^{π}$ , and $d_{ℓ i} = r_{ℓ i}^{π} - s_{ℓ i}^{π}$ . Note that $T_{ℓ i}^{π}$ is the inter-generation time of packets that are transmitted by policy $π$ , and $Z_{ℓ i}^{π} = W_{ℓ i}^{π} + d_{ℓ i} = r_{ℓ i}^{π} - g_{ℓ i}^{π}$ is equal to the age of packet $ℓ_{i}^{π}$ at the instant it is received at the monitor.

Remark 9

Note that $d_{ℓ i}$

is a random variable that denotes the transmission delay of packet

ℓ_{i}^{π}

. By definition,

d_{ℓ i} \sim D_{ℓ}

is independent of policy

π

Fig. 6: Sample AoI plot of source

ℓ

in a multi-source setting. Here, a packet

ℓ_{i}^{π}

, generated at time

g_{ℓ i}^{π}

, is received at the monitor at time

r_{ℓ i}^{π}

Figure 6 shows a sample AoI plot for source $ℓ$ , labelled with the quantities defined above. As evident from Figure 6 (and shown in detail in [kaul2012status]), AAoI (1) of source $ℓ$ can be expressed in terms of the defined quantities as follows

lim t \to \infty {¯ ¯¯¯ ¯ Δ}_{ℓ}^{π} (t) (a) = lim t \to \infty \frac{\sum_{i = 1}^{R_{ℓ}^{π} (t)} ((T_{ℓ i}^{π})^{2} / 2 + T_{ℓ i}^{π} Z_{ℓ i}^{π})}{t},

(22)

where $R_{ℓ}^{π} (t)$ denotes the number of packets transmitted by source $ℓ$ under policy $π$ , until time $t$ . Also, $t = \sum_{i = 1}^{R_{ℓ}^{π} (t)} T_{ℓ i}^{π}$ (shown in detail in [saurav2021minimizing]). Therefore,

lim t \to \infty \frac{\sum_{i = 1}^{R_{ℓ}^{π} (t)} T_{ℓ i}^{π}}{R_{ℓ}^{π} (t)} = lim t \to \infty \frac{t}{R_{ℓ}^{π} (t)} = {¯ ¯¯ ¯ T}_{ℓ}^{π},

(23)

where ${¯ ¯¯ ¯ T}_{ℓ}^{π}$ denotes the average inter-generation time of packets transmitted by policy $π$ .

From (23), it follows that when $t \to \infty$ , $\sum_{ℓ = 1}^{R_{ℓ}^{π} (t)} T_{ℓ i}^{π} = R_{ℓ}^{π} (t) {¯ ¯¯ ¯ T}_{ℓ}^{π}$ . Therefore,

R_{ℓ}^{π} (t) \sum i = 1 T_{ℓ i}^{π} - R_{ℓ}^{π} (t) {¯ ¯¯ ¯ T}_{ℓ}^{π} = R_{ℓ}^{π} (t) \sum i = 1 (T_{ℓ i}^{π} - {¯ ¯¯ ¯ T}_{ℓ}^{π}) = R_{ℓ}^{π} (t) \sum i = 1 δ_{ℓ i}^{π} = 0

(24)

where $δ_{ℓ i}^{π} = T_{ℓ i}^{π} - {¯ ¯¯ ¯ T}_{ℓ}^{π}$ . This also implies that when $t \to \infty$ ,

R_{ℓ}^{π} (t) \sum i = 1 T_{ℓ i}^{π} = R_{ℓ}^{π} (t) \sum i = 1 δ_{ℓ i}^{π} + R_{ℓ}^{π} (t) \sum i = 1 {¯ ¯¯ ¯ T}_{ℓ}^{π} = R_{ℓ}^{π} (t) \cdot {¯ ¯¯ ¯ T}_{ℓ}^{π} .

(25)

Further, squaring both sides of $T_{ℓ i}^{π} = δ_{ℓ i}^{π} + {¯ ¯¯ ¯ T}_{ℓ}^{π}$ , we get $(T_{ℓ i}^{π})^{2} = (δ_{ℓ i}^{π})^{2} + ({¯ ¯¯ ¯ T}_{ℓ}^{π})^{2} + 2 {¯ ¯¯ ¯ T}_{ℓ}^{π} δ_{ℓ i}^{π}$ . Hence,

$R_{ℓ}^{π} (t) \sum i = 1 (T_{ℓ i}^{π})^{2}$	$= R_{ℓ}^{π} (t) \sum i = 1 (δ_{ℓ i}^{π})^{2} + R_{ℓ}^{π} (t) \sum i = 1 ({¯ ¯¯ ¯ T}_{ℓ}^{π})^{2} + 2 {¯ ¯¯ ¯ T}_{ℓ}^{π} R_{ℓ}^{π} (t) \sum i = 1 δ_{ℓ i}^{π},$
	$(a) = R_{ℓ}^{π} (t) \sum i = 1 (δ_{ℓ i}^{π})^{2} + R_{ℓ}^{π} (t) \sum i = 1 ({¯ ¯¯ ¯ T}_{ℓ}^{π})^{2},$
	$= R_{ℓ}^{π} (t) \sum i = 1 (δ_{ℓ i}^{π})^{2} + R_{ℓ}^{π} (t) \cdot ({¯ ¯¯ ¯ T}_{ℓ}^{π})^{2},$	(26)

where we get (a), because $\sum_{i = 1}^{R_{ℓ}^{π} (t)} δ_{ℓ i}^{π} = 0$ (from (24)).

Substituting (25) and (B) in (22), and using the relation $t = \sum_{i = 1}^{R_{ℓ}^{π} (t)} T_{ℓ i}^{π} = R_{ℓ}^{π} (t) \cdot {¯ ¯¯ ¯ T}_{ℓ}^{π}$ (for $t \to \infty$ ), we get

	$lim t \to \infty {¯ ¯¯¯ ¯ Δ}_{ℓ}^{π} (t)$	$= lim t \to \infty ⎛ ⎜ ⎝ \frac{\sum_{i = 1}^{R_{ℓ}^{π} (t)} (δ_{ℓ i}^{π})^{2}}{2 R_{ℓ}^{π} (t) \cdot {¯ ¯¯ ¯ T}_{ℓ}^{π}} + \frac{{¯ ¯¯ ¯ T}_{ℓ}^{π}}{2} + \frac{\sum_{i = 1}^{R_{ℓ}^{π} (t)} T_{ℓ i}^{π} Z_{ℓ i}^{π}}{\sum_{i = 1}^{R_{ℓ}^{π} (t)} T_{ℓ i}^{π}} ⎞ ⎟ ⎠,$
		$= \frac{β_{ℓ}^{π}}{2 {¯ ¯¯ ¯ T}_{ℓ}^{π}} + \frac{{¯ ¯¯ ¯ T}_{ℓ}^{π}}{2} + ϕ_{ℓ}^{π},$		(27)

where

	$β_{ℓ}^{π}$	$= lim t \to \infty \frac{\sum_{i = 1}^{R_{ℓ}^{π} (t)} (δ_{ℓ i}^{π})^{2}}{R_{ℓ}^{π} (t)}, and$		(28)
	$ϕ_{ℓ}^{π}$	$= lim t \to \infty \frac{\sum_{i = 1}^{R_{ℓ}^{π} (t)} T_{ℓ i}^{π} Z_{ℓ i}^{π}}{\sum_{i = 1}^{R_{ℓ}^{π} (t)} T_{ℓ i}^{π}} .$		(29)

Taking expectation on both sides of (B), and using Tonelli’s Theorem [patrick1995probability] to exchange limit and expectation (by definition, each term in (B) is non-negative and measurable), we get

	$lim t \to \infty$	$E_{π} [Δ_{ℓ}^{π} (t)] = E_{π} ⎡ ⎣ \frac{β_{ℓ}^{π}}{2 {¯ ¯¯ ¯ T}_{ℓ}^{π}} + \frac{{¯ ¯¯ ¯ T}_{ℓ}^{π}}{2} ⎤ ⎦ + E_{π} [ϕ_{ℓ}^{π}]$
		$= \frac{E_{π} [β_{ℓ}^{π}]}{2 {¯ ¯¯ ¯ T}_{ℓ}^{π}} + \frac{{¯ ¯¯ ¯ T}_{ℓ}^{π}}{2} + lim t \to \infty E_{π} ⎡ ⎢ ⎣ \frac{\sum_{i = 1}^{R_{ℓ}^{π} (t)} T_{ℓ i}^{π} Z_{ℓ i}^{π}}{\sum_{i = 1}^{R_{ℓ}^{π} (t)} T_{ℓ i}^{π}} ⎤ ⎥ ⎦ .$		(30)

B-a Proof of Lemma 1

Recall that $Z_{ℓ i}^{π} = W_{ℓ i}^{π} + d_{ℓ i}$ , where $W_{ℓ i}^{π} = s_{ℓ i}^{π} - g_{ℓ i}^{π} \geq 0$ , and $d_{ℓ i}$ ’s are independent and identically distributed according to distribution $D_{ℓ}$ . Also, $T_{ℓ i}^{π}$ ’s are non-negative (by definition). Therefore, $T$

3-Competitive Policy for Minimizing Age of Information in Multi-Source M/G/1 Queuing Model

Authors

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This week in AI

I Introduction

Ii System Model

Definition 1

Definition 2

Remark 1

Definition 3

Iii Capacity Region C

Lemma 1

Proof:

Remark 2

Lemma 2

Remark 3

Corollary 1

Proof:

Iv Randomized Policy πR

Lemma 3

Proof:

Theorem 1

Proof:

V Weighted Sum Expected AAoI Minimization

Remark 4

Theorem 2

Proof:

Vi Numerical Results

Remark 5

Remark 6

Remark 7

Remark 8

Vii Conclusion

References

Appendix A Existence of an online policy for which the expected AAoI of each source is finite

Appendix B Proof of Lemma 1 and Lemma 3

Remark 9

B-a Proof of Lemma 1

Iii Capacity Region $C$

Iv Randomized Policy $π_{R}$