A General Formula for the Stationary Distribution of the Age of Information and Its Application to Single-Server Queues

04/17/2018 ∙ by Yoshiaki Inoue, et al. ∙ Kyoto University Osaka University 0

This paper considers the stationary distribution of the age of information (AoI) in information update systems. We first derive a general formula for the stationary distribution of the AoI, which holds for a wide class of information update systems. The formula indicates that the stationary distribution of the AoI is given in terms of the stationary distributions of the system delay and the peak AoI. To demonstrate its applicability and usefulness, we analyze the AoI in single-server queues with four different service disciplines: first-come first-served (FCFS), preemptive last-come first-served (LCFS), and two variants of non-preemptive LCFS service disciplines. For the FCFS and the preemptive LCFS service disciplines, the GI/GI/1, M/GI/1, and GI/M/1 queues are considered, and for the non-preemptive LCFS service disciplines, the M/GI/1 and GI/M/1 queues are considered. With these results, we further show comparison results for the mean AoI in the M/GI/1 and GI/M/1 queues among those service disciplines.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

We consider an information update system composed of an information source equipped with a sensor, a processor (a server), and a monitor. The state of the information source changes over time, which is observed by the sensor occasionally. Whenever the state is sensed, the sensor generates a packet that contains the sensed data and its time-stamp, and sends the packet to the server. The server processes the received data, appends the result to log database, and updates information displayed on the monitor. The age of information (AoI) is a primary performance measure in information update systems, which is defined as the length of time elapsed from the time-stamp of information being displayed on the monitor.

The information update system described above is an abstraction of various situations where the freshness of data is of interest, e.g., status monitoring in manufacturing systems, satellite imagery for weather report, tracking trends in social networks, and so on. The AoI has recently attracted a considerable attention due to its applicability to a wide range of information and communication systems. Readers are referred to [1] for a detailed introduction and review of this subject.

Information update systems are usually modeled as queueing systems, where packets arriving at a queueing system correspond to information packets. In most previous work on the analysis of the AoI, the mean AoI is of primary concern, which is defined as the long-run time-average of the AoI. To be more specific, consider a stationary, ergodic queueing system, and let () denote the AoI at time :

(1)

where () denotes the time-stamp of information being displayed on the monitor at time . The mean AoI is defined as

and under a fairly general setting, is given by [2]

(2)

where and

denote the mean and the second moment of interarrival times and

denotes the mean product of the interarrival time of the st and the th packets and the system delay of the th packet. This formula has been the starting point in most previous work on the analysis of the AoI. As stated in [1, Page 170], however, the calculation of the mean AoI based on (2) is cumbersome because and

are dependent in general and their joint distribution can take a complicated form.

Reference Service discipline Interarrival time dist. Service time dist. # of servers Buffer capacity
[2] FCFS Exponential Exponential
FCFS Exponential Constant
FCFS Constant Exponential
[3] FCFS Exponential Exponential
FCFS Exponential Exponential
[6] FCFS Exponential General
[7] FCFS Exponential Exponential
FCFS Exponential Exponential
[8] Preemptive LCFS Exponential Exponential
[9] Preemptive LCFS Exponential Gamma
[6] Preemptive LCFS Exponential General
[8, 3] Non-preemptive LCFS (C) Exponential Exponential
[9] Non-preemptive LCFS (C) Exponential Erlangian
Table 1: Summary of Known results for the AoI in standard queueing systems.

The purpose of this paper is twofold. The first one is the derivation of a general formula for the stationary distribution () of the AoI in ergodic information update systems, which is defined as the long-run fraction of time in which the AoI is not greater than an arbitrarily fixed value :

where denotes an indicator function. Although the mean AoI is a primary performance metric, it alone is not sufficient to characterize the long-run behavior of the AoI and its related processes. First of all, if the stationary distribution of the AoI is available, we can evaluate the deviation of the AoI from its mean value. To support our claim further, we provide two examples, which show that the stationary distribution of the AoI plays a central role in the analysis of AoI-related processes.

Example 1.

In [4], the Cost of Update Delay (CoUD) metric is introduced to expand the concept of the AoI. For a non-negative and non-decreasing function () with , CoUD at time is defined as (). Clearly, the time-average of CoUD is given in terms of the stationary distribution () of the AoI:

Example 2.

Consider a remote estimation of a stationary Wiener process

via a channel modeled as a stationary queueing system [5]. We define as an estimator of (see (1)). As shown in [5], if a sequence of sampling times is independent of ,

where

denotes a normal random variable with mean

and variance

. Therefore, the mean square error of the estimator is equal to the mean AoI for the Wiener process.

As an extension of this result, we consider the stationary distribution of the estimation error. It is readily verified that the characteristic function of the estimation error is given by

where () denotes the Laplace-Stieltjes transform (LST) of the stationary distribution of the AoI.

In [3], a performance metric called the peak AoI is introduced, which is defined as the AoI immediately before information updates. The formulation of the peak AoI is simpler than that of the AoI, and it can be used as an alternative metric of the freshness of data. In particular, one of the primary motivations of introducing the peak AoI in [3] is that one can characterize its stationary distribution using standard methods in queueing theory; Specifically, the stationary distribution of the peak AoI is given in terms of that of the system delay.

In this paper, we show that under a fairly general setting, the stationary distribution of the AoI is given in terms of those of the system delay and the peak AoI. As we will see, this formula holds sample-path-wise, regardless of the service discipline or the distributions of interarrival and service times.

Therefore, the analysis of the stationary distribution of the AoI in ergodic queueing systems is reduced to the analysis of those of the peak AoI and the system delay, which can then be analyzed via standard techniques in queueing theory. An important consequence of this observation is that the peak AoI is not merely an alternative performance metric to the AoI but rather an essential quantity in elucidating properties of the AoI. Furthermore, this observation leads to an alternative formula for the mean AoI in terms of the second moments of the peak AoI and the system delay.

The second purpose of this paper is the derivation of various analytical formulas for the AoI in single-server queues, which demonstrates the wide applicability of our general formula. More specifically, we consider the following four service disciplines:

  • First-come, first-served (FCFS),

  • Preemptive last-come, first-served (LCFS),

  • Non-preemptive LCFS with discarding, and

  • Non-preemptive LCFS without discarding.

Under the FCFS service discipline, all packets are served in order of arrival, while under the LCFS service disciplines (B)–(D), the newest packet is given priority. In the preemptive LCFS discipline, newly arriving packets immediately start receiving their services on arrival, interrupting the ongoing service (if any). In the non-preemptive LCFS service discipline, on the other hand, arriving packets have to wait until the completion of the ongoing service, and waiting packets are also overtaken by those which arrive during their waiting times.

Note that the non-preemptive LCFS service discipline has two variants, (C) and (D): overtaken packets are discarded in the service discipline (C), while overtaken packets remain in the system and they are served eventually in the service discipline (D). Although (D) yields a larger AoI than (C), this service discipline is also of interest in evaluating the logging overhead caused by sending all generated packets to the database. Note that for the preemptive LCFS discipline, the treatment of overtaken packets (i.e., discarding them or not) does not affect the AoI performance.

Table 1 summarizes known results for the AoI in standard queueing systems. To the best of our knowledge, no closed formula for the AoI under the service discipline (D) has been reported in the literature.

Our contribution to the analysis of the AoI in single-server queues is summarized as follows.

  • For the FCFS GI/GI/1 queue, we show that the stationary distribution of the AoI is given in terms of the system delay distribution. We also derive upper and lower bounds for the mean AoI in the FCFS GI/GI/1 queue. In addition, we derive explicit formulas for the LST of the stationary distribution of the AoI in the FCFS M/GI/1 and GI/M/1 queues.

  • For the preemptive LCFS GI/GI/1, M/GI/1, and GI/M/1 queues, we derive explicit formulas for the LST of the stationary distribution of the AoI. In addition, for the preemptive M/GI/1 and GI/M/1 queues, we derive ordering properties of the AoI in terms of the service time and the interarrival time distributions.

  • For the non-preemptive M/GI/1 and GI/M/1 queues with and without discarding, we derive explicit formulas for the LST of the stationary distribution of the AoI.

In Appendix A, we also present specialized formulas for the M/M/1, M/D/1, and D/M/1 queues.

Remark 3.

Throughout this paper, we strictly distinguish between the symbols ‘G’ and ‘GI’ in Kendall’s notation: ‘GI’ represents that interarrival or service times are independent and identically distributed (i.i.d.) random variables, while ‘G’ represents that there are no restrictions on the arrival or service processes.

Taking the derivative of the LST of the AoI, we can obtain the mean and higher moments of the AoI. In all of the above models, we provide formulas for the mean AoI. We also derive formulas for higher moments of the AoI when they take simple forms. Furthermore, we obtain comparison results for the mean AoI among the four service disciplines in the M/GI/1 and GI/M/1 queues.

The rest of this paper is organized as follows. In Section 2, we derive a general formula for sample-paths of the AoI, and using it, we obtain various formulas for the AoI in a general information update system. In Sections 3 and 4, we consider the FCFS and the preemptive LCFS GI/GI/1, M/GI/1, and GI/M/1 queues. In Sections 5 and 6, we consider the non-preemptive LCFS M/GI/1 and GI/M/1 queues with and without discarding. Furthermore, Section 7 provides some comparison results for the mean AoI among the four service disciplines in the M/GI/1 and GI/M/1 queues. Finally, the conclusion of this paper is provided in Section 8.

2 Sample-Path Analysis in a General Setting

We consider a sample path of the AoI process , where is defined in (1). Note that the process of the time-stamp of the information being displayed on the monitor is a step function of , i.e., it is piece-wise constant and has discontinuous points at which information is updated. Therefore, any sample path of the AoI process is piece-wise linear with slope one and it has (downward) jumps when information is updated. In what follows, we assume that is right-continuous, i.e., .

Any sample path of the AoI process can be constructed as follows. Let denote a marked point process on , where () denotes the time instant of the th information update and () denotes the AoI immediately after the th update. For simplicity, we assume , and (). In these settings, is given by

(3)

The AoI process is thus determined completely by . We define () as the th peak AoI (i.e., the AoI immediately before the th update).

Figure 1 shows an example of sample paths of the AoI process .

Figure 1: A sample path of the AoI process.

In what follows, we first show a general formula for sample paths of the AoI process and then, we discuss the AoI in a general information update system.

2.1 A general formula for sample paths of the AoI

Suppose a sample path of is given. Let , , and () denote the asymptotic frequency distributions (see, e.g., [10, Section 2.6]) of , , and , respectively, on the sample path:

(4)
(5)
(6)

if the limits exist. Let () denote the total number of information updates by time .

(7)

The following lemma is a sample-path analogue of the elementary renewal theorem.

Lemma 4 ([10, Lemma 1.1]).

For ,

To proceed further, we make the following assumptions.

Assumption 5.

for some .

Assumption 6.

The limits in (5) and (6) exist for each .

Assumption 5 implies that , so that is well-defined for .

Theorem 7.

Under Assumptions 5 and 6, the limit (4) exists for each and it is given by

Proof.

By definition, we have

(8)

where () and are given by

(9)
(10)

Thus, to prove the theorem, it suffices to show that

(11)
(12)

We first consider (12). From Lemma 4 and Assumption 5, we have

(13)

and thus

(14)

It then follows from Assumption 5, (10), (13), and (14) that

which proves (12).

Next we consider (11). Substituting (3) into (9) yields

(15)

where the last equality follows from the following relation: For any and ,

Applying the bounded convergence theorem to (15), and using (5) and (6), we obtain

Combining this with (13) yields

which completes the proof. ∎

Remark 8.

In the proof of Theorem 7, we do not use the inequality () that should hold in the context of the AoI.

2.2 Sample path analysis of the AoI in information update systems

In this subsection, we consider Theorem 7 in the context of information update systems. In general, two types of packets exist in information update systems; informative and non-informative packets. Informative packets update information being displayed on the monitor, while non-informative packets do not. If arriving packets are processed on an FCFS basis, all arriving packets are informative. On the other hand, if the order of processing packets is controllable, it might be reasonable to give priority to the newest packet, because it is expected to improve the AoI performance. In such a case, older (overtaken) packets do not update information on the monitor, so that they are regarded as non-informative. In the rest of this paper, we assume that time-stamps of packets are identical to their arrival times. If we ignore all non-informative packets and observe only the stream of informative packets, they are processed and depart from the system in a FIFO (first-in first-out) manner. We thus consider a general FIFO G/G/1 queue, where only informative packets are visible.

Remark 9.

As noted in Remark 3, G/G/1 queues are more general than GI/GI/1 queues. In what follows, we consider the FIFO G/G/1 queue of informative packets, so that we obtain general results which are applicable to information update systems with any service disciplines.

Let () denote the arrival time of the th informative packet. We assume that , , and (), where () denotes the departure (i.e., information update) time of the th informative packets. Let () denote the interarrival time of the st and the th informative packets and let () denote the system delay of the th informative packet.

(16)

By definition, we have

(17)
Assumption 10.

The mean arrival rate of informative packets is positive and finite, i.e.,

(18)

and the system is stable, i.e., the mean departure rate of informative packets is equal to the mean arrival rate of informative packets. More concretely,

(19)

where () is defined in (7).

In the rest of this paper, we refer to as the mean arrival rate of informative packets. Let denote the mean interarrival time of informative packets.

It then follows from Lemma 4 that under Assumption 10, we have

(20)

We define () as the asymptotic frequency distribution of , if it exists.

(21)

Note that corresponds to in (6).

Assumption 11.

The limits in (5) and (21) exist for each .

Corollary 12.

In the FIFO G/G/1 queue satisfying Assumptions 10 and 11, the limit in (4) exists for each and it is given by

Proof.

Corollary 12 immediately follows from Theorem 7 and (20). ∎

2.3 Results for stationary, ergodic stochastic models

Information update systems are usually modeled as stationary, ergodic stochastic models. We thus interpret Corollary 12 in such a context. In the rest of this paper, we use the following conventions. For a non-negative random variable , let (

) denote the probability distribution function (PDF) of

and let () denote the LST of .

Furthermore, let (, ) denote the th derivative of :

(22)

Let , , and denote generic random variables for the stationary AoI, peak AoI, and system delay, respectively. Since the system is ergodic, the following equalities hold with probability 1:

Theorem 14 presented below is immediate from Corollary 12 and basic properties of LST.

Assumption 13.

Assumption 10 holds with probability 1 and both and are stationary, ergodic stochastic processes.

Theorem 14.

In the FIFO G/G/1 queue satisfying Assumption 13,

  • the density function () of the AoI is given by

    (23)
  • the LST () of the AoI is given by

    (24)

    and

  • the th () moment of the AoI is given by

    (25)

    if and .

Remark 15.

Letting in (25), we obtain

(26)

The formula for the mean AoI in (2) is thus reproduced from (17), (20), and (26).

As shown in (22), the mean AoI can also be obtained by taking the derivative of and letting , which we will use repeatedly in the rest of this paper.

Remark 16.

Theorem 14 can be applied to information update systems with any service disciplines: Recall that we introduced the FIFO G/G/1 queue as a virtual system where only informative packets are visible, and the original system (with non-informative packets) is not required to be FIFO.

3 Applications to FCFS Queues

In this section, we apply Theorem 14 to FCFS GI/GI/1, M/GI/1 and GI/M/1 queues. Let and denote generic random variables for interarrival times and service times, where . Let denote the traffic intensity. Throughout this section, we assume , so that the system is stable. We also assume that the system is stationary and ergodic.

3.1 The FCFS GI/GI/1 queue

We consider the FCFS GI/GI/1 queue. Because all arriving packets are informative in this model, the mean arrival rate of informative packets is given by

(b) .(a) .
Figure 2: , , , and .

We first consider the peak AoI in the FCFS GI/GI/1 queue. We define () as the service time of the th packet. See Figure 2. If , i.e., no packet arrives in the system delay of the st packet, equals to . On the other hand, if , equals to . Therefore, we have an alternative formula for (cf. (17)).

(27)

Note here that is independent of . We thus obtain the following lemma from (27).

Lemma 17.

In the stationary, ergodic FCFS GI/GI/1 queue, the LST of the peak AoI is given by

(28)

Theorem 14 and Lemma 17 imply that the distributions of the AoI and the peak AoI in the stationary, ergodic FCFS GI/GI/1 queue are given in term of the distribution of the system delay .

In what follows, we derive upper and lower bounds of the mean AoI . We rewrite (2) to be

(29)

where denotes the coefficient of variation and denotes the covariance of and . Note that does not depend on because of the stationarity of the system.

Lemma 18.

In the stationary, ergodic FCFS GI/GI/1 queue, is bounded as follows.

The proof of Lemma 18 is provided in Appendix B.

Theorem 19.

In the stationary, ergodic FCFS GI/GI/1 queue, the mean AoI is bounded as follows.

(30)
(31)
Proof.

Theorem 19 follows from (29) and Lemma 18. ∎

Remark 20.

The bounds in Theorem 19 are tight in the sense that both equalities hold in the D/GI/1 queue.

Remark 21.

It is known that the mean delay in the FCFS GI/GI/1 queue is bounded by [11, 12]

(32)

Bounding in (30) and (31) by these inequalities, we can obtain a bound for in terms of only the distributions of and .

Because (17) implies

(33)

the following corollary is immediate from (31).

Corollary 22.

In the stationary, ergodic FCFS GI/GI/1 queue, if , then .

Remark 23.

does not hold in general, which might sound counterintuitive. A simple counter example is the case that and . Because leads to , we have . The inequality then follows from (29), (33), and . We will provide another instance of in Remark 30.

3.2 The FCFS M/GI/1 queue

In this subsection, we consider the AoI in the stationary FCFS M/GI/1 queue, where interarrival times are i.i.d. according to an exponential distribution with parameter

, and service times are i.i.d. according to a general distribution (). Note that , , and in this model.

It is well known that the LSTs and () of the waiting time and system delay in the stationary FCFS M/GI/1 queue are given by [13, Page 199]

(34)

We have the following lemma from (28) and .

Lemma 24.

In the stationary FCFS M/GI/1 queue, the LST () of the peak AoI is given by

It then follows from (24) and Lemma 24 that

(35)

where denotes the LST of residual service times.

(36)

Furthermore, noting (34), we can rewrite the first term on the right hand side of (35) to be

We thus obtain the following result from Theorem 14.

Theorem 25.

In the stationary FCFS M/GI/1 queue, the LST () of the AoI is given by

(37)

We can obtain moments of the AoI by taking the derivatives of . In particular, the first two moments of the stationary distribution of the AoI are given as follows.

Corollary 26.

The first two moments of the AoI in the stationary FCFS M/GI/1 queue are given by

(38)
(39)

where

3.3 The FCFS GI/M/1 queue

In this subsection, we consider the stationary FCFS GI/M/1 queue, where interarrival times are i.i.d. according to a general distribution () and service times are i.i.d. according to an exponential distribution with parameter . Note that , , and in this model.

It is well known that the PDF and the LST of the system delay in the stationary FCFS GI/M/1 queue are given by [13, Page 252]

(40)
(41)

where denotes the unique solution of

(42)

Noting that , we have the following lemma from (28), (40), and (41).

Lemma 27.

In the stationary FCFS GI/M/1 queue, the LST of the peak AoI is given by

We thus obtain the following result from Theorem 14.

Theorem 28.

In the stationary FCFS GI/M/1 queue, the LST of the AoI is given by

where denotes the LST of residual interarrival times.

(43)

Taking the derivatives of , we obtain the first two moments of the AoI as follows.

Corollary 29.

In the stationary FCFS GI/M/1 queue, the first two moments of the AoI are given by

(44)
(45)
Remark 30.

We can derive a sufficient condition for in the stationary FCFS GI/M/1 queue. It follows from (32), (33), and that

On the other hand, from (44) and