I Introduction
Freshness of status information is crucial in realtime status update systems seen, for example, in weather reports, autonomous driving, stock market trading and so on. Age of information (AoI) has been widely accepted in this decade as a measure quantifying the freshness of information in status update systems where information sources transmit packets containing status updates to destination monitors through a communication network. Specifically, the AoI is defined as the elapsed time since the information currently displayed on a monitor is generated and timestamped at the source. In many of such systems, multiple sources share a limited network resource and therefore the AoIs defined for the individual sources should be correlated with each other. However, to the best of the knowledge of the authors, there are no results studying the correlation of two or more AoIs in a status update system with multiple sources (except for the first author’s preliminary work [1]). To investigate the correlation is essential when, for example, we consider a nonlinear penalty function of multiple AoIs. In this work, we consider a multisource system sharing a common service facility and provide a framework to investigate joint performance of the AoIs defined for the individual sources.
Ia Related work
Since the advent in [2, 3], a large amount of literature has emerged on development of the AoI concept due to its importance and availability in a wide range of information communication systems. A complete review falls out of the reach of this paper and we only highlight some relevant results to ours. Interested readers are referred to recent monographs [4, 5] and references therein.
The AoI was first introduced in [2] in the context of vehicular networks and queueingtheoretic technique was applied in [3] to analyze the mean AoI under the ergodicity assumption. These results were then extended to various multisource systems in [6, 7, 8]. A more tractable metric, peak AoI (PAoI)—the AoI immediately before an update, was introduced in [9, 10]
, which characterized not only the mean but also the probability distribution of the PAoI for various queueing systems under the ergodicity. Not only the mean, expected nonlinear functions of the AoI were examined in
[11, 12].In the early stage of the development, they assumed that time intervals of packet generations and their service times are either exponentially distributed or deterministic; that is, they considered M/M, M/D or D/M inputs in the queueing notation. Recently, some researchers have challenged to incorporate more general probability distributions. In
[13], gamma distributed service times were assumed for Poisson arrival systems and
[14, 15] considered more general service time distributions in multisource systems with independent Poisson arrivals; that is, they treated independent M/G input processes. Furthermore, [16, 17, 18] studied more general frameworks of packet arrival and service time processes and derived general formulas satisfied by the stationary distribution and its Laplace transform of the AoI, where [16, 17] adopted the technique of sample path analysis while [18] worked based on the Palm calculus within the stationary framework (see, e.g., [19] for the Palm calculus). However, there are no results confronting joint performance of multiple AoIs in multisource systems within general frameworks.IB Contribution
In this paper, we consider a multisource system sharing a service facility as in [6, 7, 8, 14, 15] and provide a framework to investigate joint performance of the AoIs defined for the individual sources. We first consider a general multisource system, where the time sequence representing service completions (and status updates) follows a stationary point process on the real line, and derive a formula satisfied by the joint Laplace transform of the stationary AoIs. A tool for our analysis is the Palm calculus within the stationary framework as in [18], where we do not require the ergodic assumption but, once the ergodicity is assumed, we can obtain the same results as those from the sample path analysis by the ergodic theorem. Our formula is so general and is applicable to many multisource systems. We then apply this formula to a simple pushout server, where the system has a single server and each generated packet is immediately started for service without waiting; that is, the ongoing service of another packet (if any) is interrupted and replaced by the new one. In the case with independent M/G input processes, we derive a closedform formula of the joint Laplace transform of the AoIs. Furthermore, we reveal some properties of the correlation coefficient of the AoIs in the twosource system.
IC Organization
The rest of the paper is organized as follows. In the next section, we describe a general multisource system and derive a formula satisfied by the joint Laplace transform of the multiple AoIs. We also provide a formula for a singlesource system, which corresponds to the review of the results in [16, 17] within our stationary framework. In Section III, we apply our formula to a multisource pushout server. We first confirm in IIIA that our multisource pushout server is definitely within our framework and then derive a closedform formula of the joint Laplace transform of the AoIs in the case with independent M/G inputs in IIIB. Some properties of the correlation coefficient of the AoIs in the twosource system are revealed in IIIC. These properties of the correlation coefficient are confirmed through numerical experiments in Section IV. Finally, concluding remarks and future work are discussed in Section V.
Ii General multisource system
In this section, we consider a general multisource singleserver system. There are () sources generating packets with different kinds of status information and each source has its dedicated monitor. The set of the sources is denoted by . The system has a single server and, after a packet from source is processed by the server, the status information in the packet is immediately displayed on monitor
. We note that not all packets are completed for service and some may be discarded and lost (due to buffer overflows, packet deadlines or pushouts). We, for the moment, ignore such lost packets and focus on those being completed for service.
Let denote a point process on counting the times at each of which the service of a packet is completed and the status information on one of the monitors is updated, and let denote the corresponding time sequence, where . For , let and denote respectively the source and the delay of the packet whose service is completed at ; that is, the status information updated at is generated and timestamped at by source . For each , let denote a subprocess of counting the times of service completions of source packets; that is,
(1) 
where denotes the indicator function of a set . Clearly, , , are mutually disjoint and satisfy . We impose the following assumption on the marked point process corresponding to .
Assumption 1

The point process is simple almost surely in a probability (a.s.), where the rule of subscripts is such that conventionally (see, e.g., [19]).

The marked point process is stationary in with mark space .

The point process has positive and finite intensity , where denotes the expectation with respect to .

for all , where denotes the Palm probability for .

for all , where denotes the Palm probability for .
Note that the Palm probability is well defined under Assumptions 12) and 3) (see [19]) and that . Furthermore, under Assumption 12), is stationary in . Assumption 14) does not restrict us since we can redefine by if for some . The intensity of the subprocess is given by for each , so that the Palm probability is also well defined under 12)–4) and satisfies . Let , , denote the time sequence corresponding to satisfying , and let also denote the delay of the source packet whose service is completed at (note that a.s.). Then, the AoI process for source is defined as (see [7, 8] for AoI in multisource systems)
(2) 
This definition indicates that the AoI of source represents the elapsed time since the information currently displayed on monitor is generated and timestamped; that is, it is set to the delay of a source packet at its service completion time and increases linearly until the status information from source is next updated.
Lemma 1
For each , the marked point process corresponding to and also the AoI processes are jointly stationary with the marked point process under Assumption 1. Furthermore, is a.s. finite under the same assumption.
Proof:
The proof relies on a technical discussion within the stationary framework and is given in Appendix A.
While the AoI just after an update is equal to the delay , that just before an update is called the PAoI (see [9, 10]); that is, for each , the sequence of PAoIs is defined as , , and is also stationary in . Prior to the joint performance analysis of , , we review the results of [16, 17] for a singlesource singleserver system within our stationary framework, which is also useful in the marginal performance analysis of multisource systems.
Proposition 1 (Cf. [16, 17])
Consider a singlesource system satisfying Assumption 1 with , where , and in (2). Then, the stationary distribution of the AoI satisfies
(3) 
Let , and , , denote the Laplace transforms of , and , respectively, where denotes the expectation with respect to the Palm probability . Then, the following relation holds;
(4) 
for such that the Laplace transforms on both the sides exist.
Proof:
Applying the Palm inversion formula (see [19, p. 20]),
where for and the second equality follows from (2); that is, increases linearly from to for . The last expression above immediately derives (3) since is distributionally equal to in . The formula (4) is obtained from (3) as
using
for a random variable
and such that is finite.Remark 1
Formulas (3) and (4) respectively correspond to the results of Theorem 14 (i) and (ii) in [17]. As opposed to [16, 17], however, Proposition 1 does not require the ergodicity of . These formulas suggest that we can obtain the stationary distribution of the AoI and the corresponding Laplace transform once the stationary distributions of the delay and PAoI are available. The formula (3) also implies that the stationary distribution of the AoI has the density function , . Taking in (3), we have , which intuitively makes sense since a PAoI is the sum of a delay and its subsequent interdeparture time (see (2)).
Proposition 1 can be applied to the marginal distribution of each AoI in a system with .
Corollary 1
For a system with sources satisfying Assumption 1, the marginal stationary distribution of the AoI for source satisfies
(5) 
Let , and , , denote the Laplace transforms of , and , respectively, where denotes the expectation with respect to . Then, the following relation holds;
(6) 
for such that the Laplace transforms on both the sides exist.
Next, we consider the joint performance of multiple AoIs. Let on denote the joint Laplace transform of ; that is,
(7) 
Note that does not always exist on the whole space of . We derive a general formula satisfied by as far as it exists, which is applicable to the analysis of many multisource singleserver systems satisfying Assumption 1. Let denote a random permutation of satisfying
(8) 
That is, represents the source such that the status information on its monitor at time is the th newest among (the information on monitor is most recently updated while monitor displays the oldest information). Note that satisfy the relation;
(9) 
In the following, we write for a nonempty subset with and given the random permutation satisfying (8).
Theorem 1
For a source singleserver system satisfying Assumption 1, the joint Laplace transform of the stationary AoIs satisfies
(10) 
for such that the expectation on the righthand side exists.
Proof:
Remark 2
We can easily confirm that (10) agrees with (4) in Proposition 1 when ( in this case) since and a.s. On the other hand, it is not so straightforward to show that (10) with for all agrees with (6) in Corollary 1 because (10) is based on the Palm inversion formula with respect to (with the integral on ) while (6) is on that with respect to (with the integral on ). Therefore, Corollary 1 still makes sense in marginal analysis of multisource systems. Note that the terms in the product in (10) are evaluated in mutually disjoint intervals and on the event . This property can make formula (10
) useful for analysis of a class of multisource systems such that the sequence of service completion times forms a regenerative process or an embedded Markov chain.
Iii Application to a multisource pushout server
In this section, we apply the results in the preceding section to a pushout server with sources. We first confirm that the system satisfies Assumption 1 in the preceding section and then derive a closedform formula for the joint Laplace transform of the AoIs in the case with independent M/G input processes. We further reveal some properties of the correlation coefficient of the AoIs in the twosource system.
Iiia Multisource pushout server
We here describe the system consisting of a pushout server and sources with the dedicated monitors. Let denote a point process on counting the times at each of which a packet is generated and timestamped by any one of the sources and let denote the corresponding time sequence. For each , and denote respectively the source and the required service time of the packet generated at . For each , let denote the subprocess of counting the generation times of source packets; that is,
We impose the following assumption on the marked point process corresponding to .
Assumption 2

The point process is a.s. simple, where is numbered as conventionally.

The marked point process is stationary in with mark space .

The point process has positive and finite intensity .

for all , where denotes the Palm probability for .

for all , where , , and denotes the Palm probability for .
The Palm probability in Assumption 24) is well defined under 22) and 3) and so are , , under 22)–4) since the intensity of is given by . Note here that holds for .
The system has a single server and each generated packet is immediately started for service without waiting. If another one is in service at the generation time of a packet, the service is interrupted and replaced by the new one (the interrupted packet is pushed out and lost). There is no priority among the sources and the service of any packet can be interrupted by the next generated one from the same or other sources. The probability that a packet generated at source is completed for service without interruption is then given by , which is positive for all under Assumption 25). When the service for a packet is completed without interruption, the status information carried by the packet is displayed on the monitor dedicated to the source of that packet. Then, the marked point process , representing the service completions considered in the preceding section, is expressed in terms of as
(12) 
Lemma 2
Proof:
The proof is also based on the stationary framework and is given in Appendix B.
IiiB Multisource M/G/1/1 pushout server
In this subsection, by specifying the input point process as independent homogeneous Poisson processes and assuming independence in the service times, we derive a closedform formula for the joint Laplace transform of the AoIs . We assume that are mutually independent homogeneous Poisson processes with positive and finite intensities . The superposition theorem for Poisson processes (see, e.g., [20, p. 20], [21, p. 36]) then implies that is also a homogeneous Poisson process with intensity . We further assume that service times , , depend only on their sources and, when the sources of packets are given, the service times are mutually independent and independent of , . Namely, for any , , , and , we have
Let denote the Laplace transform of service times of source packets; that is, , (it may be infinite for some ), where we note that and . In this setup, Assumption 2 is satisfied and the probability that a source packet is completed for service without interruption is given by
where the second equality follows from the independent increments property of a Poisson process. Then, (13) and (14) in Lemma 2 are respectively rewritten as and with . Furthermore, we use the following notation such that, for a nonempty subset ,
with . Note that and for .
First, we consider the marginal Laplace transform of the AoI for each source.
Proposition 2
For the source M/G/1/1 pushout server described above, the marginal Laplace transform of the stationary AoI of source is given by
(15) 
Proof:
We use (6) in Corollary 1. Note that by (2) and the definition of the PAoI. In our M/G/1/1 pushout server, since and are mutually independent and is also independent of on the event , (6) is reduced to
(16) 
First, Neveu’s exchange formula (see [19, p. 21]) implies that
(17) 
where denotes the subsequence of corresponding to satisfying and . The second equality in (IIIB) follows from the observation that there exists at most one service completion of a source packet during and it occurs only when the packet generated at is completed for service without interruption.
Next, we consider in (16). Note that there may be one or more service completions during , but if any, they must be of the sources in . Since for (where ) and the server is always reset at , , we have
(18) 
We solve above. Let , , denote the time length of the busy period starting at and ending at the next service completion. Since is independent of the event due to the independent increments property of a Poisson process and is initialized at , we have
(19) 
Here, is immediate since is a homogeneous Poisson process with intensity . In considering , we note that there may be one or more pushedout services in a busy period. Let , which represents the index of the first packet completed for service after . Note that a.s. since . Then, and . Since , , are mutually independent and identically distributed, we have
(20) 
Here, a similar way to obtaining (IIIB) leads to
and on the other hand,
Plugging these into (IIIB), and then to (IIIB), we obtain
(21) 
The same discussion as above except for replacing by derives
(22) 
and further plugging (21) and (22) into (IIIB), we have
(23) 
Finally, substitution of (IIIB) and (23) into (16) yields (15).