## 1 Introduction

Consider a queueing system transmitting messages. If is the arrival time of the last completely served message before time , then the quantity is called “age of information” (AoI), e.g., [3, 7] and references therein. The distribution function of AoI plays an important role in quality-of-service requirements of message transmission and queueing systems for certain latency-critical applications of, e.g., remote sensing and control; i.e., for a threshold and tolerance require (Note that, abusing notation,

will typicall stand for a random variable that is distributed according to

for some, and hence all, , when the process is stationary.)Service preemption, last-in-first-out (LIFO) service, or queue push-out may or may not be practically feasible, but blocking typically is. In [4], the stationary AoI distribution was derived for GI/GI/1/1-type systems, with () or without () service preemption. In [7] for M/M/1/2, the steady-state mean AoI was derived for blocking () and queue pushout () policies. In [5] for M/GI/1/2, the stationary AoI distribution was found for and . has pathwise equal AoI as the infinite-buffer LIFO system with service preemption. Similarly, has pathwise smaller AoI than . When service times are deterministic, was shown to have lower mean AoI than , though the converse is true when service times are exponential. Also, is known to have smaller AoI than FIFO upon successful departure times of the former, see Observation 2 of [4] and the discussion in [5].

In this paper, we derive the AoI distribution for the stationary M/GI/1/2 system with a “dynamic” service preemption or queue-pushout policy depending on the amount of service received so far by the in-service message. As such, it is a causal queueing policy that generalizes both and . The approach conditions on a well-known Markov-renewal embedding [2, 6] which can be employed to compute the AoI distribution for other such queueing systems [5].

## 2 System definition and preliminaries

There is a buffer consisting of two cells. Cell 1 is reserved for the message receiving service and cell 2 for the message waiting. If there is a message in cell at time we let be the amount of service received by this message up to ; if the system is empty, we set . Fix . If a message arrives at time and then the arriving message pushes-out the message in cell 1 and takes its place. Otherwise, if then the arriving message occupies cell 2 (pushing out the message sitting there, if any). We call this system . Note that and make sense too and that the collection , , is a “homotopy” between these two systems. In fact, in the terminology of [5, 4], and . (In the latter system, cell 2 will never be occupied and so, effectively, it has buffer of size 1.)

Thus, a contiguous service interval that ends with a message departure is a sequence of preempted message-service periods followed by a completed/successful message-service period. Prior to the successful message-service period, there are no queued messages waiting for service. During the successful message-service period, any arriving messages obviously fail to preempt and, under queue pushout, the last such arriving message is queued and begins service once the successful message-service period ends.

We assume that the arrival process is Poisson with rate and that messages have i.i.d. service times (independent of arrivals) distributed like a random variable such that a.s. with expectation . We let be the distribution function of and set . Under these assumptions, we will assume that the system is in steady-state (taking into account that there is a unique such steady-state, the reasons for which are classical and will not be discussed here).

Let

denote successful departure epochs. Let

be the number of mesages in the system immediately after . Given , consider Figure 1 at right.when the interarrival time commencing at the start of the successful service period (green dot) of duration satisfies Thus,

Given , consider Figure 2 below.

. Use the memoryless property of interarivals to obtain

Here, if there are no service preemptions in , then is the residual interarrival time after which is by the memoryless property. So, is an i.i.d. Bernoulli sequence with

The queueing process over consecutive intervals and are conditionally independent given . Thus, is Markov-renewal with renewal times and the queueing process is semi-Markov [2, 6]. In particular, and are conditionally independent given .

## 3 Stationary AoI distribution

###### Proposition 1.

The Laplace transform of the stationary AoI distribution is

(1) |

where and

are respectively probability and expectation given

.###### Proof.

To calculate the terms in (1) we need to follow the steps outlined in the lemmas below. Let and where for and, for

###### Lemma 1.

(2) | |||

(3) |

###### Proof.

See Figure 1 at left and consider the interval . Let be first message arrival time in this interval minus , so that by the memoryless property. Note that there is a geometric number of interarrival times each of which is smaller than both and the associated service time; in Figure 1. The probability of such unsuccessful service is

So, for . Let so that . Finally, let be the duration between the arrival time (green dot) of the message that departs at and the next arrival time. The service time (from the green dot to ) is independent of . Considering the prior unsuccessful service completions, we are given that or . Given , . Let which has distibution

with . So,

a sum of independent terms with . Also in this case. ∎

Define

###### Lemma 2.

(4) | |||

(5) |

###### Proof.

See Figure 1 at right. The difference between this and the previous case is that here . So, the distribution of is , as defined above. ∎

Define next

###### Lemma 3.

(6) | |||

(7) |

###### Proof.

See Figure 3 below.

. For there are two subcases depending of whether there are initial unsuccessful arrivals in the interval , i.e., whether . When (Figure 3 right), this case is just like when . Otherwise (Figure 3 left),

where is the duration between the last arrival before (green dot), and and are independent given . Starting from , look backward in time until the first Poisson point appears (green dot) and condition on the event that this occurs at least units of time before the service time ends. Thus, with distribution . Also, is distributed as for the case where except the first interarrival time is absent. ∎

###### Lemma 4.

(8) | |||

(9) |

###### Proof.

The final stage: The formulas obtained in the lemmas above must now be substituted into (1) as follows:

(10) | |||

(11) |

So,

Moreover,

## 4 Stationary mean AoI,

The stationary mean AoI can be obtained from (1) and numerically minimized over for a given set of model parameters for an arbitrary service-time distribution . For example, for exponential service times, , and () achieves minimal [7, 4]. For constant service time , the system with has

which is generally smaller than (and also smaller than for traffic loads [4, 5]).

Consider next the mixture service-time distribution defined by for , and so . From Figure 4, which was obtained numerically using (1), we see that in some cases is not minimized at either or . That is, in some cases, the policy for is better than both and .

Consider the policy where a message arriving at time does not preempt the in-service message (but joins the queue in cell 2) if , otherwise the message captures the server. The stationary AoI distribution of this policy is similarly derived and, for some service-time distributions, it may have lower AoI than .

## References

- [1] F. Baccelli and P. Brémaud (2003). Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences, 2nd Ed. Springer-Verlag, Berlin.
- [2] E. Çinlar (1975). Introduction to Stochastic Processes. Dover.
- [3] S. Kaul, R. Yates and M. Gruteser (2012). Real-time status: how often should one update? Proc. 31st IEEE INFOCOM, Orlando, Florida, pp. 2731-2735.
- [4] G. Kesidis, T. Konstantopoulos and M.A. Zazanis (2020). The new age of information: a tool for evaluating the freshness of information in bufferless processing systems. Queueing Systems 95, 203-250. http://arxiv.org/abs/1904.05924; https://arxiv.org/abs/1808.00443
- [5] G. Kesidis, T. Konstantopoulos and M.A. Zazanis (2021). Age of information distribution without service preemption. http://arxiv.org/abs/2104.08050
- [6] L. Kleinrock. Queueing Systems Volume I: Theory (1975). Wiley.
- [7] A. Kosta, N. Pappas and V. Angelakis (2017). Age of information: a new concept, metric, and tool. Foundations and Trends in Networking 12, No. 3, 162-259.

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