Status Updates in a multi-stream M/G/1/1 preemptive queue

# Status Updates in a multi-stream M/G/1/1 preemptive queue

Elie Najm LTHI, EPFL, Lausanne, Switzerland
Email: elie.najm@epfl.ch
Emre Telatar LTHI, EPFL, Lausanne, Switzerland
Email: emre.telatar@epfl.ch
###### Abstract

We consider a source that collects a multiplicity of streams of updates and sends them through a network to a monitor. However, only a single update can be in the system at a time. Therefore, the transmitter always preempts the packet being served when a new update is generated. We consider Poisson arrivals for each stream and a common general service time, and refer to this system as the multi-stream M/G/1/1 queue with preemption. Using the detour flow graph method, we compute a closed form expression for the average age and the average peak age of each stream. Moreover, we deduce that although all streams are treated equally from a transmission point of view (they all preempt each other), one can still prioritize a stream from an age point of view by simply increasing its generation rate. However, this will increase the sum of the ages which is minimized when all streams have the same update rate.

## I Introduction

Previous work on status update, e.g. [1, 2, 3, 4, 5, 6], used an Age of Information (AoI) metric in order to assess the freshness of randomly generated updates sent by one or multiple sources to a monitor through the network. In these papers, updates are assumed to be generated according to a Poisson process and the main metric used to quantify the age is the time average age (which we will call average age) given by

 Δ=limτ→∞1τ∫τ0Δ(t)dt, (1)

where is the instantaneous age at the receiver of the information about the source status. If the last successfully received update was generated at time then the age of the source status at time is . When the system is idle or an update is being transmitted then the instantaneous age increases linearly with time. Once an update generated at time is received by the monitor at , drops to the value . This results in the sawtooth sample path seen in Fig. 1.

Moreover, in [7] the authors introduce another age metric: the average peak age defined as the time average of the maximum value of the instantaneous age right before the reception by the monitor of a new update. In Fig. 1 the peak age right before the reception of the successfully transmitted update is denoted by . Hence the average peak age is given by

 Δpeak=limN→∞1NN∑j=1Kj. (2)

In this paper, we assume that an ‘observer’ (which we will call source), generating updates according to a Poisson process with rate , observes streams of data. At each generation instant, the source chooses to ‘observe’ stream and send its observation (update) of this stream with probability , . This probability distribution is a design parameter that one can control. Moreover, we assume that the system can handle only one update at a time without any buffer to store incoming updates. This means that whenever a new update is generated and the system is busy, the transmitter preempts the packet being served and starts sending the new update instead. Since we consider a general service time distribution for the updates, we denote this transmission scheme by M/G/1/1 preemptive queue. It has been shown that for a single-stream source and exponential update service time, preemption ensures the lowest average age [2]. However, the work in [5] suggests that under the assumption of gamma distributed service time, preemption might not be the best policy. In [8], the authors derive a closed form expression for the average age of a single-stream source and M/G/1/1 preemptive queue.

As a generalization of the result in [8], we derive in this paper a closed form expression for the average age and average peak age per stream of the multi-stream source M/G/1/1 preemptive queue. To that end we use the detour flow graph method which is also used to find an upper bound on the error probability of a Viterbi decoder (see [9]). A special case of this problem was studied in [10] where the service time is assumed to be exponentially distributed. In this paper the average age of each stream was obtained in closed form using a stochastic hybrid system. Another related work, [11], gives closed form expressions for the average peak age of multi-stream source M/G/1 queues as well as M/G/1/1 queues with blocking. In this last model, if a newly generated update finds the system busy, it is discarded.

In addition, given a fixed total update rate , we show in this work that if we want to decrease the age of a certain stream with respect to other streams we need to increase its update rate (by increasing its choice probability ) and thus decreasing the update rates of the other streams. Moreover, if we choose the sum of the ages as our performance metric and we wish to minimize it then we prove that we need to adopt a fair strategy: all streams should be given the same update rate.

This paper is structured as follows: in Section II, we start by defining the model and the different variables needed in our study. In Section III we derive the closed form expressions of the average age and average peak age and state the conditions necessary to minimize the sum of the ages.

## Ii System Model

In this model a source generates updates according to a Poisson process with rate and send them through the network. However, we assume that the updates belong to different streams, each stream being chosen independently at generation time with probability , . This setup is equivalent to having independent Poisson sources with rates , , and (see [12]). Moreover, we consider an M/G/1/1 queue with preemption. This means that only one update can be in the system at a time and thus the different streams preempt each others and even the same stream preempts itself. This setup was analyzed in [10] where the authors considered an exponential service time. In this paper, we assume a service time with general distribution. Given that the system is symmetric from the point of view of each stream, we will focus — without loss of generality— on stream as the main stream. Hence, unless stated otherwise, all random variables correspond to packets from stream .

Moreover, in this paper we follow the convention where a random variable with no subscript corresponds to the steady-state version of which refers to the random variable relative to the received packet from stream . To differentiate between streams we will use superscripts, so corresponds to the steady-state variable relative to the stream.

It is important to note that in M/G/1/1 queues with preemption, some updates might be dropped. Hence we call the updates that are not dropped, and thus delivered to the receiver, as “successfully received updates” or “successful updates”. We also define: to be the interdeparture time between the and successfully received updates, to be the interarrival time between two consecutive generated updates from stream , , (which may or may not be successfully transmitted), so , to be the service time random variable for any update (from any stream) with distribution , to be the system time, or the time spent by the successful update in the queue and , the number of successfully received updates from stream in the interval . In our model, we assume the service time of the updates from the different streams to be independent of the interarrival time between consecutive packets (belonging to the same stream or not). These concepts are illustrated in Fig. 1, where only successfully transmitted packets from stream are shown.

## Iii Age of a multi-stream M/G/1/1 preemptive queue

We denote by , the Laplace transform of the service time distribution evaluated at , i.e. .

Before stating the main result of this section we need the following lemmas.

###### Lemma 1.

Let , and be three non-negative independent random variables with respective distributions: , and , with . Let , , , be random variables such that , , and . Then,

,

,

with being the Laplace transform of the random variable evaluated at .

###### Proof.

We will only prove the result for the variable since we can apply the same technique for the others. Denote by the complementary CDF of . Then,

 P(min(S,Λ)≥t) =P(S≥t,Λ≥t) =P(S≥t)P(Λ≥t) =¯FS(t)e−(λ−λ1)t.
 fB(t) =limϵ→0P(B∈[t,t+ϵ])ϵ =limϵ→0P(X∈[t,t+ϵ]|X≤min(S,Λ))ϵ =limϵ→0P(X∈[t,t+ϵ])P(X≤min(S,Λ)|X∈[t,t+ϵ])ϵP(X≤min(S,Λ)) =λ1e−λ1tP(min(S,Λ)≥t)P(X≤min(S,Λ))=λ1e−λt¯FS(t)P(X≤min(S,Λ)),
 P(X≤min(S,Λ)) =∫∞0λ1e−λt¯FS(t)dt=λ1λ(1−Pλ),

where the last equality is obtained using integration by parts. Thus . Using again integration by parts we find that . ∎

###### Lemma 2.

For the M/G/1/1 queue with preemption described above, the moment generating function of the system time corresponding to a stream is given by

 ϕT(i)(s)=Pλ−sPλ. (3)

Note that the right hand side of (3) does not depend on the chosen stream.

###### Proof.

Without loss of generality we will prove Lemma 2 for stream . The system time of the successfully received packet corresponds to the service time of the received packet given that service was completed before any new arrival (since any new packet from any stream will preempt the current update being served). So, in steady-state, . Hence, for ,

 fT(t) =limϵ→0P(T∈[t,t+ϵ])ϵ =limϵ→0P(S∈[t,t+ϵ]|St)P(S

where the last equality is due to the fact that is exponentially distributed with rate . Thus,

 ϕT(s) =E(esT)=∫∞0fS(t)P(S

Finally,

 P(St)dt=∫∞0fS(t)e−λtdt =Pλ. (4)

###### Lemma 3.

The moment generating function of the interdeparture time of the stream, , is

 ϕY(i)(s)=λiPλ−sλiPλ−s−s. (5)
###### Proof.

Without loss of generality, we will prove Lemma 3 for stream . We define and . Since and are the minimum of independent exponential random variables, then they are also exponentially distributed with rates and respectively. Fig. 2 shows the semi-Markov chain relative to the interdeparture time between the and received packet of the first stream. When the packet leaves the queue, the system enters the idle state where it waits for a new packet from any stream to be generated. Hence two clocks start: a clock and a clock . Clock ticks first with probability , at which point a new packet from stream will be generated first and the system goes to state . The value of the clock when it ticks has distribution . Clock ticks first with probability , at which point a new packet from one of the other streams is generated first and the system goes to state . The value of this second clock when it ticks has distribution .

When the system arrives in state , this means a packet from stream is starting service. Thus, due to the memoryless property of , three clocks start: a service clock , clock and clock . The service clock ticks first with probability and its value has distribution . At this point the stream packet currently being served finishes service before any new packet is generated and the system goes back to state . This ends the interdeparture time . On the other hand, clock ticks first with probability and its value has distribution . At this point, a new stream update is generated before any other update from other streams and preempts the one currently in service. In this case the system stays in state . The third clock ticks first with probability and its value has distribution . At this point a new update not from stream is generated, preempts the one currently in service and the system switches to state .

When the system arrives in state , this means a packet not from stream is starting service. Thus, due to the memoryless property of , three clocks start: a service clock , clock and clock . As for state , the service clock ticks first with probability and has value . At this point packet currently being served finishes service before any new packet is generated and the system goes to state . Also like before, clock ticks first with probability and has value . At this point, a new stream update is generated before any other update from other streams and preempts the one currently in service. In this case the system switches to state . The third clock ticks first with probability and has value . At this point a new update not from stream is generated, preempts the one currently in service and the system stays in state .

Finally, when the system arrives in state , this means the system is idle but no update from stream has been delivered. Given and are memoryless, the system in state behaves exactly like if it were in state .

From the above analysis we see that the interdeparture time is given by the sum of the values of the different clocks on the path starting and finishing at . For example, for the path in Fig. 2 the interdeparture time , where all the random variables in the sum are mutually independent. This value of is also valid for the path . Hence depends on the variables and their number of occurrences and not on the path itself. Therefore, the probability that exactly occurrences of happen, which is equivalent to the probability that

 Y=i1∑k=1Ak+i2∑k=1Bk+i3∑k=1Uk+i4∑k=1Vk+i5∑k=1Zk

is given by , where is the number of paths with this combination of occurrences. Taking into account the fact that the are mutually independent, the moment generating function of is

 ϕY(s) =E(E(esY|(I1,I2,I3,I4,I5)=(i1,i2,i3,i4,i5))) =∑i1,i2,i3,i4,i5[ai1bi2ui3vi4zi5Q(i1,i2,i3,i4,i5) E(es(∑i1k=1Ak+∑i2k=1Bk+∑i3k=1Uk+∑i4k=1Vk+∑i5k=1Zk))] =∑i1,i2,i3,i4,i5[ai1bi2ui3vi4zi5Q(i1,i2,i3,i4,i5) E(esA)i1E(esB)i2E(esU)i3E(esV)i4E(esZ)i5], (6)

where are the random variables associated with the number of occurrences of respectively.

Moreover, given a directed graph with algebraic label on its edges, and a node with no incoming edges, the transfer function from to a node is the sum over of all paths from to with each path contributing the product of its edge labels to the sum (see [9, pp. 213–216]). The complete set of transfer functions can be computed easily by solving the linear equations:

 {H(u)=1H(w)=∑w′:(w′,w)∈EH(w′)L((w′,w)),w≠u.

Observe that the sum in (III) is nothing but the transfer function from to in the graph shown in Fig. 3 with

 (D1,D2,D3,D4,D5)

Solving the system of linear equations above yields the transfer function as

 H(D1,D2,D3,D4,D5) =∑i1,i2,i3,i4,i5[Q(i1,i2,i3,i4,i5)ai1bi2ui3vi4zi5 Di11Di22Di33Di44Di55] (7)

Thus

 ϕY(s)=H(E(esA),E(esB),E(esU),E(esV),E(esZ)).

From Lemma 1, we know that and . Moreover, one can notice that has the same distribution as the system time so . Simple computations show that , , , , . Finally, replacing the above expressions into (III), we get our result.

###### Theorem 1.

Given an M/G/1/1 queue with preemption and service time and a source generating packets belonging to streams according to independent Poisson processes with rates , , such that , then

1. the average age of stream is given by

 Δi=1λiPλ, (8)
2. and the average peak age of stream is given by

 Δpeak,i=1λiPλ+E(Se−λS)Pλ. (9)
###### Proof.

Due to the symmetry in the system from a stream point of view, then, without loss of generality, we will prove 1 for stream only. The same proof applies for the other streams.

From (1) and Fig. 1, the average age for stream of the M/G/1/1 queue can be also expressed as the sum of the geometric areas under the instantaneous age curve. Authors in [2] show that

 Δ1 =limτ→∞Nττ1NτNτ∑j=1Qj=λeE(Q), (10)

where , is the steady-state counterpart of and the second equality is justified by the ergodicity of the system. As shown in [10] and [8], is the rate at which successful updates are received. Given that the interarrival time of all streams are memoryless, then the interdeparture times, and , between two consecutive received updates are i.i.d. Hence forms a renewal process and by [12], , where is the steady-state interdeparture random variable. Moreover, from Fig. 1 we see that by applying same reasoning as in [5]

 E(Q)=12E(Y2)+E(TY)=12E(Y2)+E(T)E(Y).

The second equality is obtained by noticing that for any received packet , and are independent. Therefore,

 Δ1=E(T)+E(Y2)2E(Y) (11)

Moreover, from Fig. 1 we see that the peak age at the instant before receiving the packet is given by

 Kj=Tj−1+Yj−1.

Hence, given that the system is ergodic, (2) becomes at steady state,

 Δpeak,1=E(K)=E(T)+E(Y). (12)

Using Lemma 2, we get that . Using Lemma 3, we get that and . Using these expressions in (11) and (12) we obtain our result for stream . ∎

Note that for , we get back the result derived in [8] for single stream M/G/1/1 preemptive queue. Moreover, if we replace in (8) by the Laplace transform of the exponential distribution evaluated at we recover the expression stated in [10, Theorem 2(a)].

###### Corollary 1.

Let a source generate updates according to a Poisson process with fixed rate . Moreover, these updates belong to different streams, each stream chosen independently with probability at generation time. Then if we use an M/G/1/1 with preemption transmission scheme we can decrease the average age (and the average peak age) of a high priority stream with respect to the other streams by increasing the probability with which it is chosen.

###### Proof.

From Theorem 1 we know that for any two streams and , in order to have or we must have . Given that , , then we must have . ∎

Given the source generates multiple streams, one interesting performance measure of the system would be the total average age or total average peak age defined respectively as

 Δtot =M∑i=1Δi, Δpeak,tot=M∑i=1Δpeak,i. (13)

The next theorem gives the distribution over the , , that minimizes the metrics in (13) as well as their minimum achievable value.

###### Theorem 2.

For the M/G/1/1 multi-stream preemptive system described above with fixed total generation rate , the optimal strategy that achieves the smallest value for the total average age, , and the total average peak age, , is the fair strategy: all streams should have the same generation rate. This means that the probability distribution over the choices of streams should be the uniform distribution with , . Moreover, the optimal values of and are given by

 Δtot =M2λPλ, Δpeak,tot=M2λPλ+ME(Se−λS)Pλ (14)
###### Proof.

From (8), (9) and (13), we get that

 Δtot =1PλM∑i=11λi=1λPλM∑i=11pi Δpeak,tot =1PλM∑i=11λi+ME(Se−λS)Pλ =1λPλM∑i=11pi+ME(Se−λS)Pλ (15)

Given that is fixed, then minimizing and over is equivalent to minimizing . As this is a symmetric convex function, it is minimized when with the value , which proves our theorem. ∎

From Corollary 1 and Theorem 2, we see that prioritizing a stream over the others from an age point of view and minimizing the total age are two contradictory objectives.

## Iv Conclusion

In this paper we studied the M/G/1/1 preemptive system with a multi-stream updates source. We derived closed form expressions for the average age and average peak age using the detour flow graph method. Moreover, using these results we showed that, for a fixed total generation rate, one can’t prioritize one of the streams and at the same time minimize the total age. In fact, we prove that in order to optimize the total age, the source needs to generate all streams at the same rate. This means that no single stream can be given a higher rate, a necessary condition to reduce its age with respect to the other streams.

## Acknowledgements

This research was supported in part by grant No. 200021_166106/1 of the Swiss National Science Foundation.

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