Discrete-time Queueing Model of Age of Information with Multiple Information Sources
11 D ISCRETE - TIME Q UEUEING M ODEL OF A GE OF I NFORMATIONWITH M ULTIPLE I NFORMATION S OURCES
A P
REPRINT
Nail Akar
Electrical and Electronics Engineering Dept.Bilkent UniversityBilkent 06800, Ankara, Turkey [email protected]
Ozancan Doan
Electrical and Electronics Engineering Dept.Bilkent University,Bilkent 06800, Ankara, Turkey [email protected]
July 24, 2020 A BSTRACT
Information freshness in IoT-based status update systems has recently been studied through the Ageof Information (AoI) and Peak AoI (PAoI) performance metrics. In this paper, we study a discrete-time server arising in multi-source IoT systems which accepts incoming information packets frommultiple information sources so as to be forwarded to a remote monitor for status update purposes.Under the assumption of Bernoulli information packet arrivals and a common geometric servicetime distribution across all the sources, we numerically obtain the exact per-source distributions ofAoI and PAoI in matrix-geometric form for three different queueing disciplines: i) Non-PreemptiveBufferless (NPB) ii) Preemptive Bufferless (PB) iii) Non-Preemptive Single Buffer with Replace-ment (NPSBR). The proposed numerical algorithm employs the theory of Discrete-Time MarkovChains (DTMC) of Quasi-Birth-Death (QBD) type and is matrix analytical, i.e, the algorithm isbased on numerically stable and efficient vector-matrix operations. Numerical examples are pro-vided to validate the accuracy and effectiveness of the proposed queueing model. We also presenta numerical example on the optimum choice of the Bernoulli parameters in a practical IoT systemwith two sources with diverse AoI requirements.
In a networked control and monitoring IoT-based system, it is of utmost importance to deliver timely status updatesand thus keep the information fresh, for stable operation. Performance metrics using the Age of Information (AoI)and Peak AoI (PAoI) processes have first been proposed in [1, 2, 3] in continuous-time, for quantitative assessmentof information freshness in status update systems. Since then, there has been a surge of interest in AoI research interms of development of queueing models [4, 5, 6, 7, 8, 9] or AoI optimization [10, 11, 12, 13, 14]. In the discrete-time setting, for each information source, there is an underlying stochastic process which is randomly sampled, andthe sampled values are transmitted in the form of information packets towards a remote monitor via a packet-basedcommunications network. Subsequently, for each information source, there is an individual stochastic process, calledthe AoI process (or sequence), which is maintained at the monitor that keeps track of the time elapsed since thegeneration of the last successfully received update packet. Therefore, this per-source AoI process is a cyclic processthat increases in time with unit steps until the end of a cycle at which a packet reception occurs and subsequently theAoI process experiences an abrupt downward jump. On the other hand, the PAoI process is obtained by sampling theAoI sequence at the embedded epochs just before the downward jumps during each cycle of the AoI process [15]. Thefocus of this paper is on an IoT-based information update system operating in discrete-time in Fig. 1 comprising N independent information sources each equipped with a sensor, a queue-server combination local to the sources, and a Mr. Dogan is supported in part by the
5G and Beyond scholarship granted by the Information and Communication TechnologiesAuthority (ICTA) of Turkey and Vodafone Turkey. a r X i v : . [ c s . PF ] J u l PREPRINT - J
ULY
24, 2020 p p N p Server MonitorSource 1Source 2Source N Queue
Figure 1: N information sources sending status update messages through a server towards a remote monitor.remote monitor. In this framework, source- n , n = 1 , . . . , N, generates information packets according to a Bernoulliprocess with parameter p n containing sensed data and a time stamp. The server is in charge of sending the informationpackets to the monitor via a communications network which introduces random delays, i.e., service time of packets,and the monitor immediately sends back positive acknowledgments to the server. We assume in this paper that theservice times of all users are geometrically distributed with the same parameter q . Particularly, we study the followingthree queueing disciplines employed at the server:• Non-Preemptive Bufferless (NPB) system for which one of the generated information packets at each timeinstant is selected uniformly at random, or randomly in short, to be placed in service when the server is idle.Otherwise, all packets are discarded.• Preemptive Bufferless (PB) system in which one of the generated information packets at each time instant(selected randomly) is always placed in service while possibly preempting the one in service.• Non-Preemptive Single Buffer with Replacement (NPSBR) system for which we have a buffer holding oneinformation packet only and one of the generated information packets at each time instant is randomly se-lected to replace the one in the waiting room. If there are no packets in the buffer, this information packet isplaced in service.The main contribution of this paper is that, based on the theory of QBDs, we propose a novel analytical modelingtechnique to derive the exact distributions of per-source AoI and PAoI in discrete-time, for NPB, PB, and NPSBR.Although distributional results exist in the literature for continuous-time, results for discrete-time are less matureexcept for ones that provide average AoI and PAoI values in less general settings.The organization of the paper is as follows. In Section 2, related work is presented. Section 3 presents QBDs. Section 4addresses the proposed queueing model for the bufferless queueing disciplines NPB and PB as well as the single-buffersystem NPSBR. Numerical examples are provided to validate the accuracy and effectiveness of the proposed queueingmodel in Section 5. Finally, we conclude in Section 6. The AoI concept was first introduced in [2] as a single-source, single-server
M/M/ queueing model. The case ofmultiple sources is then investigated in [16] in the same setting. Variations of this single-server queueing model inFig. 1 for AoI has caught the attention of researchers and practitioners for modeling IoT-based status update systems inthe literature [2, 17]. The majority of the existing queueing models on AoI or PAoI are for continuous-time operation.2 PREPRINT - J
ULY
24, 2020Let us first briefly overview the literature on single-source continuous-time models. In [2], the mean AoI is obtained forthe single-source
M/M/ , M/D/ , and D/M/ queues with infinite buffer capacity and FCFS scheduling. Althoughcurrent packet-switched communication networks such as the Internet employ large buffers and FCFS scheduling atthe routers, their use is shown to have adverse effects on AoI performance figures in moderate to high load regimes [2].The distributions of AoI and PAoI are therefore studied in [6] for the case of small buffers including the conventional M/M/ / and M/M/ / queues, as well as the non-preemptive LCFS M/M/ / ∗ queue. Exact expressions forthe stationary distributions of AoI and PAoI for a very wide class of single-source information update systems aregiven in [9]. The reference [18] obtains the exact distributions of AoI and PAoI in a bufferless P H/P H/ / systemwith probabilistic preemption as well as a single-buffer M/P H/ / system that allows probabilistic replacement ofthe waiting packet by a newer packet arrival.Next, we give an overview of the existing literature on multi-source status update systems in continuous-time. MeanPAoI expressions for M/G/ and M/G/ / systems with heterogeneous service time requirements are presented in[11]. The reference [8] investigates the multi-source M/M/ model with FCFS as well as two disciplines: preemptivebufferless and nonpreemptive single buffer with replacement using the theory of stochastic hybrid systems (SHS) andobtain exact expressions for the mean AoI. A preemptive M/G/ / queue is considered in [19] with a commonservice time for all sources in which expressions for the mean AoI and PAoI are derived. The authors of [20] allowself-preemption in which case mean AoI expressions are derived for each source using SHS-based techniques whereasthe reference [21] considers a two-source M/M/ / queueing system in which a packet waiting in the queue canbe replaced only by a newly-arriving packet from the same source, again using SHS. A non-preemptive M/M/ /m with common service times across sources is again studied by SHS in [22] and mean AoI expressions are derived.A more general hyper-exponential service time distribution for each class is considered in [23] for an M/H / / non-preemptive bufferless queue to obtain an expression for the mean AoI per class.The existing research on discrete-time AoI queueing models is less mature in comparison with continuous-timemodels. A discrete-time queueing model with Bernoulli arrivals and geometric service times, using FCFS and non-preemptive LCFS scheduling is presented in [24] with expressions for the mean AoI and PAoI values. The authors of[25] study the FCFS-type Ber/G/ queue and derive explicit expressions for average AoI and PAoI and also mean AoIexpressions for the discrete-time LCFS queue. For the single-source case, [26] considers the FCFS Geom/Geom/ queue and obtains closed-form expressions of the generating functions and the stationary distributions of the AoI andthe PAoI and provide a methodology for analyzing more general non-linear age functions. We first present notation. Uppercase bold letters are used to denote real-valued matrices. Lowercase bold (plain)letters or symbols are used to denote real-valued vectors (scalars). The notations k × (cid:96) , I m , and n denote a matrixof zeros of size k × l , an identity matrix of size m , and a column matrix of ones of size n , respectively. When usedwithout a subscript, we leave it to the reader to infer the size information, from the context. The function u k standsfor the discrete-time unit step function, i.e., u k = 1 , k = 0 , , . . . and is zero otherwise. The function δ k stands for thediscrete-time unit impulse function, i.e., δ k = 1 for k = 0 and is zero, otherwise.A discrete random variable X is said to possess a matrix geometric (MG) distribution, i.e., X ∼ M G ( c , A , b , d ) , ifits probability mass function (pmf), denoted by p X ( (cid:96) ) , is of the form [27]: p X ( (cid:96) ) = Pr { X = (cid:96) } = (cid:26) cA (cid:96) − b , (cid:96) ≥ ,d, (cid:96) = 0 . (1)The probability generating function (pgf) of X , denoted by p ∗ X ( z ) , is then of the form p ∗ X ( z ) = ∞ (cid:88) (cid:96) =0 p X ( (cid:96) ) z (cid:96) = c ( z − I − A ) − b + d. (2)The factorial moments of MG distributions can be found by differentiating (2) successively with respect to z andsubstituting z = 1 . Consequently, the i th factorial moment of an MG-distributed random variable X can be foundthrough the following expression (see [27]): E [ X ( X − · · · ( X − i + 1)] = i ! c ( I − A ) − i − A i − b . (3)An infinite QBD-type DTMC X k = ( L k , P k ) ∼ QBD ( B , B , A , A , A ) , k ≥ , (4)3 PREPRINT - J
ULY
24, 2020is a two-dimensional chain with ( L k , P k ) ∈ { ( i, j ) : 0 ≤ i < ∞ , ≤ j ≤ m } , where L k represents the levelsequence of the QBD, P k stands for the phase sequence, and X k has an irreducible probability transition matrix P ofthe following canonical block tridiagonal form: P = B A B A A A A . . . A . . .. . . , (5)with B , B , A , A , A , being m × m non-negative matrices. Finite QBD chains where the level of the chain stayswithin a bound is outside the scope of this paper. The stationary probability vector π = [ π , π , . . . ] where π k , ofsize × m , is the solution vector for level k , is the unique solution to the following equations π = πP , ∞ (cid:88) k =0 π k = 1 . (6)The stationary solution (when it exists) has a matrix-geometric form [28], i.e., π k = π R k u k , (7)where the matrix-geometric rate matrix R (all its eigenvalues being inside the unit circle) is the unique minimalnon-negative solution of the following quadratic matrix equation: R = A + RA + R A . (8)Once R is computed, the boundary vector π in (7) can be found from the following linear matrix equation [28]: π = π ( B + RB ) , π ( I − R ) − = 1 . (9)Several algorithms with computational complexity O ( m ) are known in the literature for obtaining the rate matrix R including the logarithmic reduction procedure [29], or the invariant subspace algorithm using the ordered Schurdecomposition [30], both algorithms being computationally efficient and stable. In the current paper’s numericalexamples, we use the latter algorithm.Let L be the steady-state random variable associated with the level sequence L k of the QBD with the followingexpression for its pmf p L ( (cid:96) ) : p L ( (cid:96) ) = lim k →∞ Pr { L k = (cid:96) } = π R (cid:96) u (cid:96) . (10)Therefore, L ∼ M G ( π R , R , , π ) . Another relevant random variable is the steady-state level sequence butrestricted to a particular subset of the entire set of phases. For this purpose, we define L S to be the steady-state randomvariable associated with the level sequence L k of the QBD restricted to the particular subset S ⊂ { , , . . . , m } . Inthis case, p L S ( (cid:96) ) = lim k →∞ Pr { L k = (cid:96) | P k ∈ S} . (11)It is not difficult to show that L S ∼ M G ( α π R , R , h , α π h ) , (12)where h i = 1 if i ∈ S and is zero otherwise, and α = (cid:0) π ( I − R ) − h (cid:1) − . Therefore, factorial moments (hence also ordinary moments) of L S can easily be computed from the expression (3)for any subset S . We first describe the AoI and PAoI processes (sequences) for the three queueing disciplines NPB, PB, and NPSBR,accompanied by an illustrative example. For this purpose, a successful information packet is first defined as onewhich is received successfully by the monitor; others those that could not start service or that were preempted whilein service (this latter situation being specific to PB) are called unsuccessful packets. Let t ( n ) j denote the arrival instantof the j th , j ≥ successful source- n information packet arriving at the server and let δ ( n ) j , j ≥ denote the receptioninstant at the monitor of the j th successful source- n information packet. We denote by ∆ ( n ) k , k = 0 , , . . . , the discrete-time discrete-valued random sequence representing the AoI for source- n at discrete-time instant k with a given initialcondition ∆ ( n )0 . 4 PREPRINT - J
ULY
24, 2020• When k ≥ , ∆ ( n ) k is incremented by one at each time instant until the first source- n successful packetreception occurring at instant k = δ ( n )1 at which it is again first incremented to yield the PAoI value Φ ( n )1 where Φ ( n ) j denotes the PAoI process for source- n which is a discrete-time discrete-valued random sequenceassociated with the AoI just before the reception of the j th successful source- n information packet.• Moreover, at instant k = δ ( n )1 , just after the packet’s reception, ∆ ( n ) k is set to D ( n )1 where D ( n ) j is the timespent in service by the j th successful source- n information packet i.e., D ( n ) j = δ ( n ) j − t ( n ) j , j ≥ .• Following this packet reception, ∆ ( n ) k is again incremented by one at each time instant until the receptionof the second source- n successful packet and the AoI random sequence is obtained by repeating this patternforever.Discrete-time Markov modeling is quite different than that of continuous-time since in the latter, only one event canhappen at a given time. However, in discrete-time, multiple events can happen at the same time instant. Therefore,the sequence of events happening at each time instant is crucial for modeling. Depending on the particular applicationor implementation, different event sequences might arise. Although the methodology can be easily be extended todifferent event sequences, we focus on the particular sequence of events at time instant k, k = 0 , , . . . , which isdescribed in Table 1. As stated above, in our model, we assume that only one of the newcoming arrivals can be pickedTable 1: The assumed sequence of events at time instant k Event Description k , the AoI values ∆ ( n ) k , ≤ n ≤ N are incremented.2 The service time of the packet, say source- n inservice, being over or not is checked, if over thenthe server is placed into the idle state and the AoIvalue ∆ ( n ) k is updated as D ( n ) j if this packet turnsout to be the j th successful source n packet.3 • In NPB, if the server is idle, then one of the new-coming packet arrivals is randomly chosen forstarting service. Otherwise, all packet arrivalsare discarded.• The situation is the same in PB when the serveris idle. Otherwise, one of the packet arrivals israndomly chosen for preempting the packet inservice.• In NPSBR, one of the packet arrivals is alwaysfirst written into the waiting room (possibly re-placing the one in the waiting room). Subse-quently, the packet in the waiting room starts toreceive service if the server is idle. Otherwise,it will be held at the waiting room until the nexttime instant.and processed and it is not possible to place one of the newcoming arrivals in service and another one in the waitingroom in the case of NPSBR at the same time instant. We leave such extensions for future research.Table 2 provides the sample values of the random sequences ∆ ( n ) k , k ≥ , for n = 1 , for a two-source examplewith zero initial conditions for AoI values for the three queueing disciplines NPB, PB, and NPSBR. For this particularexample, we assume that source- packets arrive periodically at instants k = 1 , , , . . . , with the first five packetsindexed having service times 5, 2, 7, 3, and 1, respectively, and source- packets also arrive periodically atinstants k = 1 , , , . . . , with the first four packets having service times 4, 4, 2, and 5, respectively. We assumethat the packet generated by source- (source- ) is to be chosen when the two sources generate packets simultaneouslyat k = 1 ( k = 13 ). The following explanations are given for each of the three disciplines to follow the sampleevolution of the AoI and PAOI sequences. 5 PREPRINT - J
ULY
24, 2020• For NBP, at k = 1 , packet is placed in service and ∆ (1) k is initially incremented by one at each instant untilthe reception of at k = 6 leading to Φ (1)1 = 6 . In the meantime, packet is discarded at k = 5 since theserver is busy opon its arrival. Subsequently, ∆ (1) k is again incremented by one but rising from the value 5(service time of ) until the reception of at k = 18 . The packets and are discarded since they foundthe server busy and packet is not picked against packet . Similarly, packet is not picked against and packet is placed in service at k = 4 and therefore ∆ (2) k is initially incremented by one at each instantuntil the reception of at k = 11 giving rise to Φ (2)1 = 11 . Subsequently, ∆ (2) k is again incremented by onebut rising from the value 4 (service time of ) until the reception of at k = 15 .• For BP, at k = 1 , packet is placed in service but is preempted by which is received at k = 7 . Thus, ∆ (1) k is incremented by one until k = 7 leading to Φ (1)1 = 7 . Packet joins service at k = 7 which is preemptedby which is also eventually preempted by the successful packet received at k = 15 . Therefore, ∆ (2) k is incremented by one at each instant until k = 15 giving rise to Φ (2)1 = 15 . Following this, packet joinsservice at k = 17 and is received at k = 18 . Thus Subsequently, ∆ (1) k is again incremented by one but risingfrom the value 5 (service time of ) until the reception of at k = 18 . Thus, ∆ (1) k is incremented by oneuntil k = 18 leading to Φ (1)2 = 13 while rising from the value of 2 at k = 7 .• For NPSBR, at k = 1 , packet is placed in service and is received at k = 6 whereas joins the waitingroom at k = 5 and starts receiving service at k = 6 which gets to complete at k = 8 . Packet joins thewaiting room at k = 7 and starts to receive service at k = 8 . Thus, the first successful source- receptionoccurs at k = 12 at which packet starts receiving service which gets to complete at k = 19 . Packet joinsthe waiting room at k = 13 but is replaced with packet which joins service at k = 17 and subsequentlycompletes at k = 18 .Table 2: Sample evolution of the AoI and PAoI sequences for ≤ k ≤ for an illustrative example for each of thethree queueing disciplines NPB, PB, and NPSBR. NPB PB NPSBR
Instant k ∆ (1) k ∆ (2) k ∆ (1) k ∆ (2) k ∆ (1) k ∆ (2) k Φ (1)1 ← ) 6 6 6 5 ( Φ (1)1 ←
67 6 7 2 ( Φ (1)1 ← Φ (1)2 ←
89 8 9 4 9 3 910 9 10 5 10 4 1011 10 4 (Φ (2)1 ← Φ (2)1 ←
13 12 6 8 13 7 514 13 7 9 14 8 615 14 2 (Φ (2)2 ←
10 2 ( Φ (2)1 ← Φ (1)2 ← Φ (1)2 ← Φ (1)3 ← ) 1120 3 7 3 7 11 12... ... ... ... ... ... ...We use the notation ∆ ( n ) , Φ ( n ) , and D ( n ) , to denote the steady-state random variables associated with the processes ∆ ( n ) k , Φ ( n ) j , and D ( n ) j , respectively. For ease of notation, we tag a single source out of all sources, say source- , anddrop the superscript while writing the steady-state pmf for the random variables ∆ = ∆ (1) , Φ = Φ (1) , and D = D (1) ,6 PREPRINT - J
ULY
24, 2020Table 3: Description of the five phases used for NPB.
Phase Description packet is in service2 The service of first source- packet is over and theserver is idle3 The service of first source- packet is over and thesecond source- packet is in service4 The service of first source- packet is over and asource- n packet is in service where n (cid:54) = 1 packet is overand we prepare for the next cyclerespectively: p ∆ ( (cid:96) ) = lim k →∞ Pr { ∆ (1) k = (cid:96) } , (cid:96) ≥ , (13) p Φ ( (cid:96) ) = lim j →∞ Pr { Φ (1) j = (cid:96) } , (cid:96) ≥ , (14) p D ( (cid:96) ) = lim j →∞ Pr { D (1) j = (cid:96) } , (cid:96) ≥ . (15)We also denote by F ∆ ( n ) ( (cid:96) ) , F Φ ( n ) ( (cid:96) ) , and F D ( n ) ( (cid:96) ) , the corresponding cumulative distribution functions (cdf) of therandom variables ∆ ( n ) , Φ ( n ) , and D ( n ) , respectively. The goal of this paper is to devise a numerical algorithm towrite the first two pmfs given in (13) and (14) or their corresponding cdfs. The pmf given in (15) is auxiliary and isto be needed for the former two pmfs for the NPSBR scenario. If the interest is on the two pmfs related to anotherinformation source- n where n (cid:54) = 1 , the same procedure can be repeated by renumbering the sources.Our proposed methodology relies on constructing an infinite Markov chain of QBD type that produces cycles repeatingforever in such a way that one cycle begins with the arrival of a successful source- packet and evolves until thereception of the next successful class- packet. The exception to this is the PB system where a cycle is to begin withthe arrival of a source- packet which is not necessarily successful. The level is always incremented until the secondsuccessful source- packet reception occurs at which we transition to an auxiliary state where the level is alwaysdecremented until the level zero is hit in order to prepare for starting the next cycle. We will show that the steady-state distribution of this properly constructed QBD enables one to find the two pmfs given in (13) and (14) for eachqueueing discipline of interest. Recall that source- n information packet generation is governed by a Bernoulli processwith parameter p n with ¯ p n = 1 − p n . We also let p = (cid:80) Nn =1 p n . The service times of all users are geometricallydistributed with the same parameter q and ¯ q = 1 − q . The load ρ is defined as ρ = p/q . For NPB, we propose an infinite Markov chain X k = ( L k , P k ) , k = 0 , , . . . of QBD type characterized as in (4)with 5 phases. Table 3 describes each of the five phases used for the NPB system. We first define the pgf (probabilitygenerating function) of the number of information packet arrivals from all sources other than source 1: τ ( z ) = N − (cid:88) j =0 τ j z j = N (cid:89) n =2 (¯ p n + p n z ) . (16)Next, we define γ = (cid:81) Nn =1 ¯ p n to be the probability of no packet arrivals at a time instant, γ = N − (cid:88) j =0 p τ j j + 1 , to be the probability of a source- packet to be chosen (uniformly at random) among all packet arrivals, γ = 1 − γ − γ , to be the probability of a source- n packet ( n (cid:54) = 1 ) to be chosen among all packet arrivals. The proposed QBDevolves in the form of repetitive cycles each of which begins with the arrival of a source- packet into an empty systemand continues until the reception of the next successful source- packet. At the beginning of a cycle, the level is zeroand we are at phase 1. With probability ¯ q , we stay in phase 1 and we transition to phase i +2 with probability qγ i whileincrementing the level. While in phase 2, we stay at phase 2 until an information packet arrival occurs. Therefore, inphase 2, we transition to phase 3 with probability γ , or to phase 4 with probability γ . The level is still incremented7 PREPRINT - J
ULY
24, 2020in all these transitions at phase 2. At phase 3, we transition to phase 5 with probability q when the service time of thenext successful source- packet is over and we stay at phase 3, otherwise. The level is still incremented in phase 3.While in phase 4, we either stay at phase 4, or to phase 3 when the service completes and a source- packet arrival ischosen at the same instant, or to phase 2 when the service completes but there are no new packet arrivals. Again, thelevel is incremented in phase 4. When in phase 5, we stay at phase 5 while decrementing the level until the level zerois hit. When the level is zero, we make a transition to phase 1 with probability one while staying at level zero. Thisepoch is where a new cycle gets to begin. With this description, the characterizing matrices of the QBD are as follows: A = ¯ q qγ qγ qγ γ γ γ
00 0 ¯ q q qγ qγ ¯ q + qγ
00 0 0 0 0 , (17) A is a matrix of zeros, A and B are matrices of zeros except for their (5 , th entry which is one, and B is a matrixof zeros except for its (5 , th entry which is one. From the evolution of this QBD, we observe that the values thatthe level sequence L k of the proposed QBD takes in phases 2, 3, and 4, in one QBD-cycle, coincide with the samplevalues of the AoI process in its own cycles, as given in Table 2. Recall that an AoI cycle starts with the age taking itsinitial value which is the service time of a successful source- packet which is then subsequently incremented until thereception of the next successful source- packet. This observation leads us to the following main result for the NPBqueueing system. Theorem 1.
Let X k ∼ QBD ( B , B , A , A , A ) having the characterizing matrices as in (17) with its stationaryvector for level k of size × denoted by π k being in matrix geometric form π k = π R k u k . Then, the steady-statepmf of the AoI sequence for the NPB system for source- is the same as that of L S A which is the steady-state levelof the QBD restricted to the subset S A = { , , } . Moreover, the steady-state pmf of the PAoI sequence for the NPBsystem for source- is the same as that of L S P where L S P is the steady-state level of the QBD restricted to phase3, i.e., S P = { } . Proof.
The proof is based on sample path arguments. From the evolution of this QBD, we observe that the valuesthat the level sequence L k of the proposed QBD takes in phases 2, 3, and 4, in one QBD-cycle, coincide with thesample values of the AoI process in its own cycles, as given in Table 2. Similarly, one added to the values that thelevel sequence L k of the proposed QBD takes in phase 3, in one QBD-cycle, coincide with the sample values of thePAoI sequence in one AoI cycle, as given in Table 2. The pmfs of the AoI and PAoI sequences can then explicitly bewritten in matrix geometric form as in (12) and their factorial moments can explicitly be written as in (3). For PB, we propose an infinite Markov chain X k = ( L k , P k ) , k = 0 , , . . . of QBD type characterized as in (4) with5 phases, the description of the first four phases being the same as that of NPB. On the other hand, for PB, in phase 5,either the service of the second source- packet is over and we prepare for the next cycle, or the first source- packetis unsuccessful, i.e., preempted during phase 1, and after transitioning to phase 5, we prepare for the arrival of thefirst source- packet. This QBD evolves in the form of repetitive cycles each of which begins with the beginning of asource- packet’s service and continues until the preemption of this packet or reception of the next successful source- packet. If the source- packet is unsuccessful during phase 1, i.e., preempted, then the cycle is called an unsuccessfulcycle. Otherwise, the cycle continues until the reception of the second source- information packet and is called asuccessful cycle. At the beginning of a cycle, the level is zero and we are at phase 1 and the service of the source- packet is about to start. To describe the operation of the QBD, we first need the following definition: γ = γ + γ , γ = γ + γ , and γ = γ + γ . The differences from the NPB system are now listed below in relation to thebehavior at phases 1, 3, and 4:• At phase 1, with probability ¯ qγ , the first source- packet gets to be preempted and we transition to phase5 whereas the behavior at phase 5 is identical to that of the same phase in NPB. In this case, the cycle is anunsuccessful cycle. The other transitions to phases 2, 3, and 4, are identical to that of NPB.• In phase 3, if the service of source- packet successfully completes, we transition to phase 5, which occurswith probability q . On the other hand, if this packet is preempted by a source- n , n (cid:54) = 1 packet, we transitionto phase 4, which occurs with probability ¯ qγ . Otherwise, we stay at phase 3.• While in phase 4, a source- n packet, n (cid:54) = 1 , is in service. We transition to phase 3 if a source- packet ischosen (with probability γ ) irrespective of whether the service is over or not. Additionaly, we transition tophase 2 if the service completes and there are no new arrivals, which occurs with probability qγ .8 PREPRINT - J
ULY
24, 2020The characterizing matrices of the proposed QBD, X k , are thus written as: A = ¯ qγ qγ qγ qγ ¯ qγ γ γ γ
00 0 ¯ qγ ¯ qγ q qγ γ ¯ qγ + qγ
00 0 0 0 0 , (18) A is matrix of zeros, A and B are matrices of zeros except for their (5 , th entry which is one, and B is a matrixof zeros except for its (5 , th entry which is one. Theorem 1 stated for NPB also applies to PB with the exception thatthe A matrix of the QBD for PB is given as in (18) as opposed to A written as in (18) for NPB. Before introducing the QBD for NPSBR as the other two queueing systems, we first need to obtain the pmf of thewaiting time p D ( (cid:96) ) of successful source- packets in NPSBR. Next, we present the corresponding main result in thefollowing theorem. Theorem 2.
The pmf of the queue waiting time for successful source- information packets in NPSBR is a mixture ofa probability mass at zero and a geometric distribution, i.e., there exist parameters a, b, ≤ a, b ≤ such that thefollowing hold: p D ( (cid:96) ) = aδ l + (1 − a ) b (1 − b ) l − u l − . (19) Proof.
Let the random sequence Y k ∈ { , , } , k ≥ represent the number of information packets in the NPSBRsystem (service and waiting room) at instant k . It is not difficult to show that the random sequence Y k is governed bya DTMC with probability transition matrix Q = (cid:32) γ − γ qγ qγ + ¯ qγ ¯ qγ q ¯ q (cid:33) . Let x = ( x x x ) be the stationary vector of this DTMC satisfying x = xQ , x = 1 . Also let γ be theprobability of a source- packet to be picked among all other arrivals given that the source- packet has arrived.Clearly, γ = γ /p . Let p s be the success probability of a source- packet. An arriving packet joins the server withprobability γ ( x + x q + x q ) and joins the waiting room with probability γ ( x ¯ q + x ¯ q ) . A packet that joins thewaiting room is successful iff there are no other arrivals in a geometrically distributed interval with parameter q , whichoccurs with probability r . Therefore, p s = γ ( x + x q + x q ) + γr ( x ¯ q + x ¯ q ) . Consequently, we have the following closed-form expression for r : r = ∞ (cid:88) k =1 γ k ¯ q k − q = γ q − γ + γ q . The conditional probability of zero queue wait, denoted by p D (0) , conditioned on a successful source- packet is γ ( x + x q + x q ) /p s . Similarly, the conditional probability of a queue wait of (cid:96) instants, denoted by p D ( (cid:96) ) , condi-tioned on a successful source- packet is γ ( x ¯ q + x ¯ q ) γ (cid:96) ¯ q (cid:96) − q/p s . Subsequently, the following choices of the twoparameters a and b a = γ ( x + x q + x q ) /p s , b = 1 − γ ¯ q, (20)give rise to the expression (19).For the NPSBR system, we propose an infinite Markov chain X k , k = 0 , , . . . of QBD type characterized as in (4)with state-space { ( i, j ) : 0 ≤ i < ∞ , ≤ j ≤ } with 10 phases. Table 4 describes each of the ten phases used theNPSBR system. The characterizing matrix A of the proposed QBD is given in Eqn. (21). Moreover, A is matrixof zeros, A and B are matrices of zeros except for their (10 , th entry which is one, and B is a matrix of zerosexcept for its (10 , th entry which is − a and (10 , th entry which is a .This QBD evolves in the form of repetitive cycles as in the previous two queueing sytems. A cycle begins withthe arrival of a successful source- packet which then first waits in the waiting room (phase 1) which is then served(phase 2). Following phase 2, the QBD evolves until the completion of the next successful source- packet’s servicecompletion (phases 3 to 9). Finally, in phase 10, we prepare for the next cycle. From the evolution of this QBD, weobserve that the values that the level process of the proposed QBD takes in phases 5 to 9 in one QBD cycle, coincidewith the sample values of the AoI process in its own cycles, as given in Table 2. We now present the following mainresult without proof for the NPSBR queueing system. 9 PREPRINT - J
ULY
24, 2020Table 4: Description of the ten phases used for NPSBR.
Phase Description packet is in thewaiting room2 The first successful source- packet is in ser-vice and the waiting room is empty3 The first successful source- packet is in ser-vice and there is a source- packet in the wait-ing room4 The first successful source- packet is in ser-vice and there is a source- n packet in the wait-ing room with n (cid:54) = 1 packet is over andwe wait for a packet arrival6 The second successful source- packet is inservice7 A source- n ( n (cid:54) = 1 ) packet is in service andthe waiting room is empty8 A source- n ( n (cid:54) = 1 ) packet is in service andand there is a source- packet in the waitingroom9 A source- n ( n (cid:54) = 1 ) packet is in service andand there is a source- n packet ( n (cid:54) = 1 ) is in thewaiting room10 The service of the second successful source- packet is over and we prepare for the nextcycle Theorem 3.
Let X k ∼ QBD ( B , B , A , A , A ) having the characterizing matrices as in (21) with its stationaryvector for level k of size × denoted by π k being in matrix geometric form π k = π R k u k . Then, the steady-statepmf of the AoI sequence for the NPSBR system for source- is the same as that of L S A which is the steady-state levelof the QBD restricted to the subset S A = { , , , , } . Moreover, the steady-state pmf of the PAoI sequence for theNPSBR system for source- is the same as that of L S P where L S P is the steady-state level of the QBD restrictedto phase 6, i.e., S P = { } . In the first numerical example, the analytical models we have proposed for the three queueing disciplines NPB, PB,and NPSBR, are validated by simulations. For this purpose, we fix N = 3 , q = 0 . , and for a given value of the loadparameter ρ , we set the Bernoulli arrival vector of probabilities ( p , p , p ) = ( p , p , p ) where p = qρ . The cdfs ofthe AoI and PAoI random sequences for the three different queueing schemes obtained with the proposed analyticalmodel and simulations are depicted in figures 2 and 3, respectively, for ρ = 0 . and ρ = 2 , respectively. In all thecases, we have observed a perfect match between the results obtained with the proposed analytical model and thesimulation results. A = ¯ b b qγ ¯ qγ ¯ qγ qγ qγ qγ qγ ¯ qγ qγ qγ qγ ¯ qγ qγ qγ γ γ γ q q qγ qγ ¯ qγ + qγ ¯ qγ ¯ qγ
00 0 0 0 0 qγ qγ ¯ qγ ¯ qγ
00 0 0 0 0 qγ qγ ¯ qγ ¯ qγ
00 0 0 0 0 0 0 0 0 0 , (21)10 PREPRINT - J
ULY
24, 2020
100 300 50000 .
51 a) PB F ∆(1)( ℓ ) F ∆(2)( ℓ ) F ∆(3)( ℓ )
100 300 50000 .
51 b) NPB F ∆(1)( ℓ ) F ∆(2)( ℓ ) F ∆(3)( ℓ )
100 300 50000 .
51 c) NPSBR F ∆(1)( ℓ ) F ∆(2)( ℓ ) F ∆(3)( ℓ )
100 300 50000 . ℓ F Φ(1)( ℓ ) F Φ(2)( ℓ ) F Φ(3)( ℓ )
100 300 50000 . ℓ F Φ(1)( ℓ ) F Φ(2)( ℓ ) F Φ(3)( ℓ )
100 300 50000 . ℓ F Φ(1)( ℓ ) F Φ(2)( ℓ ) F Φ(3)( ℓ ) Figure 2: The cdfs of AoI and PAoI processes for three different queueing models obtained by the proposed modelsand simulations results (denoted by markers) when q = 0 . , ρ = 0 . , p = ρq and ( p , p , p ) = ( p , p , p ) .In the second numerical example, we study the impact of the choice of the per-source packet generation probabilities p and p using the analytical model only on the system cost C ( α ) for a two-source system, which is given in thefollowing form: C ( α ) = E [∆ (1) ] + αE [∆ (2) ] , ≤ α ≤ , (22)which allows one to give more importance to source- over source- with a proper choice of the cost parameter α . When α = 1 , both sources are equally important whereas when α → , the mean age of the second sourcebecomes less relevant. For a given cost parameter α , we do exhaustive search to find the optimum packet generationprobabilities p ∗ and p ∗ that minimize the system cost C ( α ) where the minimum attainable cost is denoted by C ∗ ( α ) .Table 5 tabulates the optimum values p ∗ , p ∗ , C ∗ ( α ) for three different values of the service time parameter q and foreach of the three queueing disciplines of interest. As a second scenario, for power budgeting, we impose a bound onthe overall packet generation rate while doing exhaustive search, i.e., p = p + p ≤ β , where β is power constraintparameter since packet generation requires a certain energy consumption. In Table 6, we provide our results for thespecific case of β = 0 . . Studying tables 5 and 6, we have the following observations:• The optimum values of p and p turn out to be the same for all the three queueing disciplines but the optimumcost values are not necessarily the same for each discipline.• In the lack of a power constraint, the optimum packet generation rate for source- is always one and that ofsource- is one only when α = 1 and it decreases when α decreases. Since p ∗ = 1 in all these cases, thereis always a packet arrival at each time instant and the two queueing disciplines NPB and NPSBR behaveexactly the same under the same set of traffic parameters. In this case, the PB always outperforms the othertwo disciplines when the optimum packet generation rates are employed thanks to the memoryless propertyof the geometric service time distribution.• When there is a power constraint imposed on the packet generation rates, i.e., β = 0 . , the situation is verydifferent. In this case, NPSBR always outperforms NPB and there are situations when NPSBR outperformsPB which appear to occur for lower average service times.11 PREPRINT - J
ULY
24, 2020
100 200 30000 .
51 a) PB F ∆(1)( ℓ ) F ∆(2)( ℓ ) F ∆(3)( ℓ )
100 200 30000 .
51 b) NPB F ∆(1)( ℓ ) F ∆(2)( ℓ ) F ∆(3)( ℓ )
100 200 30000 .
51 c) NPSBR F ∆(1)( ℓ ) F ∆(2)( ℓ ) F ∆(3)( ℓ )
100 200 30000 . ℓ F Φ(1)( ℓ ) F Φ(2)( ℓ ) F Φ(3)( ℓ )
100 200 30000 . ℓ F Φ(1)( ℓ ) F Φ(2)( ℓ ) F Φ(3)( ℓ )
100 200 30000 . ℓ F Φ(1)( ℓ ) F Φ(2)( ℓ ) F Φ(3)( ℓ ) Figure 3: The cdfs of AoI and PAoI processes for three different queueing models obtained by the proposed modelsand simulations results(denoted by markers) when q = 0 . , ρ = 2 , p = ρq and ( p , p , p ) = ( p , p , p ) . In this paper, we propose a discrete-time queueing model to derive the exact distributions of the AoI and PAoI se-quences in a multi-source IoT-based status update system with Bernoulli packet generations and geometrically dis-tributed service times. Three queueing disciplines are considered, namely non-preemptive bufferless, preemptivebufferless, and non-preemptive single buffer with replacement systems. The proposed method gives rise to matrix ge-ometric expressions for the related distributions and moreover, the factorial moments of AoI and PAoI of any order areexplicitly given using matrix-vector operations. Numerical examples along with simulations are presented to validatethe proposed approach. Using the proposed analytical model, we also provide a numerical example on the optimumchoice of the Bernoulli parameters in a practical IoT system with two sources with diverse AoI requirements and thethree queueing disciplines are compared and contrasted in this setting. We have shown that the PB system yields thebest performance in majority of the cases whereas it is slightly outperformed by the NPSBR discipline in scenarioswith power constraints and relatively shorter average service times.
References [1] S. Kaul, M. Gruteser, V. Rai, and J. Kenney, “Minimizing age of information in vehicular networks,” in ,June 2011, pp. 350–358.[2] S. Kaul, R. Yates, and M. Gruteser, “Real-time status: How often should one update?” in , March 2012, pp. 2731–2735.[3] S. K. Kaul, R. D. Yates, and M. Gruteser, “Status updates through queues,” in , March 2012, pp. 1–6.[4] N. Pappas, J. Gunnarsson, L. Kratz, M. Kountouris, and V. Angelakis, “Age of information of multiple sourceswith queue management,” in , June 2015, pp.5935–5940.[5] A. Kosta, N. Pappas, A. Ephremides, and V. Angelakis, “Age and value of information: Non-linear age case,” in , June 2017, pp. 326–330.12
PREPRINT - J
ULY
24, 2020Table 5: The optimum values of p and p and the optimum value of C ( α ) for three different values of q ∈{ . , . , . } for various values of α for the three queueing disciplines in the absence of a power constraint. q = 0 . PB NPB NPSBR α p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α ) .1 .
48 34 . .
48 55 . .
48 55 . .2 .
62 41 . .
62 64 . .
62 64 . .3 .
71 47 . .
71 72 . .
71 72 . .4 .
77 53 . .
77 79 . .
77 79 . .5 .
83 58 . .
83 86 . .
83 86 . .6 .
87 63 . .
87 93 . .
87 93 . .7 .
91 67 . .
91 99 . .
91 99 . .8 .
94 71 . .
94 106 . .
94 106 . .9 .
97 75 . .
97 112 . .
97 112 . .
00 80 . .
00 118 . .
00 118 . q = 0 . PB NPB NPSBR α p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α ) .1 .
48 17 . .
48 27 . .
48 27 . .2 .
62 20 . .
62 31 . .
62 31 . .3 .
71 24 . .
71 35 . .
71 35 . .4 .
77 26 . .
77 39 . .
77 39 . .5 .
83 29 . .
83 42 . .
83 42 . .6 .
87 31 . .
87 45 . .
87 45 . .7 .
91 33 . .
91 49 . .
91 49 . .8 .
94 35 . .
94 52 . .
94 52 . .9 .
97 38 . .
97 55 . .
97 55 . .
00 40 . .
00 58 . .
00 58 . q = 0 . PB NPB NPSBR α p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α ) .1 .
48 6 . .
48 10 . .
48 10 . .2 .
62 8 . .
62 12 . .
62 12 . .3 .
71 9 . .
71 13 . .
71 13 . .4 .
77 10 . .
77 14 . .
77 14 . .5 .
83 11 . .
83 16 . .
83 16 . .6 .
87 12 . .
87 17 . .
87 17 . .7 .
91 13 . .
91 18 . .
91 18 . .8 .
94 14 . .
94 19 . .
94 19 . .9 .
97 15 . .
97 20 . .
97 20 . .
00 16 . .
00 22 . .
00 22 . PREPRINT - J
ULY
24, 2020Table 6: The optimum values of p and p and the optimum value of C ( α ) for three different values of q ∈{ . , . , . } for various values of α for the three queueing disciplines. The power constraint is taken as p + p ≤ β = 0 . . q = 0 . PB NPB NPSBR α p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α )( × − ) ( × − ) ( × − ) .1 . . . . . . . . . .2 . . . . . . . . . .3 . . . . . . . . . .4 . . . . . . . . . .5 . . . . . . . . . .6 . . . . . . . . . .7 . . . . . . . . . .8 . . . . . . . . . .9 . . . . . . . . . . . . . . . . . . q = 0 . PB NPB NPSBR α p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α )( × − ) ( × − ) ( × − ) .1 . . . . . . . . . .2 . . . . . . . . . .3 . . . . . . . . . .4 . . . . . . . . . .5 . . . . . . . . . .6 . . . . . . . . . .7 . . . . . . . . . .8 . . . . . . . . . .9 . . . . . . . . . . . . . . . . . . q = 0 . PB NPB NPSBR α p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α ) p ∗ p ∗ C ∗ ( α )( × − ) ( × − ) ( × − ) .1 . . . . . . . . . .2 . . . . . . . . . .3 . . . . . . . . . .4 . . . . . . . . . .5 . . . . . . . . . .6 . . . . . . . . . .7 . . . . . . . . . .8 . . . . . . . . . .9 . . . . . . . . . . . . . . . . . . PREPRINT - J
ULY
24, 2020[6] M. Costa, M. Codreanu, and A. Ephremides, “On the age of information in status update systems with packetmanagement,”
IEEE Transactions on Information Theory , vol. 62, no. 4, pp. 1897–1910, April 2016.[7] K. Chen and L. Huang, “Age-of-information in the presence of error,” in , July 2016, pp. 2579–2583.[8] R. D. Yates and S. K. Kaul, “The age of information: Real-time status updating by multiple sources,”
IEEETransactions on Information Theory , vol. 65, no. 3, pp. 1807–1827, 2019.[9] Y. Inoue, H. Masuyama, T. Takine, and T. Tanaka, “A general formula for the stationary distribution of the ageof information and its application to single-server queues,”
IEEE Transactions on Information Theory , vol. 65,no. 12, pp. 8305–8324, 2019.[10] Y. Sun, E. Uysal-Biyikoglu, R. D. Yates, C. E. Koksal, and N. B. Shroff, “Update or wait: How to keep your datafresh,”
IEEE Transactions on Information Theory , vol. 63, no. 11, pp. 7492–7508, Nov 2017.[11] L. Huang and E. Modiano, “Optimizing age-of-information in a multi-class queueing system,” in , June 2015, pp. 1681–1685.[12] A. Arafa and S. Ulukus, “Age minimization in energy harvesting communications: Energy-controlled delays,” in , Oct 2017, pp. 1801–1805.[13] Y. Hsu, E. Modiano, and L. Duan, “Age of information: Design and analysis of optimal scheduling algorithms,”in , June 2017, pp. 561–565.[14] Q. He, D. Yuan, and A. Ephremides, “Optimal link scheduling for age minimization in wireless systems,”
IEEETransactions on Information Theory , vol. 64, no. 7, pp. 5381–5394, July 2018.[15] M. Costa, M. Codreanu, and A. Ephremides, “Age of information with packet management,” in , 2014, pp. 1583–1587.[16] R. D. Yates and S. Kaul, “Real-time status updating: Multiple sources,” in , July 2012, pp. 2666–2670.[17] A. Kosta, N. Pappas, and V. Angelakis, “Age of information: A new concept, metric, and tool,”
Foundations andTrends in Networking , vol. 12, no. 3, pp. 162–259, 2017.[18] N. Akar, O. Dogan, and E. U. Atay, “Finding the exact distribution of (peak) age of information for queues ofPH/PH/1/1 and M/PH/1/2 type,” 2020, to appear at IEEE Trans. Commun.[19] E. Najm and E. Telatar, “Status updates in a multi-stream M/G/1/1 preemptive queue,” in
IEEE INFOCOM 2018- IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS) , 2018, pp. 124–129.[20] S. Farazi, A. G. Klein, and D. Richard Brown, “Average age of information in multi-source self-preemptivestatus update systems with packet delivery errors,” in , 2019, pp. 396–400.[21] M. Moltafet, M. Leinonen, and M. Codreanu, “Average age of information for a multi-source M/M/1 queueingmodel with packet management,” 2020, arXiv:2001.03959.[22] S. K. Kaul and R. D. Yates, “Timely updates by multiple sources: The M/M/1 queue revisited,” in , 2020, pp. 1–6.[23] R. D. Yates, J. Zhong, and W. Zhang, “Updates with multiple service classes,” in , 2019, pp. 1017–1021.[24] A. Kosta, N. Pappas, A. Ephremides, and V. Angelakis, “Queue management for age sensitive status updates,” in , 2019, pp. 330–334.[25] V. Tripathi, R. Talak, and E. Modiano, “Age of information for discrete time queues,” 2019, arXiv:1901.10463.[26] A. Kosta, N. Pappas, A. Ephremides, and V. Angelakis, “Non-linear age of information in a discrete time queue:Stationary distribution and average performance analysis,” 2020, arXiv:2002.08798.[27] N. Akar, “Fitting matrix geometric distributions by model reduction,”
Stochastic Models , vol. 31, no. 2, pp.292–315, 2015.[28] M. F. Neuts,
Matrix-geometric Solutions in Stochastic Models . Baltimore, MD: Johns Hopkins University Press,1981.[29] G. Latouche and V. Ramaswami, “A logarithmic reduction algorithm for quasi-birth-death processes,”
J. Appl.Prob. , vol. 30, pp. 650–674, 1993.[30] N. Akar, N. C. Ouz, and K. Sohraby, “A novel computational method for solving finite QBD processes,”