The Multi-Source Preemptive M/PH/1/1 Queue with Packet Errors: Exact Distribution of the Age of Information and Its Peak
11 T HE M ULTI -S OURCE P REEMPTIVE
M/P H/ / Q UEUE WITH P ACKET E RRORS : E
XACT D ISTRIBUTION OF THE A GE OF I NFORMATION AND I TS P EAK
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Ozancan Doan
Electrical and Electronics Engineering Dept.Bilkent University, 06800Ankara, Turkey [email protected]
Nail Akar
Electrical and Electronics Engineering Dept.Bilkent University, 06800Ankara, Turkey [email protected]
July 24, 2020 A BSTRACT
Age of Information (AoI) and Peak AoI (PAoI) and their analytical models have recently drawnsubstantial amount of attention in information theory and wireless communications disciplines, inthe context of qualitative assessment of information freshness in status update systems. We take aqueueing-theoretic approach and study a probabilistically preemptive bufferless
M/P H/ / queue-ing system with arrivals stemming from N separate information sources, with the aim of modelinga generic status update system. In this model, a new information packet arrival from source m isallowed to preempt a packet from source n in service, with a probability depending on n and m . Tomake the model even more general than the existing ones, for each of the information sources, weassume a distinct PH-type service time distribution and a distinct packet error probability. Subse-quently, we obtain the exact distributions of the AoI and PAoI for each of the information sourcesusing matrix-analytical algorithms and in particular the theory of Markov fluid queues and samplepath arguments. This is in contrast with existing methods that rely on Stochastic Hybrid Systems(SHS) which obtain only the average values and in less general settings. Numerical examples areprovided to validate the proposed approach as well as to give engineering insight on the impact ofpreemption probabilities on certain AoI and PAoI performance figures. Timely status updates are key for stable operation in networked control and monitoring systems. Lately, there has beensubstantial amount of interest centered around Age of Information (AoI) and Peak AoI (PAoI) processes in the fieldsof information theory and wireless communications in the context of qualitative assessment of information freshnessin status update systems [1, 2, 3, 4, 5, 6, 7, 8]. The survey [6] provides a relatively recent overview of the AoI conceptand its applications. AoI performance-related studies include those that propose analytical models for AoI [9, 10, 11]or research focusing on optimization of AoI-related performance metrics [7, 12, 13, 14, 15]. Actually, the AoI metrickeeps track of the staleness of a remote monitor’s knowledge of a stochastic process randomly sampled and transmitted(in the form of information packets) by an information source where the monitor and the source reside at two separatepoints in a packet-based communications network. More formally, the AoI maintained at the monitor for a given sourceis defined as the time elapsed since the generation of the last successfully received update packet. Consequently, theAoI process turns out to be a cyclic process that increases in time with unit slope within a cycle with the exceptionthat the AoI process undergoes abrubt downward jumps at random status packet reception epochs. After such a jump, 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
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Source 1Source 2Source N (1 ) ( 2 ) ( ) N Bufferless Server Monitor
Figure 1: N information sources sending status update messages through a bufferless server to a remote monitor.a new cycle begins. Under continuous-time stationary scenarios, the AoI process is a stationary continuous-time,continuous-valued stochastic process for which our interest in this paper is in finding its exact marginal distributionin a specific scenario to be detailed. A related equally important process is the discrete-time, continuous-valued PAoIprocess that is obtained by taking the peak values during each cycle of the AoI process [16], the exact distribution ofwhich is also sought in this paper.As the specific scenario, we consider the information update system in Fig. 1 consisting of N information sources eachequipped with a sensor, a bufferless server local to the sources, and a remote monitor (or destination). The state of thesource- n , n = 1 , . . . , N changes in time which is detected by its sensor and source- n generates packets according to aPoisson process with intensity λ n that contain sensed data along with a time stamp, to be immediately forwarded to theserver. Packets are sent by the server 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. In thispaper, the server is assumed to be bufferless, i.e., no waiting room for information packets. A new arriving informationpacket immediately starts to receive service if the server is idle, but is either discarded or allowed to preempt the packetin service if the server is busy upon arrival. To make the model general, we assume the following:• While the source- n information packet is in service, a new information packet arrival from source m, m =1 , . . . , N is allowed to preempt the packet in service with probability depending on both n and m . The matrixcomposed of these preemption probabilities is called the preemption matrix of this system.• The service time of each source has a PH-type distribution [17] and the requirements of sources are hetero-geneous, i.e., each source has its own service time requirement.• At the end of the service time needed for the delivery of an information packet, a packet error is said to occurwith a probability depending on the source of this packet. This corrupt packet will be retransmitted witha per-source probability but is otherwise discarded. Therefore, when the server starts sending a packet, itstores a copy locally to potentially retransmit it. However, the system is still called bufferless since it doesnot provide a waiting room for new packets.The above system is called the multi-source (probabilistically) preemptive M/P H/ / queue with packet errors,which is the focus of this paper.The AoI and PAoI processes will now be described for the bufferless server of interest followed by an illustrativeexample. First, we define a successful information packet as one which receives service until the end of the servicetime without preemption and without a packet error whereas other information packets are deemed unsuccessful. Let t ( n ) j denote the arrival instant of the j th , j ≥ successful source- n information packet arriving at the server and let2 PREPRINT - J
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24, 2020 δ ( n ) j , j ≥ denote the reception time at the monitor of the j th successful packet belonging to source n . We denoteby ∆ ( n ) ( t ) , t ≥ , the continuous-time random process with left-continuous sample paths representing the AoI forsource- n at time t with a given initial condition ∆ ( n ) (0) . At t = 0 , ∆ ( n ) ( t ) starts to increase linearly in time witha unit slope until the first successful packet reception at t = δ ( n )1 . The right limit ∆ ( n ) ( δ +1 ) = lim t ↓ δ ( n )1 ∆ ( n ) ( t ) isset to D ( n )1 where D ( n ) j = δ ( n ) j − t ( n ) j is the time spent in service by the j th successful source- n information packet.Subsequently, the process ∆ ( n ) ( t ) increases with unit slope until the next successful class- n packet reception and thepattern repeats forever. Let Φ ( n ) j = ∆ ( n ) ( δ j ) , j ≥ , denote the PAoI process for source- n which is a discrete-timecontinuous-valued random process associated with the AoI just at the epoch of packet receptions. Fig. 2 illustrates thetwo sample paths of the random processes ∆ ( n ) ( t ) , n = 1 , in a two-source probabilistically preemptive bufferlessserver system with packet errors with the initial conditions ∆ (1) (0) = 0 and ∆ (2) (0) = 5 . The arrival epochs of theinformation packets are denoted by arrows at the bottom. The arriving packets are indexed as a, b, . . . with the notation( na ), θ indicating the first arrival from source- n with a service time requirement of θ . Let us now study Fig. 2. Thefirst packet (1 a ) from source- arrives at t = 2 with a service time of 4. During its service time, no other arrivals takeplace and at the end of the service time, the packet (1 a ) is received at t = 6 without transmission errors. Therefore,in the time interval from t = 0 to t = 6 , ∆ (1) ( t ) rises from the value 0 to ∆ (1) (6) = Φ (1)1 = 6 and ∆ (1) (6 + ) is setto 4 which is the age of this packet, i.e., current time minus the time stamp on this packet. At t = 8 , packet (2 a ) arrives and immediately starts to receive service with a service time requirement of 5, but the packet (1 b ) arriving at t = 12 preempts the service of (2 a ) with a service time of 3. At t = 15 , this packet is received with packet error andis retransmitted with a service time of 3. At t = 16 , the new arrival (2 b ) is discarded. At t = 18 , the packet (1 b ) issuccessfully received. In the time interval from t = 6 to t = 18 , the AoI process ∆ (1) ( t ) rises from the value 4 to Φ (1)2 = 16 after which ∆ (1) (16 + ) is set to 6 which is the sum of two service times 3 and 3 required for transmissionof the successfully received packet (2 b ) . For source- , the first successfully received packet is (2 c ) which arrives at t = 19 with a service time requirement of 4 and is not preempted by the packet (1 c ) and is successfully received at t = 23 without error. Therefore, ∆ (2) ( t ) rises from its initial value 5 to Φ (2)1 = 28 after which ∆ (2) (23 + ) is set to theage of this particular packet at t = 23 . These patterns repeat for both these processes.We use the notation ∆ ( n ) , Φ ( n ) , and D ( n ) to denote the steady-state random variables associated with the processes ∆ ( n ) ( t ) , Φ ( n ) j , and D ( n ) j , respectively. For the general N -source probabilistically preemptive M/P H/ / systemwith packet errors (an illustration of which is given in Fig. 2 for N = 2 ), we are interested in finding the followingsteady-state cdfs (cumulative distribution function) for the random variables ∆ ( n ) and Φ ( n ) , respectively: F ∆ ( n ) ( x ) = lim t →∞ Pr { ∆ ( n ) ( t ) ≤ x } , x ≥ , (1) F Φ ( n ) ( x ) = lim j →∞ Pr { Φ ( n ) j ≤ x } , x ≥ . (2)Note that F ∆ ( n ) (0) and F Φ ( n ) (0) must be zero since there can not be a probability mass at the origin for these two pro-cesses. Also, let f ∆ ( n ) ( x ) and f Φ ( n ) ( x ) for x ≥ denote the corresponding steady-state pdfs (probability distributionfunction) with the corresponding non-central moments: E (cid:104) (∆ ( n ) ) i (cid:105) = (cid:90) ∞ x i f ∆ ( n ) ( x ) dx, (3) E (cid:104) (Φ ( n ) ) i (cid:105) = (cid:90) ∞ x i f Φ ( n ) ( x ) dx. (4)We observe from Fig. 2 that each cycle of the AoI process ∆ ( n ) ( t ) consists of a linear curve with unit slope that startsat value D ( n ) j for some successful packet index j for source- n and terminates at the peak value Φ ( n ) j +1 . Subsequently,the sample path of the AoI process ∆ ( n ) ( t ) consists of an ordered concatenation of infinitely many cycles each ofwhich behaves as described above. For obtaining the distribution of the AoI and PAoI processes, we propose to usethe theory of Markov Fluid Queues (MFQ) [18, 19, 20]. Existing steady-state MFQ solvers that we propose to useare matrix analytical and they rely on numerically stable and efficient vector-matrix operations. The main idea isthat we construct MFQs that produce sample paths whose certain parts coincide with the sample cycles of the AoIprocess. Additionally, these MFQ-produced cycles contain sample values that coincide with the sample values of thePAoI process. Hence, the exact distributions of the AoI and PAoI processes given in (2) can be obtained out of thesteady-state solution of certain MFQs, the construction of which is the main focus of this paper. The main contributionof this paper is to obtain the exact distribution of AoI and PAoI processes in a very general M/P H/ / frameworkrepresentative of bufferless servers arising in information status update systems using the theory of MFQs.3 PREPRINT - J
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24, 2020 (1a),4 (1) ( ) t t (2a),5 (1b),3 (2b),5 (2c),4 (1c),6 (1d),4 (2) ( ) t (2d),2 (1e),5 (1f),6 (1a) successful (1b) preempts (2a)(1b) errored and retransmitted with service time of 3 (1b) successful(1c) discarded (1d) starts service (1d) successful (2e),1 (1e) discarded (1f) starts service(1f) successful(2c) successful(2c) starts service (2d) starts service(2d) errored and retransmitted with service time of 5(2d) successful (2e) starts service (2e) successful (1)1 (1)2 (1)3 (1)4 (2)1 (2)2 (2)3 (2b) discarded Figure 2: A sample path for the two AoI processes ∆ (1) ( t ) and ∆ (2) ( t ) in a 2-user bufferless system.The main contributions of this paper are the following:• We present an analytical model for bufferless servers arising in status update systems that can be globally pre-emptive, self-preemptive, non-preemptive, etc. with a unifying probabilistic preemption framework. Besidesunification, probabilistic preemption can be optimum depending on how rewards or costs are defined.• Packet errors and heterogenous service time requirements across sources makes the model even more generalthan the existing ones.• Most existing results use SHS and cope with less general settings while providing means to obtain the aver-age AoI and PAoI values while falling short in most of the cases in obtaining their exact distributions (seesubsection 2.2). In this paper, we obtain the exact distributions of the AoI and PAoI processes numericallywhich can be crucial for system design.The organization of the paper is as follows. In Section 2, related work is presented. Section 3 presents the notationthroughout the paper as well as preliminaries on PH-type distributions and MFQs. Section 4 presents the analyticalmodel. In Section 5, we provide numerical examples to validate the proposed approach as well as examples to study theimpact of preemption probabilities on system performance with respect to certain AoI- and PAoI-related performancemetrics. Finally, we conclude in Section 6. The AoI concept was first introduced in [2] in the context of a single-source, single-server
M/M/ queueing model.This model is then extended to multiple sources in [21] and since then the single-server model in Fig. 1 for AoI hasextensively been used for status update systems in the literature [2, 6]. There are many variations of this single-serverqueueing system studied in the recent literature depending on• Whether there is a single-source or multiple sources, feeding the queue with information packets,• Whether there is a transmission error or not,• Whether the interest in on the mean AoI and PAoI values, or their exact distributions are sought,4 PREPRINT - J
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24, 2020• Generality of the distributions assumed for the interarrival times and service times,• Queue capacity which represents the maximum number of packets that are allowed to be in the system,including those in the waiting room and the one in service,• Scheduling discipline to be used whether it be First Come First Serve (FCFS), Preemptive Last Come FirstServe (P-LCFS), Non-preemptive LCFS (NP-LCFS), etc.• Use of buffer management schemes in charge of packet dropping at the server.The related work on AoI analytical models is summarized below first in the single-source setting. Subsequently,related work will be presented regarding the existing multi-source queueing models.
In [2], the mean AoI is obtained for the single-source
M/M/ , M/D/ , and D/M/ queues with infinite buffercapacity and FCFS scheduling. In [22], expressions are derived for the LST transform of the stationary distributionsof the AoI and the PAoI processes in M/GI/ and GI/M/ queues. Despite the fact that relatively large buffers andFCFS scheduling are the de-facto choices in operational packet-switched communication networks, such choices havebeen shown to give rise to poor AoI performance in moderate to high load regimes. The reference [10] studies theAoI and PAoI distributions for small buffer systems, including the conventional M/M/ / and M/M/ / queues, aswell as the so-called M/M/ / ∗ queue, for which the packet waiting in the queue is to be replaced by a newer packetarrival, which actually is a non-preemptive LCFS system. The mean AoI and PAoI figures in the pre-emptive LCFS M/G/ / queueing system is studied in [23] where a new arrival preempts the packet in service and the servicetime distribution is assumed to follow a more general gamma distribution. Exact PAoI expressions are derived inan M/M/ queueing system with packet delivery errors using different scheduling policies such as FCFS, P-LCFS,and NP-LCFS in [11]. Exact expressions for the stationary distributions of AoI and PAoI for a very wide class ofsingle-source information update systems are given in [9]. A recent work in [24] also obtains the exact distributionsof AoI and PAoI in bufferless systems with probabilistic preemption and PH-type distributions for both interarrivaland service times. A similar model is also proposed in [24] for a single-buffer queueing system with Poisson packetarrivals and PH-distributed service times allowing probabilistic replacement of the waiting packet by a newer packetarrival. A discrete-time queueing model with Bernoulli arrivals and geometric service times, using FCFS and non-preemptive LCFS scheduling is presented in [25] with expressions for the mean AoI and PAoI values. In additionto exact methods, a number of studies provide bounds for certain AoI-related metrics of interest. The reference [26]derives upper bounds for the mean AoI for the G/G/ / queue as well as its preemptive version while showing thatthe bounds are close to actual values. Similarly, the authors of [27] present a method for obtaining upper bounds forthe AoI violation probability for both GI/GI/ / and GI/GI/ / ∗ systems, in addition to some exact closed-formexpressions for some sub-cases. The reference [12] derives the mean PAoI expression for
M/G/ and M/G/ / systems with heterogeneous servicetime requirements which enables one to optimize system cost, as a function of mean PAoI, by choice of the updateinterval. The authors of [8] study the multi-source M/M/ model with FCFS as well as two variations (preemptiveand nonpreemptive with replacement) of LCFS using the theory of SHS and obtain exact expressions for the mean AoI.A preemptive M/G/ / queue is considered in [28] with a common service time for all sources in which expressionsfor the mean AoI and PAoI are derived. A similar preemptive M/G/ / system is studied in [29] allowing packetdelivery errors. The authors [29] allow preemption of a source in service by a newly-arriving packet from the samesource and derive the mean AoI expressions for each source using SHS technique. The reference [30] considers a two-source M/M/ / queueing system in which a packet waiting in the queue can be replaced only by a newly-arrivingpacket from the same source, again using SHS techniques. A non-preemptive M/M/ /m with common service timesacross sources is again studied by the SHS technique in [31] and mean AoI expressions are derived. A more generalhyperexponential ( H ) service time distribution for each class is considered in [32] for an M/H / / nonpreemptivebufferless queue to derive an expression for the mean AoI per class. Uppercase bold letters are used to denote real-valued matrices. Lowercase bold (plain) letters or symbols are used todenote real-valued vectors (scalars). The ( i, j ) th th entry of A is denoted by A i,j and the j th entry of a row or columnvector α is α j . The notations k × (cid:96) , I m , and n are used to denote the matrix of zeros of size k × l , identity matrix ofsize m , and a column matrix of ones of size n , respectively. When used without a subscript, it is left to the reader to5 PREPRINT - J
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24, 2020infer the size information from the context. Let A be an n × m matrix and B a p × q matrix. The Kronecker productof the matrices A and B is denoted by A ⊗ B which is of size np × mq . The notation diag { A , B } denotes the blockdiagonal concatenation of the matrices A and B and is diagonal if the individual matrices A and B are diagonal. Asquare matrix is said to be stable (anti-stable) if each of its eigenvalues has negative (non-negative) real parts. Thenotation [ α , β ] is used for the concatenation of the two row vectors α and β . The function u ( x ) refers to the Heavisidestep function, also known as the unit step function, whereas δ ( x ) stands for the Dirac delta function, also known asthe unit impulse function. In the context of queueing systems, Phase-type (PH-type) distributions are often used for modeling independent andidentically distributed (iid) non-exponential interarrival and/or service times [17]. Using PH-type distributions givesrise to algorithmically tractable methods for finding both the steady-state and transient solutions of such queueingsystems; see [33] and the references therein. For rigorous description of PH-type distributions, we first define aMarkov process on the state-space S = { , , . . . , m, m + 1 } with m transient states, one absorbing state m + 1 , initialprobability vector [ σ , σ ] , and an infinitesimal generator of the form (cid:20) S ν (cid:21) , where σ is a row vector of size m , σ = 1 − σ is a scalar, the sub-generator S is m × m , and ν is a column vectorof size m such that ν = − S . The time to absorption to the absorbing state m + 1 , say X , is said to be PH-typecharacterized with the pair ( σ , S ) , i.e., X ∼ P H ( σ , S ) . In most typical scenarios, σ is zero. The cdf and the pdf of X ∼ P H ( σ , S ) , denoted by F X ( x ) and f X ( x ) , respectively, are given as: F X ( x ) = (1 − σ e Sx ) u ( x ) , f X ( x ) = − σ e Sx S u ( x ) + σ δ ( x ) . (5)PH-type distributions are dense in the field of all positive-valued distributions and therefore they can principally beused to approximate any positive-valued distribution [34]. Given sample data or an arbitrary pdf, one can use one of theexisting algorithms, such as the Expectation Maximization (EM) algorithm of [33] for maximum likelihood estimation,or the moment-matching algorithm of [35], or the statistical inference-based algorithm of [36], to construct a PH-typedistribution that matches data or accurately approximates the given pdf. We describe an MFQ by a joint Markovian process X ( t ) = ( X f ( t ) , X m ( t )) , t ≥ , where ≤ X f ( t ) < ∞ is thecontinuous-valued fluid level in the buffer and X m ( t ) ∈ S = { , , . . . , n } is the modulating phase process X m ( t ) which behaves as a Continuous Time Markov Chain (CTMC) with state space S and generator Q ( ˜ Q ) when X f ( t ) > ( X f ( t ) = 0 ). The parameter n is the system size. The drift (rate of fluid change) of the MFQ equals r i when themodulating process X m ( t ) visits state i and R is defined as the diagonal matrix of drifts: R = diag { r , r , . . . , r n } .When X f ( t ) = 0 and X m ( t ) = i with r i < , X f ( t ) sticks to the boundary at zero. The process X ( t ) is said to becharacterized with the matrix triple ( Q , ˜ Q , R ) , i.e., X ( t ) ∼ M F Q ( Q , ˜ Q , R ) . In most existing studies, Q = ˜ Q , forwhich stationary solutions are derived by [20],[18] by using the eigendecomposition of a certain matrix and also by [37]using the matrix sign function avoiding the computation of the eigenvectors, a problem known to be ill-conditioned[38]. The more general Q (cid:54) = ˜ Q case turns out to be a sub-case of multi-regime MFQs whose steady-state solutions canbe obtained through the ordered Schur decomposition, again avoiding ill-conditioned eigendecompositions [39]. Forother eigendecomposition-free numerically efficient and stable algorithms for solving multi-regime MFQs, we referthe reader to the matrix-analytical approaches of [40] and [41].We assume r i (cid:54) = 0 , ≤ i ≤ n and r i > , i ≤ b and r i < , i > b , since otherwise states can always be reorderedfor this purpose. We are interested in finding the steady-state joint pdf vector f ( x ) = [ f ( x ) , f ( x ) , . . . , f n ( x )] , (6) f i ( x ) = lim t →∞ ddx Pr { X f ( t ) ≤ x, X m ( t ) = i } , x > , (7)and the steady-state probability mass accumulation (pma) vector at zero: c = [ c , c , . . . , c n ] , c i = lim t →∞ Pr { X f ( t ) = 0 , X m ( t ) = i } . (8)We now describe the method of [39] without proof, in three steps, adapted to MFQs described above, to find thequantities of interest in (7) and (8), when they exist. In Step 1, we find an orthogonal matrix U such that the following6 PREPRINT - J
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24, 2020holds: U T QR − U = (cid:20) Ψ a × a ∗ A b × b (cid:21) , U T = (cid:20) ∗ H b × n (cid:21) , (9)for an anti-stable matrix Ψ with an eigenvalue at the origin, stable matrix A , and ∗ denoting an arbitrary sub-matrix.The well-known ordered real Schur form (available in Lapack, Matlab, and Octave software packages) can be usedtowards obtaining the decomposition (10) [38]. In Step 2, we solve for the × b vector g and × a vector d from thefollowing linear matrix equation: [ g d ] (cid:20) HR − A − H n − ˜ Q ∗ a (cid:21) = [ × n , (10)with ˜ Q ∗ denoting the matrix composed of the last a rows of ˜ Q . Finally, in Step 3, we write f ( x ) = g e A x H u ( x ) + c δ ( x ) , f i ( x ) = g e A x h i u ( x ) + c i δ ( x ) , (11)where h i denotes the i th column of H . M/P H/ / Queue
We consider the status update system in Fig. 1 with N sources, a server, and a monitor. The source n, n = 1 , . . . , N generates packets that carry status update information, according to a Poisson process with intensity λ n . The trafficintensity vector is denoted by λ = ( λ , . . . , λ N ) . We define the total arrival rate λ = (cid:80) Nn =1 λ n . A source- n information packet immediately starts to receive service from the server when it finds the server idle upon arrivaland its service time Θ ( n ) ∼ P H ( σ ( n ) , S ( n ) ) with order (cid:96) n , ν ( n ) = − S ( n ) , and E [Θ ( n ) ] = µ n . Let the per-sourceload be defined as ρ n = λ n µ n and the total load ρ = (cid:80) Nn =1 ρ n . We define the total order (cid:96) T = (cid:80) Nn =1 (cid:96) n . We are given apreemption matrix P so that while the source- n information packet is in service, a new information packet arrival fromsource m, m = 1 , . . . , N is allowed to preempt the packet in service with probability P n,m . The following sub-casesof a general preemption matrix P have been studied in the literature:• P = refers to a non-preemptive system [6],• P = N TN case is referred to as global preemption in [8],• P = I case is referred to as self preemption in [29].For each source n , we define the total intensity of traffic that can preempt a source- n packet in service as ¯ λ n = (cid:80) Nm =1 λ m P n,m . In these systems, preemption may potentially be beneficial for two different purposes: (i) a newinformation packet always carries more timely information than the one in service, (ii) sources can be differentiatedfrom each other by proper choice of preemption probabilities. At the end of the service time of a source- n packet, atransmission error is detected at the monitor with probability e n . We also denote the successful transmission probabil-ity of a source- n packet by q n = 1 − e n . An errored class- n packet (irrespective of how many times it was transmitted)is retransmitted with probability r n whereas it will be discarded with probability d n = 1 − r n . Recall that an infor-mation packet which receives service until the end of the service time without preemption and without a packet erroris called a successful packet.We tag a specific source, say source 1, for which the exact distributions of the AoI and PAoI processes are to beobtained. If the interest is on another information source- n where n (cid:54) = 1 , the same procedure can be repeated byrenumbering the sources. For this purpose, we construct an MFQ process X ( t ) = ( X f ( t ) , X m ( t )) by which we havea single fluid level trajectory of infinitely many cycles where each cycle comprises four stages, namely stages 1-4,that are described as follows. Every cycle begins with stage 1 at which the service of a source- packet begins. Ifthis packet is preempted or errored, we go back to stage 1 through stage 4. When a source- packet is eventuallyreceived successfully, we transition from stage 1 to stage 2 while ensuring that the fluid level at this transition epochis distributed according to D (1) which actually is the system time of successful source- packets. In stage 2, we waitfor the next packet arrival from any one of the sources upon which we transition to stage 3. In stage 3, we are withinthe service time of a source- n packet for some source n . If a source- service is successfully over, we end the cycleby transitioning to stage 1 again through stage 4. If a source- n service for n (cid:54) = 1 is successfully over during stage3, we go back to stage 2 waiting for a new packet arrival. In the case of preemption or transmission error followedby retransmission during stage 3, we stay at stage 3. For the case of error for a source- n packet with discarding, atransition to stage 2 occurs. More formally, 7 PREPRINT - J
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24, 2020• During stage 1, the modulating process X m ( t ) visits states j , ≤ j ≤ (cid:96) that keep track of the phase of theservice time of the source- packet.• Stage 2 consists of one single state waiting for an information packet arrival from one of the informationsources.• During stage 3, X m ( t ) visits the states ( i, j ) , where ≤ i ≤ N keeps track of the source index of the packetin service and j, ≤ j ≤ (cid:96) i keeps track of the phase of the service time of the packet in service.• Stage 4 consists of one single final state − by which we prepare for starting the next cycle.Let us now describe the operation of the MFQ X ( t ) . A cycle of X ( t ) begins with a visit to a state in stage 1 when thefluid level is zero. During stage 1, the fluid level rises with a unit slope. Let us assume that we are at state j , ≤ j ≤ (cid:96) in stage 1 during which the arrival processes of all sources are turned on. There are five possible transitions:• with rate S (1) j,k , a transition to state k occurs,• with rate ¯ λ = (cid:80) Nn =1 P ,n λ n , the source- packet is preempted by a new arrival; and a transition to state − occurs,• with rate ν (1) j e d , the service time of the packet is over but the packet is errored and discarded; and atransition to state − occurs,• with rate ν (1) j e r (1) σ (1) k , the service time of the packet is over and it is errored and retransmitted; and atransition to state k occurs,• with rate ν (1) j q , the service time is over and the packet is successful and a transition to state occurs.With the transitions described above, we ensure that when we are at the beginning of state , the fluid level has risento a level distributed according to D (1) . During state , the fluid level rises with unit slope and with rate λ ( i ) σ ( i ) j , atransition to state ( i, j ) occurs in stage 3.Let us now assume that we are at state ( i, j ) in stage 3 during which the fluid level continues to rise again with unitslope. Therefore, a source- i packet is in service and we are in phase j of its service time. We have the followingtransition possibilities from state ( i, j ) :• with rate S ( i ) j,k , a transition to state ( i, k ) occurs,• with rate P i,n λ n , the source- i packet is preempted by a new information packet from source n and a transitionto state ( n, l ) occurs with rate P i,n λ n σ ( n ) l ,• with rate ν ( i ) j e i d i , the service time of the packet is over but the packet is errored and discarded; and a transitionto state occurs,• with rate ν ( i ) j e i r i σ ( i ) k , the service time of the packet is over and it is errored and retransmitted; and a transitionto state ( i, k ) occurs,• when i (cid:54) = 1 , with rate ν ( i ) j q i , the service time of the source- i packet is over and is successful; a transition tostate occurs,• when i = 1 , with rate ν (1) j q , the service time of the source- packet is over and is successful giving rise to atransition to state − .When at state − , the fluid level always drops with a rate of minus one without any state changes until the fluid levelzero is hit. The arrival process is turned off in this stage and the only way to escape from this particular state is througha transition to state i with transition rate σ (1) i . With the lexicographical ordering of the states from stages 1 to 4, X ( t ) ∼ M F Q ( Q , ˜ Q , R ) with system size (cid:96) T + (cid:96) + 2 where Q equals to the matrix given in Eqn. (12)8 PREPRINT - J
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24, 2020 Q = S (1) − ¯ λ I + e r V (1) q ν (1) · · · ¯ λ + e d ν (1) − λ λ σ (1) λ σ (2) · · · λ n σ ( N ) e d ν (1) S (1) − ¯ λ I + λ P , F (1) + e r V (1) λ P , F (2) · · · λ N P ,N F ( N ) q ν (1) q ν (2) + e d ν (2) λ P , F (1) S (2) − ¯ λ I + λ P , F (2) + e r V (2) · · · λ N P ,N F ( N ) ... ... ... ... . . . ... ... q N ν ( N ) + e N d N ν ( N ) λ P N, F (1) λ P N, F (2) · · · S ( N ) − ¯ λ N I + λ N P N,N F ( N ) + e N r N V ( N ) , (12)and V ( j ) = ν ( j ) ⊗ σ ( j ) , F ( j ) = ⊗ σ ( j ) , (13)the matrix ˜ Q is the same as Q except for the block entry in the south-west corner which is set to σ (1) , and for thescalar at the south-east corner which is set to − . Moreover, R = diag { I (cid:96) T + (cid:96) +1 , − } . (14)Fig. 3(a) illustrates one sample cycle of the AoI process ∆ (1) ( t ) which starts to rise from the value D (1) to the PAoIvalue Φ (1) . Fig. 3(b) illustrates one sample cycle of the fluid level process X f ( t ) which takes the value D (1) at theepoch of transition from stage 1 to stage 2. The fluid level rises from the value D (1) to Φ (1) during stages 2 and 3.Therefore, we observe that one sample cycle of the process ∆ (1) ( t ) coincides with part of the sample cycle of theprocess X f ( t ) associated with stages 2 and 3 only. Moreover, one sample value of the process Φ (1) coincides withone sample value of the process X f ( t ) taken at the epoch of transition from stage 3 to stage 4. On the basis of theseobservations, we are now ready to state the following theorem which provides an expression for the pdfs of the AoIand PAoI processes in terms of the steady-state joint pdf vector f ( x ) of the MFQ X ( t ) given in the form (11) withthe matrix A being of size b = (cid:96) T + (cid:96) + 1 which is the number of positive drift states. Theorem 1.
Consider the process X ( t ) ∼ M F Q ( Q , ˜ Q , R ) with order (cid:96) T + (cid:96) + 2 with the characterizing matricesas defined in (12) and (14) with its steady-state joint pdf vector f ( x ) given in the form (11) . Then, the pdf of the AoIprocess, f ∆ (1) ( x ) , and the pdf of the PAoI process, f Φ (1) ( x ) , are given by the following closed form expression: f ∆ (1) ( x ) = g A e A x h A u ( x ) , h A = H (cid:96) × (cid:96) T , g A = 1 − gA − h A , (15) f Φ (1) ( x ) = g P e A x h P u ( x ) , h P = H ( (cid:96) +1) × q ν (1) ( (cid:96) T − (cid:96) +1) × , (16) g P = 1 − gA − h P , Moreover, the associated non-central moments of the AoI and PAoI processes are given as follows: E (cid:104) (∆ (1) ) i (cid:105) = ( − i +1 i ! g A A − ( i +1) h A , i = 1 , , . . . , (17) E (cid:104) (Φ (1) ) i (cid:105) = ( − i +1 i ! g P A − ( i +1) h P , i = 1 , , . . . . (18)The expression (15) stems from sample path arguments and requires censoring out the states in stages 1 and 4. This isachieved by the choice of h A by which we sum up the joint pdfs in states belonging to stages 2 and 3 only. The choice9 PREPRINT - J
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24, 2020 (1) ( ) t ( ) f X t t t (1) D successful packetpreempted packet (1) one sample cycle of the fluid level process one sample cycle of the AoI process stage 1 stage 2 stage 3 stage 4 (a)(b) ( ) f X t preempted packet (1) ( ) t (1) D (1) Figure 3: A sample path of the AoI process ∆ (1) ( t ) with emphasis on one of its sample cycles in given in subfigure(a) whereas subfigure (b) presents a sample path of the fluid level process X f ( t ) with emphasis on one sample cycle.Observe that the sample cycle of the former coincides with part of that of the latter.of g A is for normalization. The expression (15) similarly follows sample path arguments and requires the pdf of thefluid level just at the epoch of a transition from a state in stage 3 to state -1 in stage 4. For this purpose, h P is chosenso as to sum up the joint pdfs belonging to states in stage 3 that have a transition to state -1, namely the states (1 , j ) instage 3, according to the transition rate vector q ν (1) . Again, the choice of g P is for normalization. Remark.
While obtaining the steady-state joint pdf vector f ( x ) of the MFQ X ( t ) through the algorithm defined inthree steps in (9) - (11) , we need an orthogonal matrix U satisfying (9) . Let v be a column vector of ones (of systemsize) except for the last entry which is minus one. Note that v is a right eigenvector of the matrix QR − . Also, let w bea column vector of zeros except for the first entry which is one. Let u = v − || v || w . Then, the symmetric orthogonalmatrix U defined by U = I − uu T u T u gives rise to the factorization (9) with the matrix Ψ reducing to a scalar which isactually zero. This process is called the Householder transformation in [38] and significantly reduces the complexityof (9) . Remark.
In the most general case, the MFQ X ( t ) has a system size of (cid:96) T + (cid:96) + 2 . However, the system sizeis significantly reduced in some important sub-cases. For example, consider the global preemption case, i.e., P = N TN and when the service time requirements of the sources are homogeneous, i.e., Θ ( n ) ∼ Θ ∼ P H ( σ , S ) for n = 1 , . . . , N with order (cid:96) and homogeneous transmission errors and retransmission policies, i.e., e n = e, r n = r, n = 1 , . . . , N. From the perspective of AoI and PAoI of the tagged source- , one can solve an auxiliary two-sourcesystem where the second source in the auxiliary system stands for the superposition of all the sources indexed from2 to N in the original system and P = T . The arising MFQ will now have a system size of (cid:96) + 2 which doesnot depend on the number of users N . Similar reductions are possible for non-premptive systems with homogeneousservice time requirements. PREPRINT - J
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24, 2020 . . . .
81 a) Self Preemption F ∆(1)( x ) F ∆(2)( x ) F ∆(3)( x ) . . . .
81 b) Global Preemption F ∆(1)( x ) F ∆(2)( x ) F ∆(3)( x ) . . . . F ∆(1)( x ) F ∆(2)( x ) F ∆(3)( x ) . . . . x F Φ(1)( x ) F Φ(2)( x ) F Φ(3)( x ) . . . . x F Φ(1)( x ) F Φ(2)( x ) F Φ(3)( x ) . . . . x F Φ(1)( x ) F Φ(2)( x ) F Φ(3)( x ) Figure 4: The cdf of both the AoI and PAoI processes of a system with 3 sources obtained by the proposed analyticalmethod and simulations (shown by markers) for three cases (a) self preemption, (b) global preemption, (c) prioritizedpreemption, when ( λ , λ , λ ) = (1 , , , ρ = 2 / , and c = 1 / . In the first set of numerical examples, we validate the proposed approach by comparing the obtained cdfs for AoI andPAoI processes against the empirical cdfs obtained with simulations in the context of a system with 3 sources withhomogeneous service time requirements. For this purpose, we fix the intensity vector ( λ , λ , λ ) and for the servicetimes, we use a PH-type distribution with mean ρ/λ for given system load ρ and the squared coefficient of variationof the service times is fixed to a given value c according the following procedure. For c = 1 /j ≤ for a positiveinteger j , the E ( µ − , j ) distribution is used which refers to an Erlang distribution with mean µ − and with order j .If j is not an integer, then we resort to a mixture of two appropriate Erlang distributions [42]. When c > , thenwe propose to use a hyper-exponential distribution with balanced means to fit the first two moments [42]. In the firstexample, we fix ( λ , λ , λ ) to (1 , , and the system load ρ to / . Moreover, we study three preemption policies:global preemption, self preemption, and prioritized preemption in which a newcoming class- i packet preempts a class- j packet in service if only if i ≤ j , P is a lower-triangular matrix of ones at and below the main diagonal. The cdfs ofthe AoI and PAoI processes are depicted in figures 4 and 5, respectively, for the choice of c = 1 / and c = 1 / ,respectively. We have the following observations:• Perfect match with the simulation results are obtained in all cases.• For symmetric preemption systems such as self preemption and global preemption, low intensity sources arepenalized in terms of AoI and PAoI. However, with preferential treatment to low traffic intensity sources bymeans of non-symmetric policies such as prioritized preemption, this situation can be mitigated.• When c increases, the overall performance of the system improves which can be inferred from the sharperincrease of the cdf curves for all the three sources in Fig. 5 than in Fig. 4. In this subsection, we will compare our findings with existing closed-form expressions in the existing literature forvalidation of the numerical accuracy of our proposed method. As the first example, the reference [8] provides closedform expressions for the mean AoI for each of the sources in an
M/M/ / system with homogeneous service time11 PREPRINT - J
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24, 2020 . . . .
81 a) Self Preemption F ∆(1)( x ) F ∆(2)( x ) F ∆(3)( x ) . . . .
81 b) Global Preemption F ∆(1)( x ) F ∆(2)( x ) F ∆(3)( x ) . . . . F ∆(1)( x ) F ∆(2)( x ) F ∆(3)( x ) . . . . x F Φ(1)( x ) F Φ(2)( x ) F Φ(3)( x ) . . . . x F Φ(1)( x ) F Φ(2)( x ) F Φ(3)( x ) . . . . x F Φ(1)( x ) F Φ(2)( x ) F Φ(3)( x ) Figure 5: The cdf of both the AoI and PAoI processes of a system with 3 sources obtained by the proposed analyticalmethod and simulations (shown by markers) for three cases (a) self preemption, (b) global preemption, (c) prioritizedpreemption, when ( λ , λ , λ ) = (1 , , , ρ = 2 / , and c = 1 / .Table 1: Mean AoI for each of the three sources obtained with the closed-form expressions in Ref. [8] and the proposedmethod for various choices of the traffic intensity vector and the load parameter ρ . E [∆ (1) ] E [∆ (2) ] E [∆ (3) ]( λ , λ , λ ) ρ Ref. [8] Proposed Ref. [8] Proposed Ref. [8] Proposed (1 , , (1 , , ρ , the mean AoI values are tabulated in Table 1 using the expressionsin [8] and the proposed method for the case of global preemption. The two sets of results perfectly match up to fourdigits. For a similar M/M/ / system allowing transmission errors and employing the self preemption policy, thereference [29] provides closed form expressions for the mean AoI for each of the sources. For two different choicesof the traffic intensity vector, for various values of the load parameter ρ and the transmission error parameter e whichis intact for each source, the mean AoI values are tabulated in Table 2 using the expressions in [29] and the proposedmethod for the case of self preemption with no retransmissions, i.e., r n = r = 0 , n = 1 , . . . , N . The two sets ofresults perfectly match up to four digits except for one single instance in which the match is up to three digits. In this subsection, we study the impact of the choice of preemption probabilities (using the analytical model only)on the system cost C ( α ) for a two-source preemptive M/P H/ / system with homogeneous service times, which is12 PREPRINT - J
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24, 2020Table 2: Mean AoI for each of the three sources obtained with the closed-form expressions in Ref. [29] and theproposed method for various choices of the traffic intensity vector, the load parameter ρ , and the error parameter e when the discarding parameter r is set to zero. E [∆ (1) ] E [∆ (2) ] E [∆ (3) ]( λ , λ , λ ) ρ e Ref. [29] Proposed Ref. [29] Proposed Ref. [29] Proposed (1 , , (1 , , C ( α ) = E [∆ (1) ] + αE [∆ (2) ] , ≤ α ≤ , (19)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 = 0 , the age of the second source is irrelevant. Fora given traffic intensity vector, load ρ , and c for the homogeneous service times, we employ a preemption matrix P such that P , = P , = P d and we perform brute-force optimization to find the optimum choices of P d , P , ,and P , , denoted by P ∗ d , P ∗ , , and P ∗ , , respectively, so that the cost function in (19) is minimized for a given costparameter α (with a resolution of . for each parameter). Table 3 provides our findings. We have the followingobservations:• The optimum value P ∗ d appears to depend on c and not on the specific choices of α and the traffic mix.When c is small (large), P ∗ d is zero (one).• The optimum values P ∗ , and P ∗ , appear to depend on c , α , and the traffic mix. All other input param-eters being fixed, P ∗ , decreases and P ∗ , increases with increased cost parameter α but their actual valuesthemselves depend on the other parameters.Table 3: Optimum preemption parameters P ∗ d , P ∗ , , and P ∗ , , which minimize C ( α ) for various values of trafficintensity vector, c , and α . ( λ , λ ) c α (1 ,
2) (1 ,
1) (2 , P ∗ d P ∗ , P ∗ , P ∗ d P ∗ , P ∗ , P ∗ d P ∗ , P ∗ , PREPRINT - J
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24, 2020 . . . . . . . ρ C ( . ) (a) Global PreemptionSelf PreemptionNo PreemptionOptimum Preemption 0 . . . . . . . ρ C ( . ) (d) Global PreemptionSelf PreemptionNo PreemptionOptimum Preemption0 . . . . . . . ρ C ( . ) (b) Global PreemptionSelf PreemptionNo PreemptionOptimum Preemption 0 . . . . . . . ρ C ( . ) (e) Global PreemptionSelf PreemptionNo PreemptionOptimum Preemption0 . . . . . . . ρ C ( . ) (c) Global PreemptionSelf PreemptionNo PreemptionOptimum Preemption 0 . . . . . . . ρ C ( . ) (f) Global PreemptionSelf PreemptionNo PreemptionOptimum Preemption1 Figure 6: The cost function C ( α ) as a function of the load parameter ρ for an M/M/ / system with four preemptionpolicies when µ = 1 , r = 0 . , e = 0 . for three values of α ∈ { . , . , } . For the subfigures (a)-(c), ρ = ρ whereas for the subfigures (d)-(f), ρ = 2 ρ .Finally, we depict the cost C ( α ) as a function of the system load ρ for four preemption policies, namely non-preemptive, self-preemptive, globally preemptive, and optimum preemptive policies (as obtained using the brute-forceapproach outlined above) in Fig. 6. We observe that optimum preemption significantly outperforms all the other poli-cies with the level of performance improvement increases with decreased cost parameter α . When α = 1 and thetraffic mix is even, global preemption and optimum preemption yield the same performance. When α = 1 and thetraffic mix is not even, i.e., ρ = 2 ρ , then there is still a substantial gain attained with the optimum preemptive policywhen compared with the other preemption policies. For homogeneous exponential service times, we observe thatglobal preemption performs better than self preemption which also outperforms the non-preemptive policy for all thecases we studied. 14 PREPRINT - J
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In this paper, we propose a novel method to obtain the exact distributions of the AoI and PAoI for a probabilisticallypreemptive bufferless heterogeneous
M/P H/ / queueing system with packet errors using Markov fluid queues. Thismodel is more general than many recent existing models arising in status update systems and obtaining the distributionsin addition to mean values is the major contribution of this paper. Numerical examples are provided to validatethe proposed approach and its numerical accuracy. We also provide a number of examples for which probabilisticpreemption is substantially beneficial when compared to conventional non-preemptive, self-preemptive, or globally-preemptive policies in terms of AoI. Such probabilistic schemes can also be used to provide source differentiation inmulti-source status update systems. 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.[6] A. Kosta, N. Pappas, and V. Angelakis, “Age of information: A new concept, metric, and tool,”
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