Age of Information in an Overtake-Free Network of Quasi-Reversible Queues
AAge of Information in a network of queues
Ioannis KoukoutsidisMay 29, 2020
Abstract
We show how to calculate the Age of Information in an overtake-free network of quasi-reversible queues, with exponential exogenousinterarrivals of multiple classes of update packets and exponential ser-vice times at all nodes. Results are provided for any number of M/M/1First-Come-First-Served (FCFS) queues in tandem, and for a networkwith two classes of update packets, entering through different queuesin the network and exiting through the same queue. The results areextensible for other quasi-reversible queues for which sojourn time dis-tributions are known, such as M/M/c FCFS queues and processor-sharing queues.
The Age of Information (AoI, or simply “age”) equals the time from the gen-eration of a piece of information, until that piece is received by its intendeddestination. In packet networks, we consider that information updates arereceived in packets, and at each time instant t , the destination observes anage H ( t ) = t − u ( t ), where u ( t ) is the generation time of the last packetreceived. If the process H ( t ) is ergodic, then its expectation E [ H ( t )] con-verges to the average ¯ H of the ages of all packets, as seen by the destinationwhen the packets are received. The latter metric will be meant when refer-ring to the AoI metric in this paper. Packets may also be of different typesor classes, in which case we are interested in the AoI ¯ H c separately for eachclass c .The AoI can serve as a metric in numerous applications, where we areinterested in the freshness of the received information. For example, in theInternet of Things, where sensor devices can transmit updates of environ-mental parameters, or the values of technical parameters such as location We use the letter H for the AoI, from the greek word ‘ Hλικι (cid:48) α ’, which means ‘Age’. a r X i v : . [ c s . PF ] M a y nd velocity in autonomous vehicles; in the storage of data in computer sys-tems, where we are interested in the freshness of imformation in the cachememory, or in robotics and control systems, where the fast interchange ofinformation plays a prominent role.These applications justify the necessity for the AoI as a metric, but per-haps its importance stems more from the fact that it motivates a redesign ofcomputer and networking systems, to also take it into account, as the tra-ditional metrics of throughput and delay were shown from the very start topoorly reflect freshness [6]. Information freshness also trade-offs with energyefficiency, this also motivating the redesign of communication protocols totake AoI into account, subject to energy constraints [12].Mathematically, the main tool for theoretically calculating the AoI isqueueing theory. Update packets are seen as arriving to a queue, where theywait for processing by a monitor, which reads the related information. So far,results have mainly been derived for single-server queues, with the exceptionof [13], where the AoI was calculated for M/M/1 queues in tandem, underLast-Come-First-Served (LCFS) service discipline with service preemption,and the more general method in [14] (see Sect. 2 for a description of relatedwork). The method in [14] can theoretically be applied for any network inwhich the movements of updates are described by a finite-state continuous-time Markov chain, but it is a matrix-based method which can lead tocombinatorial explosion for systems with a large number of states, such asnetworks with infinite queues.In this paper, we derive results for the AoI in networks, starting from theinitial analysis in [6], but presenting it in a new, more simplified manner, andthen relying on classical queueing theory results from [8] and [11], which werederived more than three decades ago. The model (in Sect. 3) and analysis(in Sect. 4) allow to calculate the AoI for different types or classes of updatepackets, both at the output of each node, where the packets pass from, andat the exit of the network.The concept that allows us to extend the calculation of the AoI to aqueueing network is that of a quasi-reversible queue. A Markovian queue is called quasi-reversible if the state of the process at time t is independent ofthe arrival process after time t and is also independent of the departure pro-cess prior to time t [10]. The important property of a quasi-reversible queue,which will be of great use here, is that Poisson streams passing through thequeue are statistically unchanged, hence producing Poisson output streams A queue is called Markovian if its state process is a stationary and ergodic Markovchain.
Poisson-in-Poisson-out property .When quasi-reversible queues are connected into a network, the networkis called a quasi-reversible network . Then the network itself possesses thePoisson-in-Poisson-out property. However, whether or not internal traffic isPoisson depends on the topology of the network [11]. More specifically, inter-nal traffic is Poisson in overtake-free , or order-preserving networks . Theseare networks where the order of transmission is preserved and jobs sentlater in time cannot arrive at the destination earlier than previously sentjobs. Note that if there is a possibility for update packets to arrive throughmultiple routes to a destination, then a packet that was sent earlier may ar-rive later than a subsequent packet, which carries fresher information. Thedestination could simply discard the packet that arrived later, but in thiscase calculating the AoI is not known. Therefore, we will use the notion ofovertake-free paths as the basis for the results in this paper.We extend the AoI results in the literature by calculating the AoI for anarbitrary number of M/M/1 queues in tandem, as well as a simple networkwith two classes of update packets, where the different class update packetsdo not directly share an output queue, but pass from other nodes first (seeSect. 5). The use of FCFS M/M/1 queues might appear simplistic, but apartfrom its theoretical value, it also has practical significance in networks ofqueues with multiple classes of update packets, where different class packetsshould share common resources, without one suffocating the other. Forexample, it was shown in [16] that for two classes of packets directly sharingan output M/M/1 queue, the FCFS discipline can minimize the sum AoIof the two classes for a large range of load values, outperforming LCFSdisciplines both with and without preemption in service. Nevertheless, asdiscussed in Sect. 6, the work in this paper can also be extended to otherservice disciplines, such as processor-sharing queues.
The basic method for calculating the AoI in a single server queue withFirst-Come-First-Served (FCFS) service discipline was shown in [6], wherethe authors derived results for M/M/1, M/D/1 and D/M/1 queues. Thispaper builds upon this basic method for deriving results for networks ofqueues. The next important paper was [3], where the authors derived resultsfor M/M/1/1 (at most one packet being served, no packets waiting) andM/M/1/2 (at most one packet being served and one packet waiting) queueswith FCFS service discipline. This paper also included the definition of the3 eak AoI , or peak age metric, which reflects the average maximum age seenby the destination, prior to receiving updates.The intuition that waiting may not be efficient when we are interested inreceiving update packets as soon as possible, led to the exploration of LCFSservice disciplines. Results for a single server M/M/1 queue were derivedin [7], where the authors showed that LCFS, with or without service pre-emption, always outperforms FCFS for a single class of update packets. En-forcing packet deadlines, or timeouts, after which an update packet waitingin the queue is discarded, was studied in [5] for an M/M/1/2 queue, show-ing that using a deadline can outperform both the M/M/1/1 and M/M/1/2without deadline.Results for multiple sources sharing a single-server M/M/1 queue withFCFS service discipline were first derived in [15]. This also produced insightsinto how a source can choose its update rate in the presence of interferingtraffic, basically showing that the minimum AoI is achieved at a smallerupdate rate, compared to when only one source sends packets to the queue.For the peak AoI, results for more general multi-class M/G/1 and M/G/1/1queues were presented in [4].A breakthough in the method of calculating the AoI was achieved in [13],where the author used a Stochastic Hybrid System (SHS) to model thesystem and derived AoI results for M/M/1 queues in tandem, under LCFSservice discipline with service preemption. The use of SHS leads to a generalmethod for calculating the AoI for a variety of queues with different servicedisciplines. In [16], the authors showed the application of the method forcalculating the AoI for two-sources sharing an M/M/1 queue under FCFSand LCFS service disciplines. The technique was presented more generallyin [14] for any network described by a finite-state continuous-time Markovchain, and led to a system of ordinary linear differential equations thatdescribe the temporal evolution of the moments and moment-generatingfunctions of the age process. For a line network of preemptive memorylessservers, the author showed that the age at a node is identical in distributionto the sum of independent exponential service times.
We consider a quasi-reversible network consisting of a set of nodes M = { , . . . , m } and a ficticious node 0 that represents both the source and the4ink of the network. The network admits different types, or classes ofupdate packets; these will be the jobs circulating in the network, and maycorrespond to different applications that share the network for transmission.External update packets of type c arrive at the network from the source node0 as a Poisson process with rate λ c . Each arrival joins node i with probability r c ,i . On service completion at node i , it is routed to node j with probability r ci,j , or leaves the network with probability r ci, = 1 − (cid:80) j (cid:54) =0 r ci,j . Generally,an update packet of type c follows a path or itinerary r c := ( r c , . . . , r cn ),defined by the sequence of nodes r c , . . . , r cn visited by the packet, from itsentry in the network through node r c , until its exit from the network vianode r cn . The number of nodes n is called the length of the path. When thepath and the packet type are obvious, we will simply denote a path by itsstart and end nodes, as r → r n .In the queueing network, the total equilibrium rate of type- c updatepackets through node i (including both external arrivals and internal tran-sitions) can be found by solving the system of traffic equations : λ ci = λ c r c ,i + (cid:88) j (cid:54) = i λ cj r cj,i , i, j = 1 , . . . , m . (1)The rate at which update packets of type c exit the network from node i ( i = 1 , . . . , m ) is (cid:80) i λ ci r ci, . Additionally, the total rate of update packets(of all types) in node i is λ i = (cid:80) c λ ci .We assume that all nodes have a FCFS queueing discipline and do notadmit batch processing, neither go on vacations. The service times of allpackets at each node i are exponentially distributed with parameter µ i .Additionally, the network is assumed to contain only overtake-free paths.An overtake-free path is defined as a path: • where every node is overtake-free (this is already assumed by the FCFSservice discipline), • which is cycle-free, i.e. every node in the path is distinct, and • which does not contain any forward short-circuits, i.e. packets sharingparts of an itinerary in the forward direction cannot overtake eachother by taking a shorter route (a route which intersects two non-contiguous nodes of another packet’s route). This section partially follows the notation in [9] (although results from that paper arenot used).
Without loss of generality, we will calculate the AoI for a type of updatepackets at the output of an overtake-free path, by looking at the path asa black box and only being interested in the input and output processes.Consider the path of length n shown in Fig.1, where update packets of type c arrive at the first node in the path as a Poisson process at time instants( a c , a c , . . . ), and exit the path at time instants ( d c , d c , . . . ). There mayalso be traffic of other types sharing the path, either wholly or partially,depicted by the diagonal arrows in the figure. The total traffic at each node i ( i = 1 , . . . , n ) is λ i , and the service rate is µ i . Given that λ i < µ i (so thatall queues are stable), we will calculate the AoI at the output of the path.We will further assume that all update packets arrive fresh at the entry ofthe path, i.e. their initial ages are zero. . . . a c , a c , ... d c , d c , ... Figure 1: A view of a path as a “black-box”For an overtake-free path, it was shown in [8] that the end-to-end so-journ time of a customer is distributed in the limit as a sum of independentexponential sojourns at each node i , ( i = 1 , . . . , n ), with respective param-eters µ i − λ i . That is, the sojourn time of a job in the network depends onits type only through the itinerary followed, and sojourn time of differenttype jobs in the same node have the same distribution.6o illustrate the method of calculation, consider an example run of theage process at the output of the path, as shown in Fig.2. At time d ci , theage of the type- c update packet received at the output of the path is equalto its end-to-end sojourn time S c, → ni through the whole path. . . . d cN ( t ) d c d c S c, → n d c S c, → n S c, → nN ( t ) − S c, → nN ( t ) S c, → n d cN ( t ) − Figure 2: Sample run of the age update processThe AoI at the receiver is equal to the time average¯ H c = lim t →∞ t N ( t ) − (cid:88) i =1 ( S c, → ni + ( d ci +1 − d ci ) / d ci +1 − d ci ) , (2)where N ( t ) is the number of packet departures in time t (the receiver per-ceives a mean age S c, → ni + ( d ci +1 − d ci ) / d ci , d ci +1 ]).Similarly, we can define the left and right limits bounding the AoI:¯ H cleft = lim t →∞ t N ( t ) − (cid:88) i =1 S c, → ni ( d ci +1 − d ci ) , (3a)¯ H cright = lim t →∞ t N ( t ) − (cid:88) i =1 ( S c, → ni + d ci +1 − d ci )( d ci +1 − d ci ) . (3b)7n Fig.2 the age process is depicted as the sawtooth curve in bold, andis right-continuous with left limits. The right limits (solid black circles)correspond to the lower (left) bounds of the age process (Eq. 3a), while theleft limits (solid white circles) correspond to the upper (right) bounds of theupdate process (Eq. 3b). Note that ¯ H cright is also referred to as the peak age ,which was first introduced in [3].It is straightforward to see that¯ H c = ¯ H cleft + lim t →∞ t N ( t ) − (cid:88) i =1 ( d ci +1 − d ci ) / , (4a)¯ H cright = ¯ H cleft + lim t →∞ t N ( t ) − (cid:88) i =1 ( d ci +1 − d ci ) . (4b)The D ci := d ci +1 − d ci correspond to the interdeparture times of the type- c update packets exiting the path, and, for an overtake-free path of quasi-reversible queues, in the limit t → ∞ successive D ci ( i = 1 , . . . , N ( t ) − t →∞ N ( t ) /t = 1 /E [ D ci ]. Moreover, sincethe interdeparture process is ergodic,lim t →∞ t N ( t ) − (cid:88) i =1 ( d ci +1 − d ci ) = lim t →∞ N ( t ) − t (cid:32) (cid:80) N ( t ) − i =1 ( d ci +1 − d ci ) N ( t ) − (cid:33) = E [( D ci ) ] /E [ D ci ] . Therefore, equations (4) become:¯ H c = ¯ H cleft + E [( D ci ) ]2 E [ D ci ] , (5a)¯ H cright = ¯ H cleft + E [( D ci ) ] E [ D ci ] . (5b)Similarly, from (3a): ¯ H cleft = E [ S c, → ni D ci ] E [ D ci ] . (6)In summary, the AoI of a type of update packets only depends on the de-parture rate of that type and the expected value of the product betweenthe sojourn time of a packet and the interdeparture interval between the8eparture of that type of packet and the one of the same type following itin the path.We calculate the E [ S c, → ni D ci ] by noting that, since the network (con-sisting of all queues in the path) is quasi-reversible, in the time-reversedprocess, the distribution of interarrivals is the same as the distribution ofinterdepartures. More specifically, the interval between the departure of apacket and the one that followed it in the forward-time system correspondsto the interval between the arrival of a packet and the one that preceded itin the reverse-time system. Denoting the interarrival interval ( a ci − a ci − ) by A ci , it therefore holds that D ci ∼ A ci , and S c, → ni D ci ∼ S c, → ni A ci .When the parameter of the exponential interarrival distribution for type- c update packets is λ c , we have E [( D ci ) ] = 2 /λ c and Eqs. 5a, 5b, 6 become:¯ H c = ¯ H cleft + 1 λ c , (7a)¯ H cright = ¯ H cleft + 2 λ c , (7b)¯ H cleft = λ c E [ S c, → ni D ci ] . (7c)Denoting by S c,ji the sojourn time of type- c update packet i at node j ,we have S c, → ni = (cid:80) nj =1 S c,ji . We also decompose S c,ji to W c,ji and X c,ji ,the waiting time and service time of this packet at node j . When dealingwith distributions or expectations of S c,ji , W c,ji and X c,ji , we can also dropthe index c , because these distributions are independent of packet type. Wehave: E [ S c, → ni D ci ] = E [ S c, → ni A ci ] = n (cid:88) j =1 E [ S ji A ci ]= n (cid:88) j =1 E [( W ji + X ji ) A ci ]= n (cid:88) j =1 ( E [ W ji A ci ] + E [ X ji ] E [ A ci ]) . (8)(The last step occurs because of the independence of service times and in-terarrival times.) 9rom the law of total expectation, we have for W ji A ci ( j = 1 . . . n ): E [ W ji A ci ] = E [ E [ W ji A ci | A ci = x ]]= E [ xE [( S ji − − x )]]= (cid:90) ∞ (cid:90) ∞ x x ( t − x ) f S ji − ( t ) f A ci ( x ) dtdx . (9)(Note again that, although a packet of different type may be ahead of thetype- c packet in the queue, they all have the same sojourn time distribution.)For a path with input (and output) rate of type- c update packets λ c ,total arrival rate (of all type update packets) λ j and service rate µ j at eachqueue j (irrespective of the type of update packets), we have the probabilitydensity functions f S ji − = ( µ j − λ j ) e − ( µ j − λ j ) t , and f A ci ( x ) = λ c e − λ c x ∀ i .Hence E [ W ji A ci ] == (cid:90) ∞ λ c e − λ c x (cid:90) ∞ x x ( t − x )( µ j − λ j ) e − ( µ j − λ j ) t dtdx = (cid:90) ∞ λ c xe − λ c x e − ( µ j − λ j ) x µ j − λ j dx = λ c ( µ j − λ j )( λ c + µ j − λ j ) . (10)Combining Eqs. 8 and 10, we finally get E [ S c, → ni D ci ] = n (cid:88) j =1 (cid:18) λ c ( µ j − λ j )( λ c + µ j − λ j ) + 1 µ j λ c (cid:19) . (11)It is noted that since each of the successive sojourn, interarrival and servicetimes are i.i.d, the above expectations are the same for all successive packetsin the limit, and the subscripts i can be dropped for notational simplicity.Finally, from Eqs. 7c and 7a, we get the AoI for type- c update packetsat the output of the path:¯ H c = n (cid:88) j =1 (cid:18) λ c ( µ j − λ j )( λ c + µ j − λ j ) (cid:19) + n (cid:88) j =1 µ j + 1 λ c . (12)Summarizing, this section has showed that the AoI for a type of updatepackets at the output of a quasi-reversible, overtake-free path depends onlyon the distribution of sojourn times at each node in the path, and the rate ofincoming (and outgoing) packets of that type in and out of the path. In thenext section we provide some examples to show how the method is applied.10 Examples
Consider the simple network shown in Fig. 1, where n queues are connectedin tandem and there is only one type of update packets, which arrive at thenetwork with rate λ . For simplicity, it is assumed that the queues have thesame service rate µ . In order for the queues to be stable, we must have ρ = λ/µ <
1. It is noted that so far results have only been taken for anetwork of M/M/1/1 preemptive LCFS queues in tandem [13].Taking λ c = λ j = λ and µ j = µ ∀ j , Eq. 12 becomes:¯ H = nρ µ − λ + nµ + 1 λ . (13)Fig. 3 shows results for the mean AoI for µ = 1 and varying values of λ , n . The same pattern appears, as in the case of a single queue: the minimumis achieved for some intermediate load value. This value also decreases withthe number of nodes; for n = 1, the minimum is achieved approximatelyat ρ = 0 .
53, for n = 2 at ρ = 0 .
46, for n = 5 at ρ = 0 .
37 and for n = 10at ρ = 0 .
31. We also remark that the minimum AoI does not increasesignificantly with the number of nodes. The effect of the added queues ismore apparent as the load increases, and the age can increase tremendouslyfor large values. In the figure, the AoI increase for ρ = 0 .
99 is equal to 891time units between n = 1 and n = 10. λ ¯ H n = 1 n = 2 n = 5 n = 10 Figure 3: AoI for n identical M/M/1 queues in tandem, for µ = 1 andvarying values of the arrival rate λ .2 A simple network with two classes Consider the network with two classes shown in Fig. 4. There are two typesof update packets α and β , entering the network as Poisson processes withrates λ α and λ β , through nodes 1 and 2 respectively. Both types of packetsexit the network through node 3. The service times at each node i areexponential with mean µ − i . We will calculate the AoI for each class ofupdate packets at the exit of the network. It is noted that the case wheretwo classes of traffic directly share an output queue (without passing througha network) was studied in [15]. λ α λ β
12 3 λ α λ β λ α + λ β µ µ µ Figure 4: A simple network with two classesWe assume that the stability conditions λ α < µ , λ β < µ and λ α + λ β <µ hold. Since all queues are quasi-reversible and all paths are overtake-free,internal traffic is Poisson and the rates of class- α and class- β update packetsat the output of nodes 1 and 2 are also λ α and λ β , respectively, and theoutput rate at node 3 is λ α + λ β . The mean sojourn time at nodes 1 and 2are ( µ − λ α ) − and ( µ − λ β ) − , respectively, while the mean sojourn timeof any type update packet at node 3 is ( µ − ( λ α + λ β )) − .From Eq. 12, substituting λ ≡ λ α , λ ≡ λ β , λ ≡ λ α + λ β , and taking λ c ≡ λ α , we can calculate the AoI of class- α update packets at the outputof node 3:¯ H α = ρ µ − λ α + λ α ( µ − ( λ α + λ β ))( µ − λ β ) + 1 µ + 1 µ + 1 λ α , (14)where ρ = λ /µ = λ α /µ is the load at queue 1. By symmetry, the AoIof class- β update packets in the path 2 →
3, ¯ H β , can then be found by12nterchanging λ α and λ β , and µ and µ in the expression for ¯ H α , and alsodefining ρ = λ /µ = λ β /µ :¯ H β = ρ µ − λ β + λ β ( µ − ( λ α + λ β ))( µ − λ α ) + 1 µ + 1 µ + 1 λ β . (15)Fig. 5a shows how the AoI of update packets in the output of the path1 → λ α , λ β values (the service rates µ i are fixed at1 for all queues). The AoI is represented by color intensities in the differentparts of the feasible region. It can be seen that ¯ H α primarily depends on thevalue of λ α ; the arrival of class- β packets effects by increasing ¯ H α , but onlywhen λ β is high enough, so that the queue is close to saturation. Fig. 5bshows a scatter plot of ¯ H α , ¯ H β values in the region defined by λ α , λ β . Itcan be seen that ¯ H α , ¯ H β do not vary continuously, and there are subregionswhere the AoI of one class of update packets stays relatively stable. Thisis in agreement with Fig. 5a, where increasing the arrival rate of one classalways effects on the AoI of that class, but the other class is affected onlywhen the output queue is close to saturation.The minimum of ¯ H α equals 4.97 and is achieved when ρ = 0 .
46 and ρ = 0 (and vice-versa for ¯ H β ). This is anticipated, as it coincides withthe minimum ρ value for n = 2 in the case of queues in tandem. Theminimum value for ¯ H α + ¯ H β equals 11.78 and is achieved for ρ = ρ = 0 . H α = ¯ H β = 5 .
89. This is slightly higher than the casewhere classes α and β directly share the output queue, which was studiedin [15], and where the sum was minimized at ρ = ρ = 0 . H α = ¯ H β = 5 . This paper presented a method for calculating the AoI in quasi-reversible,overtake-free networks of queues. The presented analysis simplified andextended the original analysis in [6] for FCFS single-server queues, and re-vealed the relationship between the AoI and its upper and lower bound (theaverage maximum age seen be the destination before reception of an update,known as the peak age, and the average minimum age right after receptionof an update).Results for n identical M/M/1 queues in tandem showed that, althoughthe minimum AoI value (achieved for some intermediate load value, whichdecreases with n ) does not increase significantly with the number of queues,the effect of the added queues becomes apparent for high load values, and the13 a) AoI of class- α packets ¯ H α ¯ H β (b) Scatter plot of ¯ H α , ¯ H β Figure 5: Results for the AoI in the network of Fig. 4, for µ i = 1 and varying λ α , λ β .age can then increase tremendously. Results for a simple network with twoclasses of update packets, entering through different queues in the networkbut sharing the output queue, revealed that changing the arrival rate of oneclass always effects on the AoI of that class, but the other class is significantlyaffected only when the output queue is close to saturation.The use of the method for calculating the AoI requires that queues inthe network path are Markovian and quasi-reversible, the path is overtake-free, and that the sojourn and interarrival time distributions at each queue14re known. Quasi-reversibility for Markovian queues is equivalent to thePoisson-in-Poisson-out property and holds more generally for M/M/c queueswith exponential constant-rate arrivals, as well as for processor-sharing queuesin BCMP networks. Sojourn time distributions are well-known for the first(see e.g. [1]), and also sometimes exist for the latter [2]. References [1] IJBF Adan and JAC Resing.
Queueing Systems: Lecture Notes . Eind-hoven University of Technology, 2017.[2] Sem Borst, Onno Boxma, and Nidhi Hegde. Sojourn times in finite-capacity processor-sharing queues. In
Next Generation Internet Net-works, 2005 , pages 53–60. IEEE, 2005.[3] Maice Costa, Marian Codreanu, and Anthony Ephremides. Age of in-formation with packet management. In , pages 1583–1587. IEEE, 2014.[4] Longbo Huang and Eytan Modiano. Optimizing age-of-information ina multi-class queueing system. In , pages 1681–1685. IEEE, 2015.[5] Clement Kam, Sastry Kompella, Gam D Nguyen, Jeffrey E Wieselth-ier, and Anthony Ephremides. On the age of information with packetdeadlines.
IEEE Transactions on Information Theory , 64(9):6419–6428,2018.[6] Sanjit Kaul, Roy Yates, and Marco Gruteser. Real-time status: Howoften should one update? In , pages2731–2735. IEEE, 2012.[7] Sanjit K Kaul, Roy D Yates, and Marco Gruteser. Status updatesthrough queues. In , pages 1–6. IEEE, 2012.[8] Benjamin Melamed. Sojourn times in queueing networks.
Mathematicsof Operations Research , 7(2):223–244, 1982.[9] Benjamin Melamed and D Yao. The asta property.
Advances in Queue-ing: Theory, Methods and Open Problems , pages 195–224, 1995.1510] Randolph D Nelson. The mathematics of product form queuing net-works.
ACM Computing Surveys (CSUR) , 25(3):339–369, 1993.[11] Jean Walrand.
An introduction to queueing networks . Prentice Hall,1988.[12] Xianwen Wu, Jing Yang, and Jingxian Wu. Optimal status up-date for age of information minimization with an energy harvestingsource.
IEEE Transactions on Green Communications and Network-ing , 2(1):193–204, 2017.[13] Roy D Yates. Age of information in a network of preemptive servers.In
IEEE INFOCOM 2018-IEEE Conference on Computer Communica-tions Workshops (INFOCOM WKSHPS) , pages 118–123. IEEE, 2018.[14] Roy D Yates. The age of information in networks: Moments, distribu-tions, and sampling. arXiv preprint arXiv:1806.03487 , 2018.[15] Roy D Yates and Sanjit Kaul. Real-time status updating: Multiplesources. In , pages 2666–2670. IEEE, 2012.[16] Roy D Yates and Sanjit K Kaul. The age of information: Real-timestatus updating by multiple sources.