Latency Analysis of Multiple Classes of AVB Traffic in TSN with Standard Credit Behavior using Network Calculus
Luxi Zhao, Paul Pop, Zhong Zheng, Hugo Daigmorte, Marc Boyer
11 Latency Analysis of Multiple Classes of AVBTraffic in TSN with Standard Credit Behaviorusing Network Calculus
Luxi Zhao *a , Paul Pop a , Zhong Zheng b , Hugo Daigmorte c , and Marc Boyer da Department of Applied Mathematics and Computer Science, Technical University of Denmark, Denmark b Department of Electronics and Information Engineering, Beihang University, Beijing, China c RealTime-at-Work, Toulouse, France d Department on Information Processing and Systems at ONERA, Toulouse, France
Abstract —Time-Sensitive Networking (TSN) is a set of amend-ments that extend Ethernet to support distributed safety-criticaland real-time applications in the industrial automation, aerospaceand automotive areas. TSN integrates multiple traffic types andsupports interactions in several combinations. In this paper weconsider the configuration supporting Scheduled Traffic (ST)traffic scheduled based on Gate-Control-Lists (GCLs), Audio-Video-Bridging (AVB) traffic according to IEEE 802.1BA thathas bounded latencies, and Best-Effort (BE) traffic, for which noguarantees are provided. The paper extends the timing analysismethod to multiple AVB classes and proofs the credit boundsfor multiple classes of AVB traffic, respectively under frozen andnon-frozen behaviors of credit during guard band (GB). Theyare prerequisites for non-overflow credits of Credit-Based Shaper(CBS) and preventing starvation of AVB traffic. Moreover, thispaper proposes an improved timing analysis method reducing thepessimism for the worst-case end-to-end delays of AVB traffic byconsidering the limitations from the physical link rate and theoutput of CBS. Finally, we evaluate the improved analysis methodon both synthetic and real-world test cases, showing the significantreduction of pessimism on latency bounds compared to relatedwork, and presenting the correctness validation compared withsimulation results. We also compare the AVB latency bounds inthe case of frozen and non-frozen credit during GB. Additionally,we evaluate the scalability of our method with variation of theload of ST flows and of the bandwidth reservation for AVB traffic.
I. I
NTRODUCTION E THERNET is a well-established network protocol thathas excellent bandwidth, scalability, compatibility and costproperties [1]. However, it was not suitable for real-time andsafety critical applications [2]. Distributed safety-critical appli-cations, like those found in the aerospace, automotive and indus-trial automation domains, require certification evidence for thecorrect real-time behavior of critical communication. Therefore,several extensions to the Ethernet protocol have been proposed,such as, ARINC 664 Specification Part 7 [3], TTEthernet [4],and EtherCAT [5]. In 2012, the IEEE 802.1 Time-SensitiveNetworking (TSN) Task Group [6] has been founded to definestandard real-time and safety-critical enhancements for Ethernet.
E-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected].
TSN integrates multiple traffic types and supports interactionsin several combinations. In this paper we consider a TSN solu-tion supporting Credit-Based Shaper (CBS), previously definedfor Audio-Video Bridging (AVB) in 802.1BA [8], currentlyin [7, § § § a r X i v : . [ c s . PF ] M a y have proposed an eligible interval based formal analysis onthe calculation of worst-case performance bounds for multipleclasses of AVB traffic. However, they do not consider the effectof ST traffic on AVB traffic, and focus only on the “relativedelay” on a single hop. Network Calculus-based analyses havebeen proposed [18] to compute the WCDs of Rate-Constrained(RC) traffic with the consideration of the ST (also named Time-Triggered, or TT) frames in TTEthernet, but the techniques arenot applicable for TSN: RC traffic has no CBS, the integrationmodes are different in TSN and TTEthernet and TSN schedulesST traffic differently from TTEthernet: individual ST frames areconsidered in TTEthernet, whereas TSN schedules ST windowswhich may include several ST frames.The timing analysis of non-ST traffic types in TSN has beenaddressed in [19], considering closed-gate blocking, strict prior-ities and a “Peristaltic Traffic Shaper”. However, their analysis isnot applicable to AVB, which uses CBS. Researchers [20],[21]have proposed the latency bounds for AVB traffic affected bycontrol-data-traffic (CDT) in TSN, but assuming CDT as leakybucket (LB) or length rate quotient (LRQ), which does not fitfor the ST traffic. The AVB Latency Math equation has beenextended to consider the ST traffic in TSN [22]. However, it doesnot consider the actual situations for AVB flows in the network,but just assumes that maximum allowable bandwidth is occupiedby the corresponding AVB traffic class, thus causing overlypessimistic, i.e., leading to overly large WCDs. In addition,[22] can only be used to determine the WCDs of AVB Class Atraffic. Researchers have extended the Eligible Interval Analysis(EIA) to calculate the delay of two classes of AVB trafficin TSN [23]. However, it does not consider relative offsetsbetween ST windows, but just assumes the ST windows alwaysarranging back-to-back. The initial idea for the two classes ofAVB analysis under the influence of ST traffic in TSN networkbased on the Network Calculus has been given by [24], and thencredit bounds for CBS are improved in [25]. However, all theseanalyses above are based on a credit behavior deviating from thestandard 802.1Q-2018 [7], i.e., assuming credit frozen duringguard band before gate closing. Moreover, the latest standard802.1Q supports any number of AVB classes, which are notsupported in previous analysis work. Thus, in this paper, we areinterested to propose a formal performance analysis method forarbitrary number of AVB classes in TSN networks, under bothbehaviors of frozen and non-frozen credit during guard band(GB, cf. Sect. III for details).The main contributions of our paper are as follows: • To the best of our knowledge, our analysis is the first one toprove the credit bounds of CBS in the cases of frozen and non-frozen credit during guard band. The analysis is able to handlewith arbitrary number of AVB classes under the influence ofST traffic for the whole TSN network. • We reduce the pessimism of the analysis and this providestighter latency bounds for AVB, by introducing the limitationsfrom the physical link rate and the output of CBS, whichare denoted as the link and CBS shaping curves with theconsideration of ST traffic. • We evaluate the proposed approach on both synthetic and real-world test cases, comparing AVB latencies in cases of frozen
Fig. 1. TSN/GCL+CBS architecture for an output port in an ES/SW and non-frozen credit during guard bands, and comparing withrelated work and simulation results to show the significantreduction of pessimism on latency bounds, and to validate thecorrectness and the scalability of our implementation.The paper is organized as follows. Sect. II presents the systemmodel. Sect. III introduces TSN. Sect. IV briefly introduces theNetwork Calculus concepts needed for the analysis. Sect. Vgives the proof of credit bounds for multiple classes of AVBtraffic respectively under frozen and non-frozen behaviors ofcredit during GB and presents tighter AVB latency bounds byintroducing shaping curves. Sect. VI evaluates the proposedanalysis and Sect. VII concludes the paper.II. S
YSTEM M ODEL
A TSN network is composed of a set of end systems (ES)and switches (SW) also called nodes, connected via physicallinks. In this paper, we assume, without loss of generality, thatall physical links have the same rate C . The links are fullduplex, allowing thus communication in both directions, andthe networks can be multi-hop. The output port of a SW isconnected to one ES or an input port of another SW.The messages are sent from ESes via flows, which havea single source and may have multiple destinations. Eachsource ES is able to send multiple flows to the network. Asmentioned, our TSN solution supports three traffic types: ST [7, § § M i ( i ∈ [ , ] ) for AVB denote respectively thedifferent AVB traffic classes. For ST traffic, we know the GCLsin each output port h of nodes, i.e., the opening and closing timeof ST traffic (named as ST window) and GCL period ( p h GCL ).For an AVB flow τ M i [ k ] ∈ τ M i , we know its frame size l M i [ k ] , theminimum frame interval p M i [ k ] in the source ES and the trafficclass M i it belongs to. The AVB Class M i has higher prioritythan the AVB Class M i + . The flows assigned the same AVBtraffic class M i are served in FIFO order. Moreover, we knowthe maximum frame size l maxBE of BE traffic.III. TSN/GCL+CBS O UTPUT P ORT
The TSN Task group has defined a set of schedulers, whichcan interact in several combinations. In this paper, we assumea specific configuration, called TSN/GCL+CBS architecture toprovide different real-time guarantees and will present how STand AVB flows are transmitted in this case, as shown in Fig. 1.Each one has eight queues for storing frames that wait to be
Fig. 2. Non-preemption integration modes forwarded on the corresponding link, one or more for ST queues,two or more ( n hCBS ) for AVB queues (respectively for Class M i )and the remaining queues are used for BE. Every queue has agate with two states, open and closed. Frames waiting in thequeue are eligible to be forwarded only if the associated gateis open.The gates for each queue are controlled by GCLs, whichare created offline and contain the times when the associatedgates are open and closed [11]. In this study, we assume thatwhen associated gate for ST traffic is open, the remaininggates for other traffic (AVB and BE) are closed, and vice versa(aka. exclusive gating). Therefore, AVB traffic is prevented fromtransmitting in the time windows reserved for ST frames.In addition, two integration modes are introduced to solvethe issue when an AVB frame is already in transmission atthe beginning of time window for ST, i.e., non-preemption [7,Annex Q] and preemption [7, Annex S] modes. In this paper,we focus on the discussion of the non-preemption integrationmode (see Fig. 2) and the analysis model is also suitable for thepreemption integration mode combined with HOLD/RELEASE.The non-preemption mode uses a “guard band” before each STwindow to prevent the AVB frame from initiating transmissionif there is insufficient time available to transmit that entireframe before the open event of ST gate. The preemption modecombined with HOLD/RELEASE also uses a guard band, butwith a rather smaller size [7, Annex S.4]. The “guard band” willlead to wasted bandwidth due to the guard band, but it ensuresno delay for ST traffic.An enqueued AVB frame is transmitted only if the associatedgate is open and the Credit-Based Shaper (CBS) [7, § sdSl M i ) during the transmissionof an AVB frame and increasing with an idle slope ( idSl M i ) whenAVB frames are waiting to be transmitted due to other higherpriority AVB frames or the negative credit. This is illustratedfor example in Fig. 3, where we have three AVB classes M i ( i = , , Fig. 3. CBS example with non-preemption mode standard to consider the credit evolution rule to be frozen.IV. N
ETWORK C ALCULUS B ACKGROUND
Network Calculus [27] is a mature theory proposed fordeterministic performance analysis. It is used to construct arrivaland service curve models for the investigated flows and networknodes. Network Calculus functions mainly belong to non-decreasing functions and null before 0: F ↑ = { f : R + → R | x < x ⇒ f ( x ) < f ( x ) , x < ⇒ f ( x ) = } . Two basic operators on F ↑ are the convolution ⊗ , ( f ⊗ g )( t ) = inf ≤ s ≤ t { f ( t − s ) + g ( s ) } , (1)and deconvolution (cid:11) , ( f (cid:11) g )( t ) = sup s ≥ { f ( t + s ) − g ( s ) } , (2)where inf means infimum and sup means supremum.An arrival curve α ( t ) is a model constraining the arrivalprocess R ( t ) of a flow, in which R ( t ) represents the inputcumulative function counting the total data bits of the flow thathas arrived in the network node up to time t . We say that R ( t ) is constrained by α ( t ) iff for all s ≤ t , R ( t ) − R ( s ) ≤ α ( t − s ) . (3)A service curve β ( t ) models the processing capability of theavailable resource. Assume that R ∗ ( t ) is the departure process,which is the output cumulative function that counts the total databits of the flow departure from the network node up to time t .There are several definitions for service curve. We say that thenetwork node offers the min-plus minimum service curve β ( t ) for the flow iff R ∗ ( t ) ≥ inf ≤ s ≤ t { R ( s ) + β ( t − s ) } = ( R ⊗ β )( t ) , (4)and offers the strict service curve β ( t ) iff R ∗ ( t + ∆ t ) − R ∗ ( t ) ≥ β ( ∆ t ) , (5)during any backlog period ( t , t + ∆ t ] . In addition, in order toevaluate service curves we will use the non-decreasing non-negative closure defined by [ f ( t )] + ↑ = max ≤ s ≤ t { f ( s ) , } . A shaping curve σ ( t ) characterizes the maximum number ofbits that are served during a period of time ∆ t , which meansthat the departure process R ∗ ( t ) from the server always respectsthe shaping curve. A server offers a shaping curve σ ( t ) iff σ ( t ) could be an arrival curve for all output cumulative function R ∗ ( t ) (Eq. 3). Note that the shaping curve σ ( t ) from the output canalso be said as an arrival curve for the subsequent nodes.If a flow R ( t ) of arrival curve α ( t ) crosses a server with theservice curve β ( t ) , then the output flow R ∗ ( t ) can be boundedby the arrival curve α (cid:48) ( t ) , α (cid:48) ( t ) = α (cid:11) β ( t ) = sup s ≥ { α ( t + s ) − β ( s ) } , (6)It can also be taken as an arrival curve of input flow for thenext node.Let us assume that the flow constrained by the arrival curve α ( t ) traverses the network node offering the service curve β ( t ) .Then, the latency experienced by the flow in the network nodeis bounded by the maximum horizontal deviation between thegraphs of two curves α ( t ) and β ( t ) , h ( α , β ) = sup s ≥ { inf { τ ≥ | α ( s ) ≤ β ( s + τ ) }} . (7)The network end-to-end delay of a frame along its route isdefined as the time duration between the instant it is transmittedby the source ES on the link and the instant it is fully receivedby the destination ES. The worst-case end-to-end delay of theflow can be bounded by the sum of latency bounds in eachnetwork nodes along its route.V. W ORST - CASE A NALYSIS FOR
AVB T
RAFFIC
A. Service Curve for AVB Traffic
In this section, we will extend the service curve in [24] tomultiple AVB Class M i ( i ∈ [ , n hCBS ] ) with respectively (i) non-frozen and (ii) frozen credit during guard band. The servicecurve for AVB traffic depends on the working mechanism ofCBS. As can be seen from Fig. 3, no matter the case (i) or(ii) for each AVB traffic Class M i , any time interval ∆ t can bedecomposed into ∆ t = ∆ t + + ∆ t − + ∆ t , (8)where ∆ t + = ∑ i ∆ t + i (resp. ∆ t − = ∑ j ∆ t − j ) represents the accu-mulated length of all period where the credit increases (resp.decreases), and ∆ t = ∑ k ∆ t k is the frozen time of credit. Theservice could only be supplied for AVB traffic during the descenttime ∆ t − of credit. Then, the service curve for multiple classesof AVB traffic is given by the following theorem. Theorem 1:
The min-plus minimum service curve for AVBClass M i ( i ∈ [ , n hCBS ] ) under non-preemption mode in an outputport h is given by β hM i ( I ) ( t ) = idSl M i (cid:20) t − α hH ( I ) ( t ) C − c maxM i ( I ) idSl M i (cid:21) + ↑ , (9)where I ∈ { NF , F } represents the behavior of credit during GB,and α hH ( I ) ( t ) is the arrival curve with regard to credit frozen. Incase of I = NF , i.e., (i) non-frozen credit during GB, α hH ( I ) ( t ) = α hST ( t ) is only related to higher priority ST traffic, while in caseof I = F , i.e., (ii) frozen credit during GB, α hH ( I ) ( t ) = α hGB + ST ( t ) should additionally take guard band slots into account. α hST ( t ) and α hGB + ST ( t ) are respectively given by Lemma 1 and Lemma 2in Sect. V-C. c maxM i ( I ) is the upper bound of credit of Class M i under cases of (i) non-frozen or (ii) frozen credit during GB,and given by Eqs. (11) or (12). The proof of credit bound has to be extended to an arbitrary number of AVB classes and toconsider different behaviors of credit during guard band, whichis one of the challenges in this paper and will be discussed inSect. V-B. The proof Theorem 1 is similar as the one given in[24], hence it will not be further discussed in this paper. B. Bounding the Credit for AVB Traffic
In this section, we bound the credit for AVB traffic. Let usrecall from Sect. III how AVB is transmitted.
Theorem 2: (Lower bound of Class M i ) Let l maxM i be themaximal frame size of any flow crossing the AVB queue Q M i .Then, the credit c M i ( t ) of Class M i is lower bounded by [14],[24], c M i ( t ) ≥ l maxM i C sdSl M i = c minM i . (10)Before we start to prove the upper bound of credit, let us de-fine some notations. Let Q ≤ iAVB = (cid:83) ≤ j ≤ i { Q M j } denotes the sameor higher priority AVB queues and Q > iAVB = (cid:83) i < j ≤ n hCBS { Q M j } denotes the lower priority AVB queues. Then for ∀ s , t ∈ R + , s ≤ t , the interval [ s , t ] can be partitioned into several kinds ofintervals: let ∆ t M i ( s , t ) and ∆ t < iAVB ( s , t ) respectively denote theduration of emissions of frames from the queue Q M i and ofhigh priority frames from the queue Q < iAVB ; ∆ t ST ( s , t ) is the timeslot reserved for ST traffic, i.e. , the duration where the gates ofAVB queues are closed; ∆ t LP ( s , t ) is the duration where a framein Q M j ∈ Q ≤ iAVB waits due to a lower priority frame in queue Q > iAVB or Q BE that is being transmitted cannot be preempted; ∆ t GB ( s , t ) denotes the guard band duration where for all queues Q M j ∈ Q ≤ iAVB with c M j > ∆ t Other ( s , t ) is the interval except all thedefinitions above. Then we have t − s = ∆ t M i ( s , t ) + ∆ t < iAVB ( s , t ) + ∆ t ST ( s , t ) + ∆ t LP ( s , t ) + ∆ t GB ( s , t ) + ∆ t Other ( s , t ) . Theorem 3: (Higher bound of Class M i ) Let l maxBE be themaximal frame size of a BE flow. The credit c M i ( t ) of Class M i is upper bounded by,(i) non-frozen credit during GB c M i ( t ) ≤ idSl M i · ∑ i − j = c minM j − l max > i − σ M i GB ρ M i GB + ∑ i − j = idSl M j − C = c maxM i ( NF ) ; (11)(ii) frozen credit during GB c M i ( t ) ≤ idSl M i · ∑ i − j = c minM j − l max > i ∑ i − j = idSl M j − C = c maxM i ( F ) . (12)where l max > i = max j ∈ [ i + , n hCBS ] { l maxM j , l maxBE } , c minM j is the lower boundof credit of Class M j from Theorem 2, and σ M i GB and ρ M i GB arethe parameters of upper envelope related to guard band durationand satisfy for ∀ s , t ∈ R + , s ≤ t , C · ∆ t GB ( s , t ) ≤ σ M i GB + ρ M i GB · ( t − s − ∆ t ST ( s , t )) . (13)which will be discussed in the end of Sect.V-C. Proof : Considering the evolution of the credit, the maxi-mal value of credit will only happen at the instant satisfyingthe following definition. Let t ∈ R + be a time point whenAVB gates are in the open state, and c M i ( t ) >
0. Then letus define s = sup (cid:110) u ≤ t ∀ Q M j ∈ Q ≤ iAVB , c M j ( t ) ≤ (cid:111) . It impliesthat ∀ u ∈ ( s , t ] , ∃ Q M j ∈ Q ≤ iAVB , c M j ( u ) > i.e. , there always exists at least one queue in Q ≤ iAVB with some frame to send.Otherwise, we can always find another s < s (cid:48) ≤ t that satisfies ∀ Q M j ∈ Q ≤ iAVB , c M j ( s (cid:48) ) ≤
0. As it always exists one queue in Q ≤ iAVB with some frame to send, either it succeeds (there is a frame of M i emission, or a high-priority frame in Q M j ∈ Q > iAVB emission)or is blocked (non-preemption or guard band or ST traffictransmission). Thus ∆ t Other ( s , t ) = s , t are fixed, the ∆ t X ( s , t ) will be simplified to ∆ t X .(i) For the case of non-frozen credit during GB
Consider first the evolution of the credit value of M i between s and t . The credit c M i ( t ) increases at speed idSl M i when theframe in the queue Q M i is waiting (during ∆ t < iAVB + ∆ t LP + ∆ t GB ),decreases at speed sdSl M i = idSl M i − C when the frame in thequeue Q M i is transmitting (during ∆ t M i ), is not modified whenthe gate of Q M i is closed during ST traffic transmission, andmay be reduced from some positive value P to 0 due to resets.Thus the variation of c M i ( t ) during ( s , t ] is, c M i ( t ) − c M i ( s )= ∆ t M i · sdSl M i + (cid:0) ∆ t < iAVB + ∆ t LP + ∆ t GB (cid:1) · idSl M i − P . (14)Since ∆ t < iAVB + ∆ t LP + ∆ t GB = s − t − ∆ t M i − ∆ t ST and P ≥ c M i ( t ) − c M i ( s ) ≤ − ∆ t M i · C + ( t − s − ∆ t ST ) · idSl M i . (15)Let c < i ( t ) = ∑ i − j = c M j ( t ) denote the sum of credits of AVBtraffic with the priority higher than M i . Consider three cases:first, at any instant between s and t either a frame of M i usesthe link, or a low priority frame blocks the link or during GB,the credit of each non-empty queue Q M j ∈ Q < iAVB increases withspeed idSl M j . Then the sum of credits c < i ( t ) increases at most atspeed ∑ i − j = idSl M j . Or a frame from class with higher prioritythan M i is being sent. In this case c < i ( t ) decrease at least atspeed ∑ i − j = idSl M j − C (all the classes from Q < iAVB gain creditexpect one which loses credit). The last case is like for M i , c < i ( t ) may decrease due to a set of resets. Then it is possibleto upper bound the variation of c < i ( t ) between s and t , c < i ( t ) − c < i ( s ) ≤ ( ∆ t M i + ∆ t LP + ∆ t GB ) · i − ∑ j = idSl M j + ( t − s − ∆ t ST − ∆ t M i − ∆ t LP − ∆ t GB ) · (cid:18) i − ∑ j = idSl M j − C (cid:19) = ( ∆ t M i + ∆ t LP + ∆ t GB ) · C + ( t − s − ∆ t ST ) · (cid:18) i − ∑ j = idSl M j − C (cid:19) . (16)By definition of s , it exists continuously some queue Q M j ∈ Q ≤ iAVB trying to send a frame. Hence there is at most one lowpriority frame that can get access to the link before s and goon transmission due to non-preemption. Thus ∆ t LP · C ≤ max j ∈ [ i + , n hCBS ] { l maxM j , l maxBE } = l max > i . Moreover, we assume that there exists σ M i GB and ρ M i GB which willbe discussed in Sect. V-C such that ∀ s , t ∈ R + , s ≤ t , ∆ t GB · C ≤ σ M i GB + ρ M i GB · ( t − s − ∆ t ST ) . Thus Eq. (16) is modified to obtain, c < i ( t ) − c < i ( s ) ≤ ∆ t M i · C + l max > i + σ M i GB +( t − s − ∆ t ST ) · (cid:18) ρ M i GB + i − ∑ j = idSl M j − C (cid:19) . (17)Considering the system is assumed to be not overloaded, wehave ρ M i GB + ∑ i − j = idSl M j ≤ C . Thus, from Eq. (17) t − s − ∆ t ST ≤ c < i ( t ) − c < i ( s ) − ∆ t M i · C − l max > i − σ M i GB ρ M i GB + ∑ i − j = idSl M j − C . (18)Then from (18), (15) is modified to obtain, c M i ( t ) − c M i ( s ) ≤ idSl M i c < i ( t ) − c < i ( s ) − l max > i − σ M i GB ρ M i GB + ∑ i − j = idSl M j − C . By definition of s, we have c < i ( s ) ≤ c M i ( s ) ≤ c minM j is the lower bound of credit of Class M j , then c < i ( t ) ≥ ∑ i − j = c minM j so to conclude, c M i ( t ) ≤ idSl M i · ∑ i − j = c minM j − l max > i − σ M i GB ρ M i GB + ∑ i − j = idSl M j − C . (ii) For the case of frozen credit during GB
The proof is similar to the above process, except the differ-ence of the evolution of credit of M i during GB. The credit c M i ( t ) increases at speed idSl M i during ∆ t < iAVB + ∆ t LP , decreasesat speed sdSl M i = idSl M i − C during ∆ t M i , and is not modifiedduring guard band ∆ t GB and ST traffic transmission ∆ t ST . Thenthe variation of c M i ( t ) during ( s , t ] is upper bounded by, c M i ( t ) − c M i ( s ) ≤ − ∆ t M i · C + ( t − s − ∆ t GB − ∆ t ST ) · idSl M i . (19)Additionally, for the sum of credits c < i ( t ) of AVB trafficwith the priority higher than M i , it increases at most as speed ∑ i − j = idSl M j when the frame of M i uses the link (during ∆ t M i )or a low priority frame blocks the link (during ∆ t LP ), c < i ( t ) − c < i ( s ) ≤ ( ∆ t M i + ∆ t LP ) · i − ∑ j = idSl M j + ( t − s − ∆ t GB − ∆ t ST − ∆ t M i − ∆ t LP ) · (cid:18) i − ∑ j = idSl M j − C (cid:19) ≤ ∆ t M i · C + l max > i + ( t − s − ∆ t GB − ∆ t ST ) · (cid:18) i − ∑ j = idSl M j − C (cid:19) . (20)Similarly, considering the system is assumed to be not over-loaded, we have ∑ i − j = idSl M j ≤ C . Thus, from Eq. (20) t − s − ∆ t GB − ∆ t ST ≤ c < i ( t ) − c < i ( x ) − ∆ t M i · C − l max > i ∑ i − j = idSl M j − C . (21)Then from (21), (19) is modified to obtain, c M i ( t ) − c M i ( s ) ≤ idSl M i c < i ( t ) − c < i ( x ) − l max > i ∑ i − j = idSl M j − C . Thus, the upper bound of c M i ( t ) is to conclude, c M i ( t ) ≤ idSl M i · ∑ i − j = c minM j − l max > i ∑ i − j = idSl M j − C . Fig. 4. Guard bands and ST windows
C. Arrival Curve of ST/GB duration
In this section, the arrival curve of ST/GB duration is con-structed. It is used to derive the leftover service curve for AVBtraffic of Class M i , as AVB traffic cannot be transmitted whencredit is frozen. As discussed above, credit will be frozen duringST time slots (windows) depending on the GCL, and whether itis frozen during GB intervals depends on the behavior selection.ST traffic is scheduled within specific ST windows accordingto GCLs. The GCL for the output port h is repeated after theGCL period p h GCL , see the example in Fig. 4. Let N hST be thenumber of ST windows in the GCL period p h GCL . It is assumedthat the i th ST window in output port h starts at o hi and hasduration L hST , i , and the relative offset between the starting timeof the i th and j th ST windows is o hj , i = o hj − o hi For the case ofcredit non-frozen for AVB traffic during guard bands, the arrivalcurve for the credit frozen part is only related to ST windows,which is given by the following Lemma.
Lemma 1:
The arrival curve of the credit frozen part due toST traffic in an output port h is given by, for all t ∈ R + α hST ( t ) = max ≤ i ≤ N hST − (cid:110) α hST , i ( t ) (cid:111) α hST , i ( t ) = i + N hST − ∑ j = i L hST , j C · (cid:38) t − o hj , i p h GCL (cid:39) , (22)where is α hST , i ( t ) is one possible arrival curve by selecting i th( i ∈ (cid:104) , N hST , i (cid:105) ) ST window as the reference; L hST , j · C representsthe maximum number of bits the could be transmitted duringthe ST traffic window of length L hST , j ; each staircase functionshows the upper bound of ST transmission in the periodic STwindows of length L hST , j , and the relative offsets o hj , i give therelationships between different ST windows within the GCLperiod. The proof of the lemma is similar to the proof for TTflows in TTEthernet [18].Moreover, for the case of credit frozen for AVB traffic ofClass M i during GB, the arrival curve for the credit frozen partis not only related to ST windows, but also to GB intervals.A guard band may appear before a ST window, depending onthe flow backlog. In the worst-case, the time duration of theguard band L h , M i GB , j before the j th ( j ∈ [ , N hST − ] ) ST windowequals to the minimum value of the maximum transmission time( l max ≤ i / C ) of AVB frames of Class M i competing in output port h and the idle time interval between two consecutive ST windows( ( j − + N hST ) % N hST th and j th windows). We merge the guardband and ST window together to construct the arrival curve ofthe credit frozen part, the following Lemma is given, Lemma 2:
The arrival curve of the credit frozen part due toST traffic and guard band in an output port h is given by, for all t ∈ R + α h , M i GB + ST ( t ) = max ≤ j ≤ N hST − (cid:110) α h , M i GB + ST , j ( t ) (cid:111) (23) α h , M i GB + ST , j ( t ) = j + N hST − ∑ k = j ( L hST , k + L h , M i GB , k ) C (cid:38) t − o hk , j + L h , M i GB , k − L h , M i GB , j p h GCL (cid:39) , where is α h , M i GB + ST , j ( t ) is one possible arrival curve by selectingthe j th ( j ∈ (cid:2) , N hST − (cid:3) ) ST window as the reference.In the following, we are interested to derive σ M i GB and ρ M i GB defined in Eq. (13) which is used for deriving credit upper boundof AVB Class M i . They are parameters (burst and long-term rate)related to the upper envelope of accumulated bits in guard bandduration ∆ t GB ( s , t ) in any interval [ s , t ] . Additionally, note that σ M i GB and ρ M i GB are parameters defined based on t − s − ∆ t ST ( s , t ) in Eq. (13). The following Theorem gives the upper envelope ofthe guard band defined in Eq. (13) in the form of the staircasefunction. Theorem 4:
The staircase upper bound on the GB durationdefined from Eq. (13) for AVB Class M i traffic for all t ∈ R + is as follows, α h , M i GB ( t ) = max ≤ j ≤ N hST − (cid:110) α h , M i GB , j ( t − ∆ t ST ( t )) (cid:111) , (24)where α h , M i GB , j ( t − ∆ t ST ( t )) = j + N hST − ∑ k = j L h , M i GB , k C t − o hk , j + L h , M i GB , k + ∑ kp = j L hST , p p h GCL − ∑ j + N hST − p = j L hST , p . (25) Proof:
We need to upper bound ∆ t GB ( s , s + t ) the number of L h , M i GB , j in [ s , s + t ] . Introduce ∀ j : u j = o j − L h , M i GB , j . First, considersome u j and t ≥
0, let k = max (cid:8) p u p < u j + t (cid:9) ∆ t GB ( u j , u j + t ) ≤ L h , M i GB , j + . . . + L h , M i GB , k − + L h , M i GB , k (26) = ∞ ∑ k = j L h , M i GB , k { t > u k − u j } (27)Noticing that ∆ t ST ( u j , u k ) = ∑ k − p = j L hST , p , Chasle’s relation ∆ t ST ( u j , u j + t ) = ∆ t ST ( u j , u k )+ ∆ t ST ( u k , u k + t ) and ∆ t ST ( u k , u j + t ) ≤ L hST , k leads to ∆ t GB ( u j , u j + d ) ≤ ∞ ∑ k = j L h , M i GB , k (cid:110) t − ∆ t ST ( u j , u j + d ) > u k − u j − ∑ kp = j L hST , p (cid:111) (28)But any infinite sum on any expression E ( k ) , can be decom-posed as a double sum using k = k (cid:48) + nN hST with k (cid:48) = k mod N hST ∞ ∑ k = j E ( k ) = j + N hST − ∑ k (cid:48) = j ∞ ∑ n = E ( k (cid:48) + nN hST ) (29)The system behavior is periodic, so ∀ k , u k + N hST = u k + p h GCL , L hST , k + N hST = L hST , k , L h , M i GB , k + N hST = L h , M i GB , k , leading to, ∀ t ∞ ∑ k = j (cid:110) t > u k − u j − ∑ kp = j L hST , p (cid:111) = j + N hST − ∑ k (cid:48) = j ∞ ∑ n = (cid:110) t > u k (cid:48) + np h GCL − u j − ∑ k (cid:48) p = j L hST , p − nL h ST,GCL (cid:111) (30)
Fig. 5. Linear bound on staircase function with L h ST,GCL = ∑ N hST − k = L hST , k . Now, remark that ∀ t , J , P ∈ R + , P > (cid:24) t − JP (cid:25) + = ∞ ∑ n = { t > J + nP } . (31)Combining all leads to ∞ ∑ k = j L h , M i GB , k (cid:110) t > u k − u j − ∑ kp = j L hST , p (cid:111) = j + N hST − ∑ k = j L h , M i GB , k (cid:38) t − ( u k − u j − ∑ kp = j L hST , p ) p h GCL − L h ST,GCL (cid:39) + def = F j ( t ) (32)Let s ≥
0, it exists o j such that s ∈ [ o j − , o j ) . If s ∈ [ o j − , u j ] , ∆ t GB ( s , s + t ) = ∆ t GB ( u j , s + t ) . Using eq. (28), (32), ∆ t GB ( s , s + t ) ≤ F j ( s + t − u j − ∆ t ST ( u j , s + t )) . From Chasles’s relation and ∆ t ST ( s , u j ) ≤ u j − s comes s − u j − ∆ t ST ( u j , s + t ) ≤ − ∆ t ST ( s , s + t ) . And since F j is non decreasing, it leads to ∆ t GB ( s , s + t ) ≤ F j ( t − ∆ t ST ( s , s + t )) (33)If s ∈ [ u j , o j ) , ∆ t GB ( s , s + t ) = ∆ t GB ( u j , s + t ) − ( s − u j ) . Butfor any x , y , z , z ≤ y ∆ t GB ( x , x + y ) − z ≤ ∆ t GB ( x , x + y − z ) , so ∆ t GB ( s , s + t ) ≤ ∆ t GB ( u j , u j + t ) ≤ F j ( t − ∆ t ST ( u j , u j + t )) . Con-sider now ∆ t ST : from s ∈ [ u j , o j ] , ∆ t ST ( u j , u j + t ) = ∆ t ST ( s , u j + t ) , and from Chalses’s relation ∆ t ST ( s , s + t ) = ∆ t ST ( s , u j + t ) + ∆ t ST ( u j + t , s + t ) . Since s ∈ [ u j , o j ] , s − u j ≤ o j − u j = L h , M i GB , j , then ∆ t ST ( u j + t , s + t ) ≤ ∆ t ST ( u j + t , o j + t ) ≤ L h , M i GB , j .So, − ∆ t ST ( u j , u j + t ) ≤ − ∆ t ST ( s , s + t ) + L h , M i GB , j , and since F j isnon decreasing, ∆ t GB ( s , s + t ) ≤ F j ( t + L h , M i GB , j − ∆ t ST ( s , s + t )) . (34)Last, to get rid of the position of s wrt o i and u i , one gets themaximum ∆ t GB ( s , s + t ) ≤ max j ∈ N (cid:110) F j ( t + L h , M i GB , j − ∆ t ST ( s , s + t )) (cid:111) (35) = max ≤ j ≤ N hST − (cid:110) F j ( t + L h , M i GB , j − ∆ t ST ( s , s + t )) (cid:111) (36)Last, we are looking for a bound on C · ∆ t ST ( s , s + t ) and t + L h , M i GB , j − u k + u j = t + L h , M i GB , j − ( o hk − L h , M i GB , k )+ o hj − L h , M i GB , j = t − o hk , j + L h , M i GB , k , leading to eq. (25). Lemma 3: (Linear bound on staircase function) Let P ∈ R + , { o , ..., o n } , { l , ..., l n } ∈ R + be two sets of non-negative valuessuch that o ≤ o ≤ ... ≤ o n ≤ P . Let f ( t ) = ∑ nj = l j (cid:108) t − o j P (cid:109) . Thenfor all t ∈ R + f ( t ) ≤ ρ t + max { σ , ... σ n } , (37)with σ k = ∑ kj = l j − ρ o k , ρ = ∑ nj = l j / P . Proof:
For any k , let l k : t (cid:55)→ σ k + ρ t . Since f ( t + P ) = f ( t ) + ρ P , if ∀ u ∈ [ , P ] , f ( u ) ≤ l k ( u ) , then f ≤ l k . Notice that l k ( o k ) = f ( o k ) , and ∀ u ∈ [ o k , o k + ] f ( u ) ≤ max { l k , l k + } (cf. Fig. 5). So, ∀ u ∈ [ , P ] , f ( u ) ≤ max k ∈ [ , n ] { l k ( u ) } . D. Tighter latency bounds by introducing shaping curves forarrival of AVB traffic
According to Network Calculus, the upper bound latency ofa Class M i flow τ M i [ k ] in output port h is given by the maximumhorizontal deviation between the aggregate arrival curve α hM i ( t ) of intersecting flows of AVB Class M i and the service curve β hM i ( I ) ( t ) for AVB Class M i in h , D hM i [ k ] = h ( α hM i ( t ) , β hM i ( I ) ( t )) , (38)where the service curve β hM i ( I ) ( t ) is from Theorem 1.The individual arrival curve for the flow τ M i [ k ] in the sourceES ( h ) can be given by α h M i [ k ] ( t ) = l M i [ k ] + l M i [ k ] p M i [ k ] · t (39)where l M i [ k ] is the burst of the flow τ M i [ k ] in h , l M i [ k ] / p M i [ k ] is thelong-term rate of τ M i [ k ] sending from h . The individual arrivalcurve α hM i [ k ] ( t ) for τ M i [ k ] in the intermediate node is calculatedfrom the previous node port h (cid:48) along the path of τ M i [ k ] , α hM i [ k ] ( t ) = α h (cid:48) M i [ k ] ( t ) (cid:11) δ D h (cid:48) Mi [ k ] ( t ) , (40)where D M i [ k ] is the maximum queuing latency of the flow τ M i [ k ] in output port of h (cid:48) , and δ D ( t ) is the burst-delay function [27]which equals to 0 if t ≤ D and ∞ otherwise.Moreover, we provide a solution that leads to tighter ag-gregate arrival curves for AVB traffic thus reducing pessimismof worst-case delays for AVB flows in intermediate nodes, bytaking flows τ M i [ k ] of Class M i from the same previous node port h (cid:48) as a group. On one hand, such grouped flows are limitedby the physical link speed, meaning that they cannot arrivesimultaneously. The shaping curve of physical link is given by, σ link ( t ) = C · t . (41)On the other hand, the grouped flows are constrained by CBSas well. The shaping curve of CBS σ h (cid:48) M i ( t ) will be discussed inSect. V-E with the consideration of ST effects.Thus the aggregate arrival curve α hM i , h (cid:48) ( t ) of grouped flowsin the input of h from the preceding node port h (cid:48) is constrainedby all three aspects, i.e. , sum of individual output arrival curvesfrom h (cid:48) (Eq. (40)), the shaping curve of physical link (Eq. (41))and the shaping curve of CBS (Eq. (46)). Then it is given by, α hM i , h (cid:48) ( t ) = (cid:18) ∑ τ Mi [ k ] ∈ [ h (cid:48) , h ] α hM i [ k ] ( t ) (cid:19) ∧ (cid:16) σ link ( t ) + l h , maxM i , h (cid:48) (cid:17) ∧ (cid:16) σ h (cid:48) M i ( t ) + l h , maxM i , h (cid:48) (cid:17) , (42)where x ∧ y = min { x , y } , l h , maxM i , h (cid:48) is the maximum frame size offlows of Class M i from h (cid:48) to h .Then the aggregate arrival curve α hM i ( t ) for the output port h is the sum of all grouping arrival curves of Class M i . Moreover,it is assumed that there is no limitation for arriving of flowsin the source ES ( h ). Then they can simultaneously arrive andthus the aggregate arrival curve α h M i ( t ) for the port h is thesum of all individual arrival curves of Class M i flows.By disseminating the computation of latency bounds alongthe routing of τ M i [ k ] , its WCD is obtained by the sum of delaysfrom its source ES to its destination ES, D M i [ k ] = ∑ h ∈ dr Mi [ k ] D hM i [ k ] + ( | h | − ) · d tech , (43) where | h | − d tech is the constant technical latency in a SW. E. CBS Shaping Curve for AVB trafficLemma 4:
With the non-preemption mode, the strict servicecurve for ST traffic in an output port h is given by, β hST ( t ) = min ≤ i ≤ N hST − (cid:26) i + N hST − ∑ j = i β hTDMA ( t + t , L hST , j ) (cid:27) , (44)where β hTDMA ( t , L ) = C · max (cid:26)(cid:22) tp h GCL (cid:23) L , t − (cid:24) tp h GCL (cid:25) ( p h GCL − L ) (cid:27) , and t = p h GCL − L hST , j − o h , i − o hj , i . Proof : As discussed in Sect. V-A, for the given GCL in anoutput port, there are N hST number of ST windows in the GCLperiod p h GCL . Then the service for ST traffic can be taken as N hST sets of periodic windows of known lengths and relative offsets.If considering one set of periodic ( p h GCL ) ST windows oflength L ST , its service in the port h is similar to the TDMAcommunication [28]. In the worst-case, the service cannot beguaranteed for ST traffic during any time interval 0 ≤ ∆ t < p h GCL − L hST , but can be guaranteed of C · ( ∆ t − p h GCL − L hST ) inany time interval p h GCL − L hST ≤ ∆ t < p h GCL . Then the servicecurve for periodic ST windows during ∆ t can be given by, β hTDMA ( ∆ t , L hST )= C · max (cid:26)(cid:22) ∆ tp h GCL (cid:23) L hST , ∆ t − (cid:24) ∆ tp h GCL (cid:25) ( p h GCL − L hST ) (cid:27) . (45)Then the service for all N hST sets of periodic ST windows isderived from (45) as well as the relative offsets between twoST windows from different sets. Taking the i th ( i ∈ [ , N hST − ] )ST window as the reference, we have that the first served frameis from the i th ST window during the backlogged period ∆ t .Assume that o h , i is the maximum idle time interval before theopening time of the i th window, then the service guaranteed forthe i th set of ST windows is the curve in (45) shifted to the leftwith the positive value p h GCL − L hST , i − o h , i , β hST , i , i ( ∆ t ) = β hTDMA ( ∆ t + p h GCL − L hST , i − o h , i , L hST , i ) . Moreover, for the other set of the j th ( j ∈ [ i + , i + N hST − ] )traffic windows, with the known of the relative offset o hj , i byconsidering the i th ST window as the benchmark, the serviceguaranteed for ST traffic in the j th set of windows can be givenby shifting the curve in (45) to the left with the positive value p h GCL − L hST , j − o h , i − o hj , i , β hST , j , i ( ∆ t ) = β hTDMA ( ∆ t + p h GCL − L hST , j − o h , i − o hj , i , L hST , j ) . Note that β hST , j , i ( ∆ t ) equals to β hST , i , i ( ∆ t ) if j = i .Thus if the i th ST window is as benchmark, the strict servicefor ST traffic is as follows after considering all N hST sets ofperiodic ST windows, β hST , i ( ∆ t ) = i + N hST − ∑ j = i β hST , j , i ( ∆ t ) . Then the strict service curve for ST traffic in an output port h is the lower envelope of β hST , i ( ∆ t ) by considering each ST windows in the GCL period as benchmark, β hST ( ∆ t ) = min ≤ i ≤ N hST − { β hST , i ( ∆ t ) } . Theorem 5:
The CBS shaping curve of Class M i for the non-preemption mode is given by, σ hM i ( t ) = (cid:20) t − β hST ( t ) C (cid:21) + ↑ · idSl M i + c maxM i − c minM i , (46)where β hST ( t ) is given by the Lemma 4, and c maxM i and c minM i arerespectively the upper and lower bounds of credit of M i givenby Theorems 3 and 2. Proof : For an arbitrary period of time ∆ t , the variation ofcredit during ∆ t satisfies ∆ c M i = c M i ( t + ∆ t ) − c M i ( t ) = ∆ t + · idSl M i + ∆ t − · sdSl M i = ( ∆ t − ∆ t ) · idSl M i − ∆ t − · ( idSl M i − sdSl M i ) . (47)In the best-case, the frozen duration ∆ t = ∆ t ST is only relatedto ST windows, and has nothing to do with guard bands, for thenon-preemption integration mode. Considering the strict servicecurve of ST traffic expressed in the Lemma 4, ∆ t ST is limitedby, R h ∗ ST ( t + ∆ t ) − R h ∗ ST ( t ) = ∆ t ST · C ≥ β hST ( ∆ t ) . (48)Moreover, ∆ t − can be expressed by, R h ∗ M i ( t + ∆ t ) − R h ∗ M i ( t ) = ∆ t − · C , (49)and ∆ c M i satisfies the relationship as follows, ∆ c M i ≥ c minM i − c maxM i . (50)Due to non-decreasing function R h ∗ M i ( t ) and using the expressions(48), (49), (50) and sdSl M i = idSl M i − C , (47) is modified toobtain, R h ∗ M i ( t + ∆ t ) − R h ∗ M i ( t ) ≤ (cid:20) ∆ t − β hST ( ∆ t ) C (cid:21) + ↑ · idSl M i + c maxM i − c minM i . VI. E
XPERIMENTAL R ESULTS
We have evaluated our proposed improving NetworkCalculus-based WCD analysis for multiple AVB classes withconsideration of both frozen and non-frozen credit during guardband (GB) in TSN (called impNC/TSN) as follows. impNC/TSNis implemented in C++ using the Java kernel of the RTCtoolbox [29], running on a computer with Intel Core i7-8550UCPU at 1.80 GHz and 8 GB of RAM.
A. Synthetic Test Cases
In this section, our evaluation focuses on a topology of 6ESes and 2 SWs, connected via physical links with rates of100Mb/s. The test case (TC1) has 12 ST flows and 10 AVBflows with two classes ( M and M ). Only two classes of AVBtraffic are considered in TC1, as we are interested in comparingour improved method (impNC/TSN) with the other existingmethods [23], [24] from related work, which only support theanalysis for two classes of AVB traffic. The average load of linksis about 14.4% and the maximum link load is about 33.7%. Theidle slopes of Class M and Class M are respectively set to 40%and 20% (1) . (1) More detailed information for TC1 and TC2 can be downloaded from https://zenodo . org/record/3785915 . XrDsH2j7RPZ
TABLE IC
OMPARED RESULTS BY DIFFERENT METHODS
Flow EIA17 ( µ s) NC/TSN18 ( µ s) impNC/TSN ( µ s) Sim ( µ s)AVB1 3798 3449 2456 1448AVB2 3778 3273 2379 1559AVB3 4771 3663 2466 1529AVB4 5230 5635 4442 2699AVB5 4971 4612 3865 1721AVB6 5151 5287 4421 2615AVB7 4969 3744 2331 1306AVB8 4642 4236 3490 1996AVB9 4757 5635 4442 2494AVB10 5005 5287 4421 2436 Fig. 6. Comparison of WCDs with frozen and non-frozen credit during GB
Firstly, we are interested to show the improvement of AVBWCDs in TSN network obtained with our proposed methodby considering physical link and CBS shaping curves in com-parison to existing work [23], [24]. The earliest method pro-posed for AVB analysis in TSN network is based on EligibleInterval Analysis [23], which we denote with EIA17. TheNC-based method from Zhao [24] without shaping constraintsis denoted with NC/TSN18. Additionally, we give simulationresults (denoted as Sim) in order to show the correctness ofthe proposed method to some extent. Note that here we use thesame assumption of the credit variation as related work, i.e.,credit frozen during guard band.The results for each AVB flow of TC1 are presented inTable I, calculated from different timing analysis methods. Aswe can see, compared with our impNC/TSN method, EIA17suffers the larger WCDs, with 43.6% larger on average. This isexpected, since EIA17 does not consider the relative locationsof ST windows. The pessimism of EIA17 will be even largerwith the increasing interval among ST windows. Compared withNC/TSN18, our method by introducing physical link shapingand CBS shaping curves is able to significantly reduce theWCDs compared to [24]. As we can see from the third and forthcolumns in Table I, impNC/TSN is able to reduce on averagethe WCDs obtained with [24] by 23.5% and as much as 37.7%in some cases. Additionally, we give simulation results in thelast column. The simulation results denoted with Sim are usefulfor validating our approach, as our obtained WCDs should all belarger than the maximum delays obtained by simulation, whichis the case in our experiments. However, as rare events can bemissed, a simulation approach will not be able to determine theWCDs, hence it is not useful for safety critical applications thatrequire safety guarantees.For the second experiment, to show the impact of frozen andnon-frozen credit during GB on the WCD of AVB traffic, weuse the same TC1, but just assume different credit variationsduring guard band. We show the results in Fig. 6. The values
Fig. 7. Compared WCDs with different number of ST flows on the x-axis give the AVB flows, from AVB1 to AVB10, andon the y-axis show the WCDs in microseconds. As it can seenin Fig. 6, WCDs of Class M under frozen credit assumptionare larger than the Class M results under the non-frozen creditassumption, while WCDs of Class M are just the opposite. Thisconfirms our intuition, since if considering non-frozen credit,both credits for Class M and Class M will be increased duringGB. Moreover, as Class M has lower priority than Class M ,the increased possibility of M obtaining service means that thepossibility of M getting service is reduced. B. Evaluation on a Larger Realistic Test Case
In this section, we use a larger real-world test case (TC2),adapted from the Orion Crew Exploration Vehicle (CEV) [30]by using the same topology and ST flows, and consideringrate-constrained (RC) flows as AVB flows. We investigate thescalability of our method, show the extension of the analysismethod on multiple AVB classes and investigate the influenceof varied idle slopes on multiple classes of AVB traffic. Withoutloss of generality, the credit is assumed to be frozen duringGB in this subsection. CEV has a topology consisting of 31ESes, 15 SWs, 188 dataflow routes, connected by dataflow linkstransmitting at 1 Gbps, and running 100 ST flows, 25 AVB flowsof Class M , 25 AVB flows of Class M , 24 AVB flows of Class M and 13 AVB flows of Class M (including multicast flows).In such experiment, we would like to evaluate the scalabilityof our analysis impNC/TSN with the number of ST flows. Wereduce the total number of ST flows from 100 to 50 and 20respectively. The idle slope for Classes M i ( i = , , ,
4) arerespectively 40%, 20%, 10% and 5%. The obtained results arevisually shown in Fig. 7 As shown in Fig. 7, the obtained resultsare grouped by the AVB Classes with vertical dotted lines andrespectively sorted in increasing order by results from the casewith 100 ST flows. As expected and can be seen from Fig. 7, theWCDs of the AVB flows grow with the increasing number of STflows. In addition, the average WCD (horizontal dotted lines)of Class M i is smaller than M i + , since the AVB class of higherpriority has larger bandwidth guarantee, i.e., idSl M i > idSl M i + .VII. C ONCLUSION
The TSN IEEE task group is defining new extensions ofEthernet, devoted to real-time and safety-critical applicationareas. These types of applications require a method to boundthe worst-case delays of a given configuration. This paper haspresented an improved Network Calculus-based approach by considering the limitations from physical link rate and the outputof CBS, which are modeled as shaping curves to compute tighterbounds. Moreover, the timing analysis method has be extendedto arbitrary number of AVB traffic classes, considering bothfrozen and non-frozen of credit during guard band, in the caseof a TSN/GCL+CBS architecture.Our analysis is, to the best of our knowledge, the first oneto handle with both credit frozen and non-frozen during guardbands and with any number of AVB classes under the influenceof ST traffic for the whole TSN network. We have evaluatedthe proposed approach on both synthetic and realistic test cases.The comparison results to the existing approaches show that theNetwork Calculus approach is a viable approach for the analysisof TSN. Our approach provides safe upper bounds on WCDs,reduces the pessimism of the analysis (tighter WCD bounds),and is scalable to handle large problem sizes.R EFERENCES[1] IEEE, “802.3 Standard for Ethernet,” 2015.[2] J. D. Decotignie, “Ethernet-based real-time and industrial communications,”
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