End-to-End Latency and Backlog Bounds in Time-Sensitive Networking with Credit Based Shapers and Asynchronous Traffic Shaping

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1 End-to-End Lateny and Baklog Bounds in Time-Sensitive Networking with Credit Based Shapers and Asynhronous Traffi Shaping Ehsan Mohammadpour, Eleni Stai, Maaz Mohiuddin, Jean-Yves Le Boude Éole Polytehnique Fédérale de Lausanne, Switzerland arxiv: v [s.ni] 27 Apr 208 Abstrat We ompute bounds on end-to-end worst-ase lateny and on nodal baklog size for a per-lass deterministi network that implements Credit Based Shaper CBS) and Asynhronous Traffi Shaping ATS), as proposed by the Time- Sensitive Networking TSN) standardization group. ATS is an implementation of the Interleaved Regulator, whih reshapes traffi in the network before admitting it into a CBS buffer, thus avoiding burstiness asades. Due to the interleaved regulator, traffi is reshaped at every swith, whih allows for the omputation of expliit delay and baklog bounds. Furthermore, we obtain a novel, tight per-flow bound for the response time of CBS, when the input is regulated, whih is smaller than existing network alulus bounds. We also ompute a per-flow bound on the response time of the interleaved regulator. Based on all the above results, we ompute bounds on the per-lass baklogs. Then, we use the newly omputed delay bounds along with reent results on interleaved regulators from literature to derive tight end-to-end lateny bounds and show that these are less than the sums of per-swith delay bounds. I. INTRODUCTION Time-Sensitive Networking TSN) is an emerging IEEE standard of the 802. Working Group whih defines mehanisms for bounded end-to-end lateny and zero paket loss []. It speifies a number of per-lass queuing, sheduling and shaping mehanisms. Beause the mehanisms are per-lass, one key issue in this ontext is how to deal with the burstiness asade: individual flows that share a resoure dediated to a lass may see their burstiness inrease, whih may in turn ause inreased burstiness to other flows downstream of this resoure. Computing lateny upper bounds for perlass networks is diffiult, unless flows are reshaped at every hop [2] [5]. This is why a TSN proposal is to reshape flows at every hop, using the onept of interleaved regulator introdued in [6] and analyzed in [7] alled Asynhronous Traffi Shaping, ATS, within TSN). An interleaved regulator reshapes individual flows without per-flow queuing. In [6], an end-to-end delay bound is omputed for a network of FIFO onstant rate servers with aggregate multiplexing that uses interleaved regulators to avoid the burstiness asade. However, this does not aount for the multi-lass nature of a TSN network and for a representative ombination of queuing and sheduling mehanisms proposed by TSN, speifially for the sheme alled Credit Based Shaper CBS). The first goal of this paper is to extend these alulations to a more generi TSN network. However, the alulations in [6] are very omplex; extending them seems to be intratable unless some higher level of abstration is used, as desribed below. The seond goal of this paper is to provide baklog bounds, whih an be used to dimension buffers. To address these goals, we use lassi network alulus onepts suh as a servie-urve haraterization of CBS and extend the results in [8] to inlude high-priority ontrol-data traffi CDT). We ombine this with the max-plus representation of interleaved regulators proposed in [7]. Further, we use the result of [7] that the upper bound on the delay in the ombination of an interleaved regulator following a FIFO system is no greater than the upper bound on the delay of the FIFO system. Overall, in this paper we ompute delay upper bounds for the CBS, the interleaved regulator and end-to-end delay bounds along with baklog bounds for the first two. Our main ontributions are listed below. i) We obtain a servie urve for every AVB lass at a CBS system, extending a similar result in [8] by aounting for the presene of CDT Theorem III.). The servie urves are used to deouple the interleaved regulator from CBS and are essential to obtain the other results mentioned below. ii) We obtain a novel, tight bound for the response time at a CBS subsystem when the input traffi is reshaped by an interleaved regulator Theorem III.2). iii) Using this bound and that an interleaved regulator does not inrease the delay bound of a FIFO system [7], we obtain a delay bound for the interleaved regulator Theorem III.3). iv) We use the delay bound of the interleaved regulator to derive a servie urve for the interleaved regulator and hene a baklog bound at the interleaved regulator. v) We are the first to ompute a tight end-to-end lateny bound for a TSN network of this kind. We show that the endto-end lateny bound obtained is less than the sum of delay bounds omputed at every swith along the path of a flow. Ignoring this, as is often done, leads to a gross overestimation of the worst-ase end-to-end lateny. Setion II desribes the system model. Setion III provides: a servie urve for the CBS subsystem; a novel tight bound on the response time in the CBS subsystem; a delay bound for the interleaved regulator; and a tight end-to-end delay bound. Setion IV uses these results to derive baklog bounds. Setion V provides ase studies, shows the tightness of the bounds and

2 Fig. : Arhiteture of one TSN node output port. the sub-additivity of the end-to-end delay bound. Setion VI onludes the paper. II. SYSTEM MODEL We onsider a network with a set S of nodes swithes and hosts) along with a set of flows, F, between hosts. Hosts are soures or destinations of flows. There are four types of flows, namely, ontrol-data traffi CDT), lass A, lass B, and best effort BE) [9] in dereasing order of priority. Flows of lasses A and B are together referred to as Audio-Video Bridging AVB) flows, as mentioned in [8], [0]. We fous on delay and baklog bounds for AVB traffi. We assume a subset of TSN funtions as desribed next. A. Arhiteture of a TSN node We assume that ontention ours only at the output port of a TSN node. Eah node output port performs per-lass sheduling with eight lasses: one for CDT, one for lass A traffi, one for lass B traffi, and five for BE traffi denoted as BE 0 -BE 4 TSN standard []). In addition eah node output port also performs per-flow regulation for AVB flows using an interleaved regulator. Thus, at eah output port of a node, there is one interleaved regulator per-input port and per-lass [6], [7]. The detailed piture of sheduling and regulation at a node output port is given by Fig.. The pakets reeived at a node input port for a given lass are enqueued in the respetive interleaved regulator at the output port. Then, the pakets from all the flows, inluding CDT and BE flows, are enqueued in a lass based FIFO system CBFS). The CBFS inludes two CBS subsystems [], one for eah lass A and B. As defined in [], [], the CBS serves a paket from a lass aording to the available redit for that lass. The CDT and BE 0 -BE 4 flows in the CBFS are served by separate FIFO subsystems. Then, pakets from all flows are served by a transmission seletion subsystem that serves pakets from eah lass based on its priority. All subsystems are non-preemptive. Guarantees for AVB traffi an be provided only if CDT traffi is bounded; we assume that the CDT traffi from node i to node j has an affine arrival urve r t+b. How to derive suh arrival urves involves other TSN mehanisms and is outside the sope of this paper. Fig. 2 shows a part of a TSN network with three swithes serving four flows of lass A. In swithes SW and SW 2 two flows are oming from two different input ports, thus, they use different interleaved regulators. The flows entering swith SW 3 from swith SW are going to different output ports, and use different interleaved regulators. Fig. 2: Illustration of the queuing poliy by TSN swithes for four flows of lass A. B. Flow Regulation Fig. 3: Timing Model in TSN Following [6], we assume that flows are regulated at their soure, aording to either leaky buket LB) or length rate quotient LRQ). The LB-type regulation fores flow f to onform to the arrival urve r f t+b f. The LRQ-type regulation with rate r f ensures that the time separation between two onseutive pakets of sizes l n and l n+ is at least l n /r f. Note that if flow f is LRQ-regulated, it satisfies the arrival urve onstraint r f t + L f where L f is its maximum paket size but the onverse may not hold). For an LRQ regulated flow we set b f = L f. We also all M f the minimum paket size of flow f. We assume that, at the soure hosts, the traffi satisfies its regulation onstraint, i.e. we an ignore the delay due to interleaved regulator at hosts. At every swith inside the network, flows are reshaped inside the interleaved regulator, as desribed in [7]. The regulation type and parameters for a flow are the same at its soure and at all swithes along its path. C. Other Notation and Definitions The indies for nodes, e.g., i, j, k lie in [, S ]. A direted link from node i to j is denoted by i, j) with a apaity of. Also, n N \ 0 is used as an index for pakets, f F is used as an index for flows, and x A, B, E} is used as an index for AVB lasses A and B, and BE flows, respetively. The set of pakets belonging to flow f is N f. The set of flows of lass x going from node i to node j is denoted by F x and those that ontinue to node k, by F k x. The flows in Fk x use the same CBFS in node i and interleaved regulator in node j. As mentioned in Setion II-A, for eah output port, there is a per-lass per-input-port interleaved regulator. Thereby, the interleaved regulator in node j onneted to link j, k) indiates an output port of node j onneted to node k.

3 The maximum paket size of lass x flows from node i to j is given by L x. Fig. 3 shows the various delays of a paket n of a flow in Fk x. We see five important time instants: ) A n is the arrival time of paket n in CBFS, 2) Q n is the time that paket n starts transmission from CBFS, 3) D n is the time that paket n is reeived at a node, 4) D n is the time that paket n is enqueued in the interleaved regulator, and 5) E n is the time that paket n leaves the interleaved regulator. D n D n ) is the proessing time at node j, whih is defined as the delay from the reeption of the last bit of a paket, oming from node i, to the time the paket is enqueued [ at the interleaved ] regulator. We assume that D n D n T pro,min, T pro,max. D n Q n ) is the output delay for paket n traversing form node i to node j, whih is defined as the time required from the seletion of a paket for transmission from a CBFS queue of node i to the reeption of the last bit of the paket by the node j. Also, D n Q n = l n / +T var,n, where var, min l n is the length of paket n and T var,n var, max is in [T, T ]. We ompute the following bounds for pakets of AVB flows belonging to lass x going from node i to j see Fig. 3): Sf, i, j, x): upper bound on the response time for flow f in CBFS, i.e. on D n A n ); Hf, i, j, k, x): upper bound on the response time for flow f in the interleaved regulator at node j s output port for link j, k), i.e. a bound on E n D n); Ci, j, k, x): upper bound for all flows on the response time in the ombination of the CBFS at node i and the interleaved regulator at node j for link j, k), i.e. a bound on E n A n ) for all flows. III. DELAY BOUNDS IN TSN The aim of this setion is the omputation of bounds on the delays an AVB flow experienes due to CBFS, Sf, i, j, x), and interleaved regulator at a node, Hf, i, j, k, x). To do so, in Setion III-A, we first derive a servie urve of CBFS for an AVB flow, in presene of CDT with an LB arrival urve. Then, in Setion III-B, we use this servie urve to ompute a bound on the response time for an AVB flow in the CBFS of a node, i.e., Sf, i, j, x). Using Sf, i, j, x), we ompute Ci, j, k, x). Consequently, we an ompute the bound on the delay of the interleaved regulator, Hf, i, j, k, x) in Setion III-C, and therefore we have all the elements to ompute a bound on the delay of a single TSN node. We also ompute a tight end-to-end delay bound for an AVB flow in Setion III-D. A. Servie Curve Offered by CBFS to AVB flows The following theorem provides servie urves offered by a CBFS at a TSN node, for AVB flows in presene of CDT flows with LB arrival urve. In [8], the authors ompute servie urves for AVB flows aording to the IEEE AVB standard [0], i.e., in absene of CDT. Note that servie urves for AVB flows in TSN are proposed in [2]; however in their proof redit reset is not onsidered. We obtain different servie urves than [2] and we use them to obtain tight delay bounds. Theorem III.. Assume a node i and a link i, j), where the CDT has an LB arrival urve with parameters r, b ) and the line rate is. Then, the CBFS offers to lass A flows a rate-lateny servie urve with parameters, T A = LA r + b + r L ), ) R A = IA i r ) I A, 2) SA where I A and SA are the idle slope and send slope, orrespondingly, of the CBS for lass A and link i, j), LA = maxl B, LE ), and L = maxl A, LB, LE ). Similarly for lass B flows, CBFS offers a rate-lateny servie urve with parameters, T B L A = L E + L A IA + b + r L ), 3) r R B = IB r ) I B, 4) SB where I B and SB are the idle slope and send slope, orrespondingly, of the CBS for lass B and link i, j). S A The proof is given in the Appendix A. B. Upper Bound on the Response Time in CBFS The rate-lateny servie urve offered by CBFS at node i for link i, j) to lass x A, B} has parameters R x, T x, as alulated in Theorem III.. Also, let b tot,x = f F b x f. Then, the following theorem gives an upper bound on the response time for a flow f at a CBFS of node i. Theorem III.2. A tight upper bound on the response time in the CBFS of node i following the interleaved regulator) for flow f of lass x A, B}, going from node i to j, is: Sf, i, j, x) = T x + btot,x ψ f + ψ f, 5) where the parameter ψ f depends on the flow f and the type of regulator, namely, for LRQ: ψ f = L f and for LB: ψ f = M f. The proof is given in the Appendix B. Remark. Importantly, we should note that the bound on the response time in the CBFS given by Eq. 5) improves the orresponding bound obtained by using the lassial network alulus approah see [3], Theorem.4.2 and Setion.4.3). Speifially, the latter bound is not a per-flow bound and is equal to T x + btot,x R x, whih is always larger than Sf, i, j, x) Eq. 5)) sine ψ f R + ψ f x < 0. We reahed this improved bound by ombining the min-plus representation of servie urve and max-plus representation of regulation [7].

4 It is known from [7] that for all flows belonging to lass x sharing the same CBFS queue at node i and interleaved regulator at node j e.g., for link j, k)), Ci, j, k, x) = sup Sf, i, j, x) + T pro,max. 6) f Fk x Therefore, the following Corollary is a diret result. Corollary III.2.. Assume flows of lass x A, B}, going from node i to j, and enqueued in the interleaved regulator at node j for link j, k). An upper bound, for eah flow, of the ombination of the response time in CBFS of node i following the interleaved regulator of i) and the interleaved regulator at node j for link j, k) is given by: Ci, j, k, x) =T x + btot,x + sup f F x k ψ f ψ f where for LRQ: ψ f = L f and for LB: ψ f = M f. ) + T pro,max, 7) C. Bound on the Response Time in the Interleaved Regulator The following theorem proves an upper bound on the response time in the interleaved regulator, Hf, i, j, x). Theorem III.3. An upper bound on the response time for flow f of lass x A, B} in the interleaved regulator at node j for link j, k) that follows the CBFS of node i is: Hf, i, j, k, x) = Ci, j, k, x) M f The proof is given in the Appendix C. T var,min pro, min T. Remark. It is shown numerially in Setion V, that H is tight for the flow f that aheives the maximum response time at the CBFS, i.e., for whih Sf, i, j, x) = Ci, j, k, x). D. Upper Bound on the End-to-End Delay Assume an AVB flow f routed through the nodes i,..., i k ), where the soure is i and destination is i k. It is assumed that the arrival urves of the generated flows in soure onform to the flows regulation poliies, and thus the flows do not experiene delay at the interleaved regulators of the soure nodes. An upper bound on the end-to-end delay for flow f of lass x, namely, Df x, is, k 2 Df x = Ci j, i j+, i j+2, x) + Sf, i k, i k, x). 9) j= Df x an be easily omputed by using Eqs. 5), 7). In Setion V-E, we show numerially that this bound is tight. IV. BACKLOG BOUNDS In this setion, we determine an upper bound on the baklog for eah AVB lass of interleaved regulator and CBFS. A. Baklog Bound on Interleaved Regulator In network alulus, omputing upper bounds on the baklog requires information on arrival and servie urves [3]. 8) ) Servie Curve Offered by Interleaved Regulator: It is known that a servie urve offered by a FIFO system whih guarantees a maximum delay D, is equal to the impulse funtion δ D t) [7]. The interleaved regulator is a FIFO system, for whih a delay upper bound is omputed in Setion III-C. Therefore, a servie urve offered by the interleaved regulator for lass x, at node j for link j, k), that follows a CBFS of node i is δ Di,j,k,x) t), where Di, j, k, x) = sup f F Hf, i, j, l, x) is omputed using Theorem III.3. k x 2) Arrival Curve of Interleaved Regulator Input: The output flows of the upstream CBFS node i) may not share the same interleaved regulator. Let us onsider the interleaved regulator of node j for link j, k) that follows the CBFS of node i. Suppose that r s and b s are the sum of rates and bursts of the flows f Fk x for x A, B}. In addition, r w and b w are the sum of rates and bursts of the flows that do not use the same interleaved regulator in downstream node with the previous flows. The CBFS offers a rate-lateny servie urve with parameters R x, T x ) to the lass x A, B} Theorem III.). Then, aording to [3], the output arrival urve of the former flows is an LB one, r s t + b out with b out = b s + r s T x + bw R ). x On the other hand, the upstream line has onstant rate,. Therefore, it also enfores an arrival urve to the input of the interleaved regulator equal to t + sup f F L k x f. As the CBFS follows the interleaved regulator, the input arrival urve of the interleaved regulator is, αt) = min t + sup f F x k L f }, r s t + b s + r s T x + b w ) ). 0) 3) Baklog Bound on Interleaved Regulator: The baklog bound is alulated as sup s 0 αs) βs) [3], where αt) is the arrival urve and βt), the servie urve. By replaing the arrival and servie urves obtained in the two previous subsetions, we obtain the baklog bound of the interleaved regulator for lass x A, B} at node j for link j, k) that follows the CBFS of node i, denoted as B IR,x k B IR,x k = min Di, j, k, x) + sup f F x k L f }, and given: r s Di, j, k, x) + b s + r s T x + b ) w R x ), where T x, Rx are omputed in Theorem III.. B. Baklog Bound on Class-Based FIFO System ) Consider all flows f F x. The input of the CBFS for lass x has an arrival urve equal to the sum of all α f. Using Theorem III. and following a proess similar to the one followed for the interleaved regulator, the baklog bound of the CBFS at node i for link i, j) and lass x, denoted as B CBFS,x i,j is defined as δ D t) = 0, 0 t D, else δ D t) = +.

5 Fig. 4: Pratial TSN network used for the ase study. B CBFS,x i,j = f F x b f + f F x V. CASE STUDY r f T x. 2) In this setion, we apply the results obtained in the previous setions to pratial TSN networks Fig. 4, 6). We highlight the tightness of the delay bounds obtained and the sub-additivity property of the end-to-end delay bound. A. TSN Network Setup and Flows We use the network shown in Fig. 4. It onsists of five swithes labeled -5, and five hosts as soures and destinations of flows), namely H -H 5, with five lass A flows f -f 5. Flow f is LRQ regulated with rate r f = 20 Mbps and has maximum paket length L f = Kb. Flows f 2 -f 5 are LRQ regulated with rate 20 M bps and maximum paket length 2 Kb. It is assumed that on eah output port there is a CDT flow with an LB arrival urve 20 Mbps, 4 Kb), and a BE flow with maximum paket length of 2 Kb. As is shown in the Fig. 4, due to irular dependeny among the flows, use of interleaved regulators leads to bounded end-to-end lateny for the flows. Otherwise, burstiness asade of the flows would ause unbounded end-to-end lateny. For ease of presentation, we assume that the CBFS has only three lasses: CDT, lass A, and one BE. Moreover, T var,max and T pro,max are zero in all swithes i, links i, j) and all pakets of a same flow have the same size. The line rate is equal to 00 Mbps. The parameters of CBS are I A = 50 Mbps and SA = 50 Mbps, i, j, H H 5. We are interested in studying the worst ase response time of flow f in CBFS of host H and its orresponding interleaved regulator in swith. Also, we ompute the theoretial end-to-end delay bound of this flow and show its sub-additivity property. B. Computation of Theoretial Bounds We ompute the obtained upper bounds for the response time in CBFS and interleaved regulator for flow f, and the baklog bounds for the host H and swith. Aording to Theorem III.2, the bound on the CBFS response time for flow f in the host H is Sf, H,, A) = 40 µs. Also, from Theorem III.3, the bound on the response time in interleaved regulator for flow f, enqueued in the output port for link, 2) on swith is Hf, H,, 2, A) = 30 µs. Also, the baklog bound for the same interleaved regulator Eq. )) is.4 Kb. The baklog bound for CBFS of lass A in host H is 6.2 Kb Eq. 2)). To ompute the end-to-end delay, we use Eq. 9). Using Eq. 7), we find that CH,, 2, A) = C, 2, 3, A) = C2, 3, 4, A) = C3, 4, H 4, A) = 40 µs, and Sf, 4, H 4, A) = 40 µs. Thus, for flow f of lass A we have the upper bound on delay Df A = 700 µs. C. Numerial Example of Tightness Next, we show how these bounds are tight by presenting a partiular series of paket arrivals as shown in Fig. 5. This figure shows the input and output urves related to f, f 2, CDT and BE flows in host H and swith. A step in the input urve indiates the time of reeption of the entire paket. Aording to Fig. 5a, at time 0 µs, a paket of BE arrives and starts being transmitted. At time 0 + µs, a burst of CDT traffi arrives and then for time t 0 +, CDT traffi ontinues to arrive with rate 20 Mbps up to the time 75 µs. The transmission of CDT traffi at time 0 + is bloked by the transmission of the BE paket as all swithes are nonpreemptive. At time 20 µs, CDT traffi has aumulated a baklog and starts its transmission. From Fig. 5b, we see that time 20 µs is the start of the baklog period of lass A sine a paket of flow f 2 and a paket of f arrive, with first of the two being the former. The first paket of flow f 2 reahes at time 95 µs the interleaved regulator in swith for link, 2) that implies a response time of 75 µs for flow f 2 in the CBFS of host H. The first paket of flow f finishes its transmission at time 60 µs from CBFS in H, due to its earlier blokage by the CDT and f 2 traffi. This implies a response time of 40 µs for flow f in CBFS of H, i.e., equal to the bound in Setion V-B. From Fig. 5, we notie that the worst-ase response time in the interleaved regulator for flow f at swith is for the paket that arrives at time 230 µs. This paket is delared eligible by the interleaved regulator at time 360 µs. This implies the response time of 30 µs in the interleaved regulator that is the upper bound for flow f as omputed in Setion V-B. The maximum response time seen in Fig. 5 for flow f 2 is for its paket that arrives at time 260 µs at the interleaved regulator at swith and is equal to 00 µs. Note that this paket of flow f 2 ould have been delared eligible by the interleaved regulator already at 260 µs but is bloked by preeding pakets of flow f that were not yet eligible at that time. Based on Fig. 5b and 5, we observe that pakets of flow f experiene a maximum delay of 40 µs from the time being enqueued in the CBFS of H to the time being delared eligible by the interleaved regulator at swith equal to CH,, 2, A), omputed in Setion V-B). The maximum observed baklog for lass A used in the CBFS at the output port of H is equal to 4 Kb during times 70 µs to 75µs, whih is 65% of the omputed bound. Furthermore, the maximum baklog observed in the interleaved regulator at output port of swith is equal to 5 Kb during times 230 µs to 260 µs, whih is 43% of the omputed bound. D. Sub-additivity of End-to-End Delay Bound In TSN, the ommon way of omputing the end-to-end delay bound is by adding the delay bounds of eah swith in the path of a flow. However, Eq. 9) provides a muh better upper bound. To show this, we first ompute the delay upper bound for swith i following swith j and followed by swith k in the path of a flow f of lass x, is given by,

6 Data Kb) CBFS of Host H 3 Input of CDT flow 2 Input of BE Output of CDT flow Output of BE Time µs) a) CDT and BE flows Data Kb) CBFS of Host H Input of f Input of f 2 Output of f Output of f Time µs) b) f and f 2 flows lass A) Data Kb) Interleaved Regulator of Swith Input of f Input of f 2 Output of f Output of f Time µs) ) f and f 2 flows lass A) Fig. 5: Cumulative data input and output urves for the CBFS of H and interleaved regulator of swith, with respet to Fig. 4). Figures a) and b) show the data arrival and departures from the CBFS of H, and the figure ) shows the arrival and departure of data from the interleaved regulator for flows f and f 2 in swith. Fig. 6: TSN network for tightness of end-to-end delay bound. are rigorously proven, while we provide a representative ase study that highlights the tightness of the delay bounds provided and shows the sub-additivity of the end-to-end delay bound. Future work will address other mehanisms in TSN. REFERENCES d x,f pro,max j,i,k =Hf, j, i, k, x) + Sf, i, k, x) + Ti, 3) where for i being a soure, Hf, j, i, k, A) is equal to zero. Considering T pro,max i equal to zero in this ase, an end-to-end delay bound for flow f over the path H, ),, 2), 2, 3), 3, 4), 4, H 4 ) an be omputed as ) ) = 220 µs. From Setion V-C, we know that an upper bound on the endto-end delay for flow f is 700 µs whih is 57% of 220µs, obtained using Eq. 3. A similar phenomenon ours with per-flow networks, alled pay bursts only one. E. Tightness of End-to-End Delay Bound Consider the network shown in Fig. 6, having four swithes labeled - 4, and six hosts, namely H - H 6, with six lass A flows f - f 6. The assumptions on f f 5, CDT and BE traffi are as in Setion V-A and f 6 is similar to f 2 f 5. To show the tightness of end-to-end delay bound of Eq. 9), we laim that eah pair of f, f 3 at swith, f, f 4 at swith 2, f, f 5 at swith 3, and f, f 6 at swith 4 experiene the same input/output urves as the pair of flows f, f 2 in Fig. 5 but appropriately shifted in time, so that they take plae sequentially. Thus, flow f has a delay of 40 µs from the time being enqueued in the CBFS of H to the time delared eligible from the interleaved regulator at. The same delay is experiened by flow f at the rest pairs of swithes in its path. Similar to Setion V-C, the response time of f at CBFS of swith 4 is equal to 40 µs. Therefore, the end-to-end delay for f is equal to = 700 µs, whih is equal to the bound omputed from Eq. 9). VI. CONCLUSION We have provided a set of formulas for omputing bounds on end-to-end delay and baklog for lass A and lass B traffi in a TSN network that uses CBS and ATS. The bounds [] Time-sensitive networking TSN) task group. [Online]. Available: [2] A. Charny and J.-Y. Le Boude, Delay bounds in a network with aggregate sheduling, in Quality of Future Internet Servies. Springer, 2000, pp. 3. [3] J. C. Bennett, K. Benson, A. Charny, W. F. Courtney, and J.-Y. Le Boude, Delay jitter bounds and paket sale rate guarantee for expedited forwarding, IEEE/ACM Transations on Networking TON), vol. 0, no. 4, pp , [4] M. Boyer and C. Fraboul, Tightening end to end delay upper bound for afdx network alulus with rate lateny fifo servers using network alulus, in IEEE Int l Workshop on Fatory Communiation Systems WFCS), 2008, pp. 20. [5] A. Bouillard and G. Stea, Exat worst-ase delay for fifo-multiplexing tandems, in the 6th Int l Conferene on Performane Evaluation Methodologies and Tools VALUETOOLS), 202, pp [6] J. Speht and S. Samii, Urgeny-Based Sheduler for Time-Sensitive Swithed Ethernet Networks, in the 28th Euromiro Conferene on Real-Time Systems ECRTS), Jul. 206, pp [7] J.-Y. Le Boude, A Theory of Traffi Regulators for Deterministi Networks with Appliation to Interleaved Regulators, arxiv: [s], Jan [Online]. Available: [8] J. A. R. De Azua and M. Boyer, Complete modelling of avb in network alulus framework, in the 22nd ACM Int l Conf. on Real- Time Networks and Systems RTNS), NY, USA, 204, pp [9] S. Thangamuthu, N. Coner, P. J. L. Cupers, and J. J. Lukkien, Analysis of ethernet-swith traffi shapers for in-vehile networking appliations, in Design, Automation & Test in Europe Conf. & Exhibition DATE), 205, pp [0] IEEE standard for loal and metropolitan area networks audio video bridging AVB) systems, pp. 45. [] IEEE 802.: 802.qbv - enhanements for sheduled traffi- draft 0.0. [Online]. Available: Aessed:29/03/208) [2] L. Zhao, P. Pop, Z. Zheng, and Q. Li, Timing analysis of avb traffi in tsn networks using network alulus, in Real-Time and Embedded Tehnology and App. Symp., ser. RTAS 8. IEEE, 208, pp [3] J.-Y. Le Boude and P. Thiran, Network Calulus: A Theory of Deterministi Queuing Systems for the Internet. Springer Siene & Business Media, 200, vol [4] J. Liebeherr, Duality of the max-plus and min-plus network alulus, vol., no. 3, pp , 207.

7 APPENDIX A PROOF OF THEOREM III. Proof. In this proof, for the ease of presentation, the indies i and j are negleted from the parameters. For a given lass x A, B}, V x u) stands for the amount of CBS redit at time u, whih is less than maximum redit, namely V x,max. Also, let N x u), O x u) be the umulative input, output traffi, respetively, for lass x at time u and O H u) the umulative output CDT traffi. Given a time t, we define the time s, as follows, } s = sup u t V x u) = 0 and O x u) = N x u). 4) In the rest of the proof, we use t = t s as the time interval over whih the system is analyzed. For a given lass x, we also define the time intervals t x, t x+, and t x0, as the aggregated time periods within t, over whih the redit is dereasing due to paket transmission of this lass), the redit is inreasing due to transmission of BE traffi or AVB traffi exept lass x in baklog period of this lass or redit reovery), and the redit is frozen due to transmission of CDT flows), respetively. Then, V x t) V x s) = V x t) = t x+ I x + t x S x. 5) As for a lass x A, B}, t = t x+ + t x + t x0, then, t x = Ix t V x t) I x t x0 I x S x. 6) The number of bits served during t for lass x A, B} is t x. By utilizing Eq. 6), O x t) O x s) = I x S x Ix t V x t) I x t x0 ). 7) t x0 is the aggregated transmission time periods of CDT flows during t; therefore, t x0 = OH t) O H s). 8) The servie urve offered to an aggregate of CDT flows in CBFS at time t > s is β H t) = [t L/] +, where L = maxl A, LB, LE ). As the CDT queue has an LB arrival urve denoted as α H = rt + b), the following equation holds for the output arrival urve of the CDT queue: O H t) O H s) α H β H ) t) = b + r t + L ). 9) Therefore, from Eqs. 8) and 9), L t x0 b + r t + ). 20) From Eqs. 7) and 20), and knowing that O x s) = N x s) from the assumption for s): O x t) N x s) Ix r) I x S x t V x t) I x r) b + r ) L. r 2) As V x t) V x,max, and using Eq. 2), we define the following rate-lateny servie urve for lass x A, B}: [ β x t) = Ix r) I x S x t V x,max I x r) b + r ] + L. 22) r Finally, we replae in the last equation the redit upper bound given in [8] for eah lass x A, B}. For ompleteness: V A,max = L A IA, 23) ) V B,max = IB L E + L A A IA L S A, 24) and we obtain the equations in the statement of Theorem III.. APPENDIX B PROOF OF THEOREM III.2 Proof. First we ompute the bound and then we show that the bound is tight. For the omputation of Sf, i, j, x), f, i, j, x, we use the waiting time of the n th paket assuming it belongs to the flow f at the CBFS, W n, f, i, j, x), omputed in Lemma B.: D n A n = Q n A n ) + D n Q n ) 25) W n, f, i, j, x) + D n Q n 26) W n, f, i, j, x) + l n. 27) If flow f is LB-regulated, by Lemma B. we have thus in this ase W n, f, i, j, x) = T x + btot,x D n A n T x + btot,x R x T x + btot,x R x ) + l n ) + M f l n, 28) 29), beause <. This shows Eq. 5) in this ase. If flow f is LRQ-regulated, by Lemma B. we have thus in this ase W n, f, i, j, x) = T x + btot,x D n A n T x + btot,x T x + btot,x L f L f This shows Eq. 5) in this ase. 30) L f, 3) + l n 32) + L f. 33)

8 Now we show that the bound is tight via analyzing an example illustrated in Fig. 7. For the ease of presentation, we drop the indies i, j from the terms, R x, T x, btot,x, L x, L x, Ix, Sx, b, r,. We assume L E = L = L A. At time s 0, a paket of BE with size L E arrives and starts being transmitted. At time s + 0, a burst b of CDT traffi arrives and then for time t > s + 0, CDT traffi ontinues to arrive with rate r up to the time s. The transmission of CDT traffi at time s + 0 is bloked by the transmission of the BE paket, sine it is assumed that lower priority queues are non-preemptive. At time s = s 0 + LE, CDT traffi has aumulated a baklog of b + r LE and starts its transmission. The transmission time of this amount of CDT traffi is τ = b+r LE. During τ, aording to arrival urve of CDT, rτ bits are added to CDT queue. Similarly, during rτ, r 2 τ bits are added to the CDT queue, et., and this ontinues until the time that the CDT queue empties, s. At s, lass A an start transmitting its pakets. The time interval between s and s an be alulated as follows: T s s = τ + rτ + r rτ ) = r τ = +... = τ + r + r )2 +...) E r b + r L 34) ) = b + r L E r. 35) At time s +, there is also arrival of a burst of lass A traffi with b tot,a = L A + l 2 and assuming that l 2 is a paket of flow f that omes seond in the burst. We are interested in the response time of the paket of this flow, f, in the CBFS of swith i. Finally, at time s 3 a paket of BE arrives and starts being transmitted, as lass A has still negative redit and the baklog of CDT is zero. Finally, at time s 4, CDT traffi arrives aording to its arrival urve i.e., rt s s 4 if assuming that rt s s 4 < b, with T s s 4 = LA IA S A ) + L A ) as shown I A in Fig. 7) and starts being transmitted. Similarly with T s s, we an ompute that T s4 s 5 = rts s 4 r. Due to the priority of CDT over lass A and of the latter over BE along with the traffi arrivals desribed above, the redit evolution of lass A is shown in Fig. 7. The transmission of the paket l 2 of flow f an only start at time s 5. As a result, assuming that T var,max = 0, the CBFS response time of this paket based on this example is: S exp = b + r L ) + T s s r 4 + rt s s 4 + l 2 r = b + r L r + L A) + L A I A S A ) r I A + l 2. 36) The theoretial bound on the response time of this paket is, Sf, i, j, A) = T A + btot,a ψ f = r R A b + r L + L A) + r + ψf I A S A ) I A L A + l 2 ψ f ) + ψf. 37) From our assumptions on this example, LA = L. Sine Sf, i, j, A) is proven to be the bound, it holds that and for tightness it should hold Sf, i, j, A) S exp, 38) Sf, i, j, A) = S exp. 39) Therefore, for tightness, we require that l 2 = ψ f. If the flow is LB-regulated, the bound is tight if l 2 = ψ f = M f, and if it is LRQ-regulated, the bound is tight with l 2 = ψ f = L f. In the ase of LRQ, l 2 = L f implies b tot,a = L A + L f ; i.e. for LRQ there should be at least two flows in queue of lass x so that the bound for the CBFS waiting time is ahieved sine in ase of LRQ flow f has an arrival urve at the CBFS with burstiness L f ). Lemma B.. Consider the n th paket in the CBFS queue of swith i, assuming it belongs to flow f of lass x A, B}, going from swith i to j. An upper bound on the waiting time of this paket of flow f in the CBFS of swith i following the interleaved regulator of i) is: W n, f, i, j, x) = T x + btot,x ψn f, 40) where the parameter ψ f n depends on the n th paket that belongs to flow f and the type of regulator, namely, for LRQ: ψ f n = L f and for LB: ψ f n = l n, where l n stands for the paket size of the n th paket. Proof. We do the proof for a flow of lass x going from swith i to j. For the ease of presentation, we drop the indies i, j, x from the terms, R x, T x, btot,x. Assume that, l k stands for the paket size of the k th paket in the AVB FIFO queue at the CBFS. From Lemma B.2, the following equation is true for a servie urve βt) and m n, l k βq n A m ). 4) Let β t) be the upper pseudo-inverse of a servie urve [4] βt), defined as β t) = sups 0 βs) t} = infs 0 βs) > t}. Then, by taking upper pseudo-inverse in both sides of Eq. 4), Q n A m β l k ). 42) Therefore, Q n satisfies, Q n max A m + β ) } l k. 43) m n Let F k) be the flow id of k th paket and f be the flow id of the head-of-line paket that is urrently being served in the

9 CBFS, say paket n. Then, we an express the sum of paket lengths as, l k = [ ] F k)=f }l k + F k)=f} l k. f f 44) As the input flows are regulated, for flows f f there is an arrival urve α f that satisfies, F k)=f }l k α f A n A m ). 45) Also, for flow f that is being served now, we show that F k)=f} l k α f A n A m ) ψ f n, 46) where ψn f is the parameter defined in the statement of the lemma. For the onsidered regulators, LB and LRQ, α f, f are of the form r f t + b f where for the LRQ b f = L f. ψn f is omputed as follows. For the LB, F k)=f}l k + l n α f A n A m ), thus ψn f = l n. For the LRQ, based on its definition, Therefore, or A k+ A k l k r F k). 47) F k)=f} l k r f A n A m, 48) F k)=f} l k r f A n A m ). 49) Sine for LRQ, α f t) = r f t + L f, we obtain ψn f = L f. By utilizing Eqs. 44), 45) and 46) in Eq. 43), we obtain, Q n max A m + β ) } α f A n A m ) ψn f. 50) m n By setting A n A m = t 0, we obtain, Q n A n sup t + β ) } α f t) ψn f t 0 = sup t 0 = sup t 0 f t + t + f f α f t) ψf n R f α f t) R } + T } 5) ψf n R + T 52) f α f 0) ψf n R R + T 53) = btot R ψf n + T. 54) R Fig. 7: Senario to have tight bound on response time of CBFS for lass A The right part of Eq. 54) gives W n, f, i, j, x) in the statement of the lemma. Lemma B.2. If β is a servie urve, then for eah paket n, there exists an m n suh that, βq n A m ) l k, where l k is the size of the k th paket. Proof. From the definition of a servie urve [3], we have t, s : Ot) Ns) + βt s), 55) where Ot) is the number of bits that have been served until time t and Nt) is the number of bits that have arrived until time t and β ) 0. Let m suh that, A m < s A m, then, Then, for some δ 0, we have Ns) = l l m. 56) s = A m δ 57) = βt s) = βt A m + δ) βt A m ). 58) By replaing Eq. 57) and 58) in 55), we obtain: Ot) NA m δ) + βt A m ). 59) By setting t = Q + n, we obtain t as the time instant we begin to serve paket n. This means, that all pakets before n have already been served. Thus, Ot) = l l. 60) By replaing Eqs. 56), 60) in 59), we obtain: l l l l m + βq n A m ) = βq n A m ) l m l. 6) If m > n, we obtain βq n A m ) < 0, whih is a ontradition.

10 APPENDIX C PROOF OF THEOREM III.3 Proof. Hf, i, j, k, x) is an upper bound on E n D n for all pakets n of flow f. Now E n D n = E n A n ) D n A n ), 62) and E n A n Ci, j, k, x). 63) Therefore, for a paket n of flow f: now E n D n Ci, j, k, x) inf n N f D n A n ), 64) D n A n M f + T var,min + T pro, min. 65)