4.1 Modeling of network and traffic

A real-time avionics network is composed of a set of end systems and switches which are interconnected through physical links. Each multiplexing point (i.e. output port of an end system or a switch) can be modeled as a node. A node can be seen as a FIFO buffer which is working at the maximum servicing rate G = 100 Mb/s. All these nodes constitute a node set denoted as S = {N ₁, N ₂,…, N _p}. The switching fabric delay is denoted as L which is upper-bounded to 16µs^{(Reference Kemayo, Ridouard and Bauer8,Reference Benammar, Ridouard and Bauer16)} .

Packet flows transmitting through switches are composed of a flow set denoted as Г = {v ₁,v ₂,…,v _n}. The subset ${\Gamma_h}$ includes all the flows passing through node h. The transmission path ${{\mathscr{P}}_i}$ of flow v _i is statically defined as an ordered node list from the source node (first _i) to the destination node (last _i). Here, the source node (first _i) is the transmitting end system where the flows are generated. The destination node (last _i) is not the receiving end system but the output port of the last visited switch along the transmission path ${{\mathscr{P}}_i}$ .

Generally, a flow v _i can be characterised as follows:

1. • C _i, the maximum transmission time of a packet generated by flow v _i. It is equal to the maximum packet length ${F_{max}}$ divided by the switch servicing rate G:

   \begin{align*}{C_i} = {{{F_{max}}} \over G}\end{align*}

2. • T _i, the minimum time interval between two consecutive packets generated by flow v _i. It corresponds to the BAG of a given VL in AFDX.

3. • ${{\mathscr{P}}_i}$ , the transmission path of flow v _i. An ordered list of the crossed nodes from the source node (first _i) to the destination node (last _i).

4. • L, the switching fabric delay. It is upper-bounded to 16µs.

The Forward Approach can be used for any other networks which can be abstracted as this model, only if all the flows have the same characteristics above.

4.2 Principles

Provided that m _i is a packet generated by flow v _i, in order to compute the worst-case end-to-end transmission delay, the Forward Approach (FA) iteratively analyses the maximum backlog at each node of path ${{\mathscr{P}}_i}$ . It estimates the maximum transversal delay from the generation time at its source node to the departure time from each visited node h of path ${{\mathscr{P}}_i}$ . We denote this delay $R_i^h$ . The estimation propagates until the destination node (last _i), so that the worst-case end-to-end transmission delay can be computed^{(Reference Kemayo, Ridouard and Bauer8)}.

Suppose node h belongs to path ${{\mathscr{P}}_i}$ , and node h + 1 follows node h along path ${{\mathscr{P}}_i}$ . As each node follows the FIFO scheduling policy without considering priority, packet m _i cannot be delayed by the packets arriving after itself, $S{\rm{max}}_i^h$ can be estimated iteratively as depicted in Fig. 2^{(Reference Kemayo, Ridouard and Bauer8,Reference Kemayo, Benammar and Ridouard12)} .

Figure 2. Iterative estimation of $S{\rm{max}}_i^h$ and $R_i^h$ .

We define the generation time of packet m _i at the source node (first _i) is the origin time, so that we have $S\!\max _i^{\,firs{t_i}} = S\!\min _i^{\,firs{t_i}} = 0$ . $S{\rm{max}}_i^{h{\rm{ + 1}}}$ can be obtained as below if getting the value of $S{\rm{max}}_i^h$ ^{(Reference Kemayo, Ridouard and Bauer8)}:

(1) \begin{align}S\!\max\nolimits_i^{h + 1} = S\!\max\nolimits_i^h\, + \,Bklg_i^h\, + \,{C_i}\, + \,L\end{align}

This computation propagates until the destination node (last _i), the worst-case end-to-end delay R _i of the packet m _i is equal to the maximum transversal delay $R_i^{las{t_i}}$ . It can be expressed as Eq. (2).

(2) \begin{align}{R_i} = R_i^{las{t_i}} = S\!\max\nolimits_i^{las{t_i}}\, + \,Bklg _i^{las{t_i}}\, + \,{C_i}\end{align}

To compute an upper bound of the end-to-end transmission delay of flow v _i we need to determine the maximum backlog $Bklg_i^h$ at each node h along path ${{\mathscr{P}}_i}$ .

For $S{\rm{min}}_i^h$ , it occurs if there are no pending packets in the output buffer when packet m _i arrives. Thus packet m _i can be immediately forwarded once it arrives. When there is no backlog at each visited node, $S{\rm{min}}_i^h$ can be obtained as Eq. (3) where ${\left| h \right|^i}$ denotes the number of nodes crossed by packet m _i from first _ito h:

(3) \begin{align}\left\{ \begin{array}{ll}{S{\rm{min}}_i^{h{\rm{ + 1}}} = S{\rm{min}}_i^h + {C_{{\rm{i}}}} + L} \\[5pt] {{\rm{}}S{\rm{min}}_i^h = \left( {{{\left| h \right|}^i} - 1} \right) \times \left( {{C_i} + L} \right)} \end{array} \right.\end{align}

The Forward Approach (FA) evaluates the maximum backlog $Bklg_i^h$ at each node h along path m _i using $S{\rm{min}}_i^h$ and $S{\rm{max}}_i^h$ , and the worst-case end-to-end transmission delay R _i can be computed^{(Reference Benammar, Ridouard and Bauer16)}.

4.3 Worst-Case scenario with the maximum backlog

This section presents the worst-case scenario which can construct the maximum backlog. We focus on packet m _i generated by flow v _i. The maximum backlog $Bklg_i^h$ which can delay packet m _i at node h is bounded by the time window from its earliest arrival time ( $S{\rm{min}}_i^h$ ) to its latest arrival time ( $S{\rm{max}}_i^h$ ).

The corresponding worst-case scenario is depicted as Fig. 3.

Figure 3. Worst-case scenario that packet m _jarrives at node h simultaneously with other k previous packets.

Proof To construct the maximum backlog, the interfering flow v _j should generate the maximum competing traffic. To achieve this, packets of flow v _j should be generated at the source node (first _j) as quickly as possible, hence, flow v _j should generate packets at the minimum time interval T _j, which proves item (i).

Suppose packet m _j of flow v _j arrives at node h $\varepsilon$ earlier than b. When packet m _i arrives at node h strictly at time b as depicted in Fig. 4 (a), it can be forwarded at time $b + {C_j} - \varepsilon $ . It means packet m _i suffers ${C_j} - \varepsilon $ delay which is marked as a black bar. If packet m _j of flow v _j arrives at node h strictly at time b as depicted in Fig. 4 (b), when packet m _i arrives at node h strictly at time b, it can be forwarded at time $b + {C_j}$ . It means packet m _i suffers ${C_j}$ delay (also marked as a black bar) which is longer than ${C_j} - \varepsilon $ .

Figure 4. Packet m _j delays packet m _i.

No matter how earlier than b packet m _j arrives at node h, there is always a worse scenario that packet m _j arrives at node h strictly at time b, and this scenario can cause a longer delay to packet m _i. Item (ii) is proved.

If the last packet m _j arrives at node h suffering a delay d ( $S{\rm{min}}_i^h \lt d \lt S{\rm{max}}_i^h$ ), the time difference between the arrivals of the last two packets (m _j and its previous packet) in the packet sequence $\omega$ is larger than when packet m _j arrives suffering the minimum delay $S{\rm{min}}_i^h$ , so that the backlog is not maximised. Some packets in the packet sequence $\omega$ will not arrive simultaneously with packet m _j. To ensure the maximum backlog they should suffer the maximum delay to arrive at node h. Item (iii) is proved.

Since packet $m^{\prime}_{j}$ arrives at node h in the time interval [a,b] suffering the maximum delay $S{\rm{max}}_j^h$ , to ensure more packets can arrive at node h during this time interval, all the packets before $m^{\prime}_{j}$ have to suffer the maximum delay $S{\rm{max}}_j^h$ . Item (iv) is proved.

For item (v), these k packets arrive at node h no earlier than $m^{\prime}_{j}$ , based on item (ii) they should arrive at node h simultaneously with m _j to ensure the maximum delay to packet m _i. The last item is proved.

Generally, we can analyse the packet sequence $\omega$ from back to front.

1. • The last packet m _j arrives at node h strictly at the end of the time interval [a,b], suffering the minimum delay ${\rm{}}S{\rm{min}}_j^h$ .

2. • There are k packets before packet m _j, these k packets arrive at node h simultaneously with packet m _j suffering different delays between ( $S{\rm{min}}_j^h$ , $S{\rm{max}}_j^h$ ).

3. • There is a packet $m^{\prime}_{j}$ arriving at node h earlier than the end of the time interval [a,b], suffering the maximum delay $S{\rm{max}}_j^h$ .

4. • The other packets before $m^{\prime}_{j}$ in the packet sequence $\omega$ arrive at node h suffering the maximum delay $S{\rm{max}}_j^h$ .

Proof Suppose that packet $m^{\prime}_{j}$ arrives at node h at time $\theta$ depicted in Fig. 3. From item (ii) and (iii) we have $\theta$ < b. Packet $m^{\prime}_{j}$ suffers the maximum delay $S{\rm{max}}_j^h$ . The next packet is generated T _j after $m^{\prime}_{j}$ , and it won’t suffers a delay longer than $S{\rm{max}}_j^h$ , so that the arrival time difference between those two packet ( $m^{\prime}_{j}$ and the following one) is not longer than T _j. This following packet arrives at node h simultaneously with packet m _j, thus the arrival time difference between packets $m^{\prime}_{j}$ and m _jis not longer than T _j. Theorem 2 is proved.

Proof Item (iv) demonstrates that all the packets before $m^{\prime}_{j}$ arrive at node h suffering the maximum delay $S{\rm{max}}_j^h$ . Since packet $m^{\prime}_{j}$ also suffers the maximum delay, and their generation interval is T _j, so that they arrive at node h one after another with a time interval T _j. Theorem 3 is proved.

4.4 Request bound function

The maximum backlog generated by flow v _j at node h during a time interval [a,b] has been presented in Theorem 1. In order to determine this backlog, we define the Request Bound Function (RBF) and the cumulative function W(t) in this section.

The Request Bound Function $RBF_j^h{\rm{(}}t{\rm{)}}$ defines the total transmission time of packets generated by flow v _j arriving at node h during a time interval of length $t(t{\rm{ }} = {\rm{ }}b - a)$ . The cumulative function ${W^h}{\rm{(}}t{\rm{)}}$ defines the total transmission time of packets generated by all the flows passing through node h during a time interval of length t. We have^{(Reference Kemayo, Ridouard and Bauer8)}:

(4) \begin{align}{{\rm{W}}^{\rm{h}}}{\rm{(t)}} = \sum\nolimits_{{\rm{v}}_{\rm{j}}\in _{\Gamma_{\rm{h}}}} {{\rm{RBF}}_{\rm{j}}^{\rm{h}}{\rm{(t)}}} \end{align}

The computation of $RBF_j^h{\rm{(}}t{\rm{)}}$ has to be consistent with the scenario built as Theorem 1. As depicted in Fig. 3, the backlog at node h generated by flow v _j is composed of two parts:

1. (a) The periodic part ending with packet $m^{\prime}_{j}$ ( $m^{\prime}_{j}$ is included). Packets in this part arrive at node h one after another with a time interval T _j.

2. (b) The simultaneous part ending with packet m _j (m _jis included). Packets in this part arrive at node h simultaneously with m _j.

LEMMA 1 ^{(Reference Benammar, Ridouard and Bauer16)} The maximum backlog generated by flow v _j at node h is achieved when packet $m^{\prime}_{j}$ arrives at node h at time $\theta$ . Suppose $\theta$ is ${\rm{\Delta }}_j^h$ earlier than b, we have:

(5) \begin{align}{\rm{\Delta }}_{\rm{j}}^{\rm{h}} = \left( {{\rm{k}} + 1} \right) \times {T_j} - {\rm{J}}_{\rm{j}}^{\rm{h}}\end{align}

Where $J_j^h = S{\rm{max}}_j^h - S{\rm{min}}_j^h$ is the worst-case arrival jitter of flow v _j, and k is the number of the packets arriving at node h simultaneously with packet m _j.

Proof Consider the generation time of packet $m^{\prime}_{j}$ as the reference time ${t_{ref}}$ . Its arrival time at node h can be expressed as:

(6) \begin{align}\theta = {{\rm{t}}_{{\rm{ref}}}}{\rm{}} + {\rm{Smax}}_{\rm{j}}^{\rm{h}}\end{align}

As there are k+1 time intervals T _jfrom $m^{\prime}_{j}$ to m _j as depicted in Fig. 3, and packet m _j arrives at node h suffering the minimum delay $S{\rm{min}}_j^h$ , the time instant b can be expressed as:

(7) \begin{align}{\rm{b}} = {{\rm{t}}_{{\rm{ref}}}} + {\rm{(k + 1) \times }}{{\rm{T}}_{\rm{j}}} + {\rm{Smin}}_{\rm{j}}^{\rm{h}} \end{align}

and Lemma 1 can be proved as below:

(8) \begin{align} {\rm{\Delta }}_j^h & = b - {\rm{}}\theta \nonumber\\ & = (k + 1) \times {T_j} + S{\rm{min}}_j^h - S{\rm{max}}_j^h \\ & = (\rm{k} + 1) \times {{\rm{T}}_{\rm{j}}} - {\rm{J}}_{\rm{j}}^{\rm{h}}\nonumber\end{align}

Proof If ${\rm{\Delta }}_j^h{ = \rm{ 0}}$ , it means that packet $m^{\prime}_{j}$ arrives at node h simultaneously with m _j, this is contrary to item (iii) of Theorem 1 which demonstrates packet $m^{\prime}_{j}$ arrives earlier than m _j. If ${\rm{\Delta }}_j^h{\rm{ \lt 0}}$ , it means that packet $m^{\prime}_{j}$ arrives at node h later than m _j as depicted in Fig. 5. Packet m _j is generated after $m^{\prime}_{j}$ , with the FIFO scheduling policy, it definitely arrives at node h after $m^{\prime}_{j}$ . It is impossible that ${\rm{\Delta }}_j^h{\rm{ \lt 0}}$ .

Figure 5. Packet $m^{\prime}_{j}$ arrives later than m _j.

LEMMA 3 ^{(Reference Kemayo, Ridouard and Bauer8)} Let k is the number of the packets arriving at node h simultaneously with packet m _j. In the scenario with worst-case backlog, k satisfies:

(9) \begin{align}{\rm{k}}{{\rm{T}}_{\rm{j}}}{\rm{}} \le {\rm{J}}_{\rm{j}}^{\rm{h}}{\rm{ \lt (k + 1)}}{{\rm{T}}_{\rm{j}}}\end{align}

Proof LEMMA 2 has proved that ${\rm{\Delta }}_j^h{\rm{ \gt 0}}$ , from Eq. (8) we can get ${\rm{(}}k{\rm{ + 1) \times }}{T_j} - J_j^h{\rm{ \gt 0}}$ , and $J_j^h{\rm{ \lt (}}k{\rm{ + 1)}}{T_j}$ , the right part of Eq. (9) is proved. Theorem 2 demonstrates that ${\rm{\Delta }}_j^h \le {T_j}$ , it means that ${\rm{(}}k{\rm{ + 1) \times }}{T_j} - J_j^h \le {T_j}$ , and $k{T_j} \le J_j^h$ , the left part of Eq. (9) is proved.

THEOREM 4 ^{(Reference Benammar, Ridouard and Bauer16)} The Request Bound Function $RBF_j^h{\rm{(}}t{\rm{)}}$ for flow v _j passing through node h in a time interval of length t (t > 0) is:

(10) \begin{align}{\rm{RBF}}_{\rm{j}}^{\rm{h}}{\rm{(t) = }}\left( {{\rm{1 + }}{{{\rm{t + J}}_{\rm{j}}^{\rm{h}}} \over {{{\rm{T}}_{\rm{j}}}}}} \right) \cdot {{\rm{C}}_{\rm{j}}}\end{align}

Proof We separately prove Eq. (10) for two different parts: the periodic part ending with packet $m^{\prime}_{j}$ and the simultaneous part ending with packet m _j.

When length t is shorter than ${\rm{\Delta }}_j^h$ , the maximum count of the pending packets is k+1 (packet m _j and the k packets arriving simultaneously with itself as depicted in Fig. 3). From Eq. (9), we have $k{T_j} \le J_j^h$ , so that $k{T_j} \le t + J_j^h$ . Since t is shorter than ${\rm{\Delta }}_j^h$ , we have $t + J_j^h \lt J_j^h + \Delta_j^h$ . From Eq. (8) we have $J_j^h{\rm{ + \Delta }}_j^h{ = \rm{ (}}k{\rm{ + 1) \times }}{T_j}$ , so that $t + J_j^h{\rm{ \lt (}}k{\rm{ + 1) \times }}{T_j}$ . We have:

(11) \begin{align} & k{T_j} \le t + J_j^h \lt \left( {k + {\rm{1}}} \right) \cdot {T_j}\nonumber \\ & \quad\Leftrightarrow k \le {{t + J_j^h} \over {{T_j}}} \lt \left( {k + {\rm{1}}} \right)\\ & \quad\Leftrightarrow \,\Bigg\lfloor {{{t + J_j^h} \over {{T_j}}}} \Bigg\rfloor = k \nonumber\end{align}

so that Eq. (10) can get the maximum backlog for any time interval $t{\rm{ (}}0 \lt t{\rm{ \lt \Delta }}_j^h)$ .

When the length t is ${\rm{\Delta }}_j^h$ , the maximum count of the pending packets is k+2 (packet m _j, the k packets arriving simultaneously with itself and packet $m^{\prime}_{j}$ as depicted in Fig. 3). We have:

(12) \begin{align} t + J_j^h & = \Delta _j^h + J_j^h \nonumber\\ & = \left( {k + {\rm{1}}} \right) \cdot {T_j} \\ & \Leftrightarrow \Bigg\lfloor{{t + J_j^h} \over {{T_j}}}\Bigg\rfloor = k{\rm{ + 1}} \nonumber\end{align}

Eq. (10) can get the maximum backlog for time interval $t = \Delta _j^h$ . and $t + J_j^h$ is multiple of T _j.

When the length t is longer than ${\rm{\Delta }}_j^h$ , one new packet can arrive every T _j because all the packets before $m^{\prime}_{j}$ arrive at node h one after another every T _j.

All the analysis above proves Theorem 4.

Proof By the definition, at the source node, we have $S{\rm{min}}_j^{\,firs{t_j}} = S{\rm{max}}_j^{\,firs{t_j}} = 0$ . Therefore, $J_j^{\,firs{t_j}} = 0$ and Eq. (10) can be expressed as:

(13) \begin{align}{\rm{RBF}}_{\rm{j}}^{\rm{h}}{\rm{(t) = }}\left( {{\rm{1 + }}\left\lfloor {{{\rm{t}} \over {{{\rm{T}}_{\rm{j}}}}}} \right\rfloor } \right) \cdot {{\rm{C}}_{\rm{j}}}\end{align}

As illustrated by Corollary 1, at the source node, every ${T_j}$ time interval one new packet of flow v _j can be generated. There is only the periodic arriving part at the source node.

With the Request Bound Function $RBF_j^h{\rm{(}}t{\rm{)}}$ computed by Eq. (10), we can obtain the total transmission time ${W^h}{\rm{(}}t{\rm{)}}$ for the packets generated by all the interfering flows during a time interval of length t. On one hand, the competing packets arrive forming the backlog at node h, on the other hand, the arrived packets are being forwarded leaving node h, so that the available time to forward those packets should be subtracted. Furthermore, Fig. 2 and Eq. (1) demonstrate that packet m _i is included in the computation of $S{\rm{max}}_i^{h{\rm{ + 1}}}$ , the transmission time C _i should be subtracted in the computation of the maximum backlog $Bklg_i^h$ . We have:

(14) \begin{align}{{Bklg}}_{\rm{i}}^{\rm{h}}{ = \rm{ ma}}{{\rm{x}}_{{\rm{t}} \ge {\rm{0}}}}\left( {{{\rm{W}}^{\rm{h}}}\left( {\rm{t}} \right) - {{\rm{C}}_{\rm{i}}} - {\rm{t}}} \right)\end{align}

Proof When there is only one flow v _i without any interfering flows, for the minimum time interval t = 0, we can obtain ${W^h}(0) - {C_i}{\rm{ - 0 = 0}}$ . This is the minimum backlog. However, $Bklg_i^h$ corresponds to the maximum value of this computation, so it cannot be negative.

4.5 Stop condition

To compute the maximum backlog $Bklg_i^h$ encountered by flow v _i at node h, The time interval of any possible length t should be tested. It is necessary to determine a maximum length as the stop condition of this test to ensure the maximum backlog is included. We denote this maximum length of the time interval as ${\mathscr{B}^h}$ . To determine the maximum length, we separately take into account the two parts in Section 4.4, the periodic part and the simultaneous part.

For the periodic part, we introduce the concept of busy period^{(Reference Qingfei and Xinyu25,Reference Bauer, Scharbarg and Fraboul32)} . The backlog is included in a busy period. A busy period is a time interval between two consecutive idle time instants which are respectively the starting time and the ending time of this busy period. An idle time instant means all the pending packets have been forwarded (there is no backlog at this idle time instant). Therefore, packets arrived before an idle time instant should not be included in the computation of the backlog for the following busy period. Consequently, the computation of the backlog should end with the first following idle time instant (the ending time of the busy period), as the packets arrived after the idle time instant cannot generate any backlog.

LEMMA 4 ^{(Reference Kemayo, Benammar and Ridouard12)} The length of the maximum busy period at node h generated by flow set ${\Gamma_h}$ is bounded to the first positive value satisfying the following equation if $\sum\nolimits_{j \in {{\rm{}}_h}} {{{{C_j}} \over {{T_j}}} \le {\rm{1}}} $ .

(15) \begin{align}\left\{ \begin{array}{ll} {{\mathscr{B}}{{\mathscr{P}}^{\rm{h}}}(0) = \sum\nolimits_{{\rm{j}} \in {{\rm{}}_{\rm{h}}}} {{{\rm{C}}_{\rm{j}}}} } \\ {{\mathscr{B}}{{\mathscr{P}}^{\rm{h}}}\left( {{\rm{k + 1}}} \right) = \sum\nolimits_{{\rm{j}} \in {{\rm{}}_{\rm{h}}}} {\left\lceil {{{{\mathscr{B}}{{\mathscr{P}}^{\rm{h}}}({\rm k})} \over {{{\rm{T}}_{\rm{j}}}}}} \right\rceil \cdot {{\rm{C}}_{\rm{j}}}} } \end{array} \right.\end{align}

This computation ends with the first positive value satisfying ${\mathscr{BP}}^h\left( {k{\rm{ + 1}}} \right) = {\mathscr{BP}}^h\left( k \right)$ .

This longest busy period represents the maximum time interval between two consecutive idle time instants. Therefore, for the periodic part of a flow, if the length of a time interval is larger or equal to the longest busy period ${\mathscr{BP}}^h$ , the computation of the backlog can be stopped as an idle time exists^{(Reference Kemayo, Ridouard and Bauer8)}.

For the simultaneous part, as depicted in Lemma 1, ${\rm{\Delta }}_j^h$ represents the length of the simultaneous part. Generally, ${\rm{ma}}{{\rm{x}}_{j\in{\Gamma _h}}}(\Delta _j^h)$ can include all the simultaneous parts of the flows passing through node h. ${\rm{\Delta }}_j^h$ can be determined by Lemma 5 when knowing the maximum jitter $J_j^h$ .

LEMMA 5 ^{(Reference Benammar, Ridouard and Bauer16)} For any flow v _j passing through node h, we have:

(16) \begin{align}{\rm{\Delta }}_{\rm{j}}^{\rm{h}} = {\rm{}}\left( {1 + \left\lfloor {{{{\rm{J}}_{\rm{j}}^{\rm{h}}} \over {{{\rm{T}}_{\rm{j}}}}}} \right\rfloor } \right) \cdot {{\rm{T}}_{\rm{j}}} - {\rm{J}}_{\rm{j}}^{\rm{h}}\end{align}

Proof From Eq. (8) we obtain ${\rm{\Delta }}_j^h{ = \rm{ (}}k{\rm{ + 1) \times }}{T_j} - J_j^h$ , where k is the number of the packets arriving simultaneously with packet m _j, $J_j^h$ is the maximum arrival jitter ( $J_j^h = S{\rm{max}}_j^h - S{\rm{min}}_j^h$ ). From Eq. (9) we have:

(17) \begin{align} & k{T_j} \le J_j^h \lt \left( {k + {\rm{1}}} \right) \cdot {T_j} \nonumber\\ & \quad \Leftrightarrow k \le {{J_j^h} \over {{T_j}}} \lt \left( {k + {\rm{1}}} \right) \\ & \quad \Leftrightarrow \left\lfloor {{{J_j^h} \over {{T_j}}}} \right\rfloor = k \nonumber\end{align}

Replacing k with $\left\lfloor {{{J_j^h} \over {{T_j}}}} \right\rfloor $ in Eq. (8) proves Lemma 5.

The maximum length of the time interval containing all simultaneous parts can be limited to ${\mathscr{B}}{{{S}}^{\rm{h}}}$ :

(18) \begin{align}{\mathscr{B}}{{\rm{S}}^{\rm{h}}} = \mathop {\max }\limits_{{v_j} \in {\Gamma _h}} \left\{ {\left( {1 + \left\lfloor {{{J_j^h} \over {{T_j}}}} \right\rfloor } \right) \cdot {T_j} - J_j^h} \right\}\end{align}

The two lemmas above determine the maximum length of the time interval for computing the worst-case backlog respectively considering the periodic and simultaneous part.

THEOREM 5 ^{(Reference Kemayo, Ridouard and Bauer8)} Suppose a set of flows Г_h passing through node h, if $\sum\nolimits_{j \in {{\rm{\Gamma}}_h}} {{{{C_j}} \over {{T_j}}} \le {\rm{1}}} $ and node h follows the FIFO servicing policy, the maximum length of the test time interval can be expressed as:

(19) \begin{align}{\mathscr{B}^h} = \mathscr{B}{P^h} + \mathscr{B}{S^h}\end{align}

Proof Lemma 4 and 5 prove this theorem.

For each flow $v_i{\rm{}} \in \Gamma$ and each node h of path $\mathscr{P}_{i}$ , an upper bound of the worst-case transversal delay $R_i^h$ can be obtained by

(20) \begin{align}{\rm{R}}_{\rm{i}}^{\rm{h}} = S{\rm{max}}_{\rm{i}}^{\rm{h}}+ Bklg_{\rm{i}}^{\rm{h}} + {{\rm{C}}_{\rm{i}}}\end{align}

The maximum backlog $Bklg_i^h$ can be computed according to Eq. (21)

(21) \begin{align}{{Bklg}}_{\rm{i}}^{\rm{h}} = \rm{ ma}{{\rm{x}}_{{{\mathscr{B}}^{\rm{h}}} \ge {\rm{t}} \ge 0}}\left( {{{\rm{W}}^{\rm{h}}}\left( {\rm{t}} \right) - {{\rm{C}}_{\rm{i}}} - {\rm{t}}} \right)\end{align}

With ${W^h}(t)$ being defined by Eq. (4). The computation ends with ${{\mathscr{B}}^h}$ which is obtained by Eq. (19).

The pseudo-code for the Forward Approach is given by Algorithm 1. In fact, the computation can be done in a discrete manner by testing time t only at arrival times of new packets^{(Reference Kemayo, Ridouard and Bauer8)}. This computation does not consider the serialisation effect.

Algorithm 1 Forward Approach without considering the serialisation effect.

4.6 Serialisation effect for forward approach

In packet switched networks such as AFDX, packets are necessarily serialised when being transmitted through a common physical link. Packets transmitted through a common physical link cannot arrive at the next switch simultaneously. They are serialised. This is what is commonly called the serialisation effect^{(Reference Kemayo, Benammar and Ridouard12)}.

For example, consider flow v ₁₀ in Fig. 1. Suppose packet i is generated by flow v _i. $\alpha _i^h$ indicates the arrival time at node h of packet i. According to the Forward Approach, the maximum backlog encountered by packet 10 at each node along its transmission path is depicted in Fig. 6. Packet 10 is delayed by packet 1 at its source node (e ₃). At node S ₃, it is delayed by packet 2 from node e ₄. At node S ₄, it is delayed by both packets 6 and 7 from node S ₁. But this is impossible as packets 6 and 7 are transmitted through the same physical link. They are serialised and cannot arrive at node S ₄ simultaneously. The actual worst-case end-to-end transmission delay for packet 10 is depicted in Fig. 7.

Figure 6. Maximum backlog for packet 10 on its transmission path according to the Forward Approach.

Figure 7. Actual worst-case end-to-end transmission delay for packet 10 along its transmission path.

Figure 7 shows that not both packets 6 and 7 can cause delay to packet 10. This demonstrates the maximum backlog computed by the Forward Approach is pessimistic (overestimated), thus the obtained upper bound of the end-to-end transmission delays is pessimistic (overestimated). The computation of the serialisation effect in this section can improve the Trajectory approach to ensure the tightness of the backlog as well as the delay upper bound.

Consider node h with n input physical links as depicted in Fig. 8. Flows from n physical links are transmitted to the same output physical link $O{P^h}$ . The n input physical links are respectively denoted as $IP_x^{h}$ where $x \in {\rm{}}\left\{ {{\rm{1, \ldots ,}}n} \right\}$ .

Figure 8. Node h with n input physical links.

The cumulative function W(t) can be obtained by summing the transmission time of the packets from each physical link $IP_x^h$ . Suppose $W_x^h(t)$ is the transmission time of packets from the physical link $IP_x^{h}$ during a time interval of length t, so that we have^{(Reference Benammar, Ridouard and Bauer13,Reference Benammar, Ridouard and Bauer16)} :

(22) \begin{align}{{\rm{W}}^{\rm{h}}}\!\left( {\rm{t}} \right) = \sum\nolimits_{{\rm{x}} = 1}^{\rm{n}} {{\rm{W}}_{\rm{x}}^{\rm{h}}\left( {\rm{t}} \right){\rm{}}}\end{align}

We need to determine the maximum value of each $W_x^h(t)$ where $x \in {\rm{}}\left\{ {{\rm{1, \ldots ,}}n} \right\}$ . The fixed transmission rate of each virtual link in AFDX networks ensures the maximum account of the arrived packets during a time interval of length t. The expression of $W_x^h(t)$ thus can be determined.

Figure 9. Worst-case cumulative transmission time of packets through an input link during a time interval of length t.

LEMMA 6 ^{(Reference Kemayo, Benammar and Ridouard12)} The worst-case transmission time of packets arriving at node h from the input physical link $IP_x^{h}$ during a time interval of length t is limited to Eq. (23):

(23) \begin{align}{\rm{W}}_{\rm{x}}^{\rm{h}}\left( {\rm{t}} \right){\rm{}} \le {\rm{}}{{{{\rm{G}}^{{{\rm{h}}_{\rm{x}}} - {\rm{1}}}}} \over {{{\rm{G}}^{{{\rm{h}}_{\rm{x}}}}}}} \times {\rm{t}} + {\max _{{v_i} \in\Gamma {{\rm{}}_{{\rm{h,x}}}}}}\left\{ {{\rm{C}}_{\rm{i}}^{\rm{h}}} \right\}\end{align}

where ( ${h_x} - 1$ ) is the predecessor of node h along the physical the link $IP_x^{h}$ .

Proof Nodes ${h_x} - 1$ and h are directly connected through the physical link $IP_x^{h}$ . Therefore, the transmission time of packets at node h coming through the input link $IP_x^{h}$ is determined by the count of transmitted packets^{(Reference Benammar, Ridouard and Bauer16)}:

1. (1) During the time interval of length t, the count of packets transmitted to node h through $IP_x^{h}$ is ${G^{{h_x} - {\rm{1}}}} \times t$ depicted as the shaded area in Fig. 9.

2. (2) To ensure the worst-case, we suppose that a packet with the maximum length has been completely transmitted at the very beginning of this time interval, ${\max _{{v_i} \in\Gamma {{\rm{}}_{h,x}}}}\left\{ {C_i^{{h_x} - {\rm{1}}}} \right\}$ .

The worst-case transmission time of the packets coming through the input physical link $IP_x^{h}$ during a time interval of length t is limited to Eq. (24) and Lemma 6 is proved.

(24) \begin{align}{{{{\rm{G}}^{{{\rm{h}}_{\rm{x}}}{\rm{ - 1}}}}{\rm{ \times t}}} \over {{{\rm{G}}^{\rm{h}}}}}{\rm{}} + {\max _{{{\rm{v}}_{\rm{i}}} \in\Gamma {{\rm{}}_{{\rm{h,x}}}}}}\left\{ {{\rm{C}}_{\rm{i}}^{\rm{h}}} \right\}\end{align}

THEOREM 6 ^{(Reference Kemayo, Ridouard and Bauer8,Reference Benammar, Ridouard and Bauer16)} The cumulative transmission time of the packets transmitted through the physical link $IP_x^{h}$ during a time interval of length t at node h is bounded by

(25) \begin{align}{\rm{W}}_{\rm{x}}^{\rm{h}}\left( {\rm{t}} \right) = {\min _{}}\left\{ {\sum\nolimits_{{{\rm{v}}_{\rm{i}}} \in\Gamma {{\rm{}}_{{\rm{h,x}}}}} {{\rm{RBF}}_{\rm{i}}^{\rm{h}}{\rm{(t)}}} ,{{{{\rm{G}}^{{{\rm{h}}_{\rm{x}}}{\rm{ - 1}}}}{\rm{ \times t}}} \over {{{\rm{G}}^{\rm{h}}}}} + {{\max }_{{{\rm{v}}_{\rm{i}}} \in\Gamma {{\rm{}}_{{\rm{h,x}}}}}}\left\{ {{\rm{C}}_{\rm{i}}^{\rm{h}}} \right\}} \right\}\end{align}

Proof Eq. (4) and (23) define two guaranteed upper bounds of value $W_x^h(t)$ , so it can be limited by the smaller value. Theorem 6 is proved.

There is exception for Theorem 6 at source node of flow v _i. At the source node there is no serialisation effect, and ${W^h}{\rm{(}}t{\rm{)}}$ should be computed only according to Eq. (4).

Proof At the source node (first _i) of flow v _i, all the flows are generated at this node instead of coming from input physical links. Packets are generated by different flows at the same time, and arrive at the output port simultaneously. As depicted in Fig. 10, suppose the source node (first _i) of flow v _i has three other flows such as flow j, k and l. and packets j, k and l arrive at the output port of node h simultaneously with packet i. There is no serialisation effect for these flows.

Figure 10. No serialisation effect for the packets generated at the source node.

The pseudo-code of the Forward Approach integrated with the serialisation effect is given by Algorithm 2. The computation can also be done in a discrete manner by testing time t only at arrival times of new packets.

Algorithm 2 Forward Approach integrated with the serialisation effect

5.1 Comparative analysis

An illustrative AFDX network is depicted in Fig. 11. The flow characteristics are given in Table 1. As an example, we analyse flow ${v_1}({\mathscr{P}_1} = {e_1} - {S_1} - {S_3})$ which meets flow v ₂ at switch S ₁, meets v ₃, v ₄ and v ₅ at switch S ₃.

Figure 11. Illustrative AFDX network.

Table 1 Characteristics of flows

When not considering the serialisation effect, we should analyse all the nodes along path $\mathscr{P}_{1}$ according to Algorithm 1. By definition, we have $S{\rm{min}}_{\rm{1}}^{{e_{\rm{1}}}} = S{\rm{max}}_{\rm{1}}^{{e_{\rm{1}}}}{ = \rm{ 0}}$ and the arrival jitter ${J^{{e_{\rm{1}}}}}$ is zero. As v ₁ is the only flow emitted from e ₁, the maximum backlog can be computed by Eq.(26):

(26) \begin{align}{{Bklg}}_{\rm{1}}^{{{\rm{e}}_{\rm{1}}}}{ = \rm{ ma}}{{\rm{x}}_{{\rm{0}} \le {\rm{t}} \le {{\mathscr{B}}^{{{\rm{e}}_{\rm{1}}}}}}}\left\{ {\left( {{\rm{1 + }}\left\lfloor {{{\rm{t}} \over {{{\rm{T}}_{\rm{1}}}}}} \right\rfloor } \right) \cdot {{\rm{C}}_{\rm{1}}} - {{\rm{C}}_{\rm{1}}}{\rm{ - t}}} \right\}\end{align}

According to Lemma 4 and 5, we have:

\begin{align*}\mathscr{BP}^{{{\rm{e}}_{\rm{1}}}} = {{{C}}_{\rm{1}}} = {\rm{40}}\end{align*}

\begin{align*}\mathscr{B}S^{{e_{\rm{1}}}} = \left( {{\rm{1 + }}\left\lfloor {{{{J^{{e_{\rm{1}}}}}} \over {{T_{\rm{1}}}}}} \right\rfloor } \right). T{\,_{\rm{1}}} = 4000{\rm{\mu s}}\end{align*}

where the arrival jitter ${J^{{e_{\rm{1}}}}} = 0$ .

The length of the test interval is obtained as:

\[{\rm{}}{\mathscr{B}^{{e_{\rm{1}}}}} = \mathscr{B}{\mathscr{P}^{{e_{\rm{1}}}}} + \mathscr{B}{S^{{e_{\rm{1}}}}} = {\rm{40 + 4000}} = {\rm{4040}}{\rm{\mu s}}\]

The maximum backlog is obtained at time t = 0. We have

\begin{align*}Bklg_{\rm{1}}^{{e_{\rm{1}}}} = 0{\rm{\mu s}}\end{align*}

According to Eq. (1), we have

\begin{align*}S{\rm{max}}_{\rm{1}}^{{S_1}}{ = \rm{ 40 + 16 = 56\mu s}}\end{align*}

According to Eq. (3), we can obtain

\begin{align*}S{\rm{min}}_{\rm{1}}^{{S_1}}{ = \rm{ 40 + 16 = 56\mu s}}\end{align*}

The arrival jitter ${J^{{S_1}}}$ is zero.

Similarly, we also have

\begin{align*}S{\rm{max}}_{\rm{2}}^{{S_1}} = S{\rm{max}}_2^{{S_1}} = {\rm{56\mu s}}\end{align*}

and the arrival jitter is zero, too.

At node S ₁, flow v ₁ meets v ₂, the maximum backlog ${\rm{}}Bklg_{\rm{1}}^{{S_1}}$ can be expressed as:

\begin{align*}Bklg_{\rm{1}}^{{S_1}} = ma{x_{0 \le t \le {\mathscr{B}^{{{\rm{S}}_1}}}}}\left\{ {\left( {{\rm{1 + }}\left\lfloor {{t \over {{T_{\rm{1}}}}}} \right\rfloor } \right) \cdot {C_{\rm{1}}} + \left( {{\rm{1 + }}\left\lfloor {{t \over {{T_{\rm{2}}}}}} \right\rfloor } \right) \cdot {C_{\rm{2}}} - {C_{\rm{1}}} - t} \right\}\end{align*}

According to Lemma 4 and 5, we have:

\begin{align*} \mathscr{B}{P^{{S_1}}} & = {C_{\rm{1}}} + {C_{\rm{2}}} = {\rm{80}}{\rm{\mu s}} \\ \mathscr{B}{S^{{S_1}}} & = \left( {{\rm{1 + }}\left\lfloor {{{{J^{{S_1}}}} \over {{T_{\rm{1}}}}}} \right\rfloor } \right) \cdot {T_{\rm{1}}} = 4000{\rm{\mu s}} \end{align*}

where the arrival jitter ${J^{{S_1}}} = 0$ .

The length of the test interval is obtained as:

\begin{align*}{\mathscr{B}^{{S_1}}} = \mathscr{B}{\mathscr{P}^{{S_1}}} + \mathscr{B}{S^{{S_1}}} = {\rm{80 + 4000 = 4080\mu s}}\end{align*}

The maximum backlog is obtained at time t = 0, and we have

\begin{align*}Bklg_{\rm{1}}^{{S_1}} & = {\rm{40\mu s}}\\ S{\rm{max}}_{\rm{1}}^{{S_{\rm{3}}}} & = {\rm{56 + 40 + 40 + 16}} = {\rm{152\mu s}}\end{align*}

Finally, in the last node of $\mathscr{P}_{1}$ , flow v ₁ meets v ₃, v ₄ and v ₅. The values of $S{\rm{max}}_j^{{S_{\rm{3}}}}$ and $S{\rm{min}}_j^{{S_{\rm{3}}}}\left( {j = {\rm{1,3,4,5}}} \right)$ for those four flows are depicted in Table 2.

Table 2 Durations to reach node S ₃

The maximum backlog $Bklg_{\rm{1}}^{{S_{\rm{3}}}}$ can be expressed as:

\begin{align*}Bklg_{\rm{1}}^{{S_{\rm{3}}}} = & \mathop {max}\limits_{{\rm{0}} \le {\rm{t}} \le {\mathscr{B}^{{S_{\rm{3}}}}}} \left\{ \left( {{\rm{1 + }}\left\lfloor {{{t + J_1^{{S_{\rm{3}}}}} \over {{T_{\rm{1}}}}}} \right\rfloor } \right) \cdot {C_{\rm{1}}} + \left( {{\rm{1 + }}\left\lfloor {{{t + J_3^{{S_{\rm{3}}}}} \over {{T_3}}}} \right\rfloor } \right) \cdot {C_3} + \left( {{\rm{1 + }}\left\lfloor {{{t + J_4^{{S_{\rm{3}}}}} \over {{T_4}}}} \right\rfloor } \right) \cdot {C_4}\right.\\ & \left. + \left( {{\rm{1 + }}\left\lfloor {{{t + J_5^{{S_{\rm{3}}}}} \over {{T_5}}}} \right\rfloor } \right) \cdot {C_5} - {C_{\rm{1}}} - {\rm{t}} \right\}\end{align*}

According to Lemma 4 and 5, we have:

\begin{align*}\mathscr{B}{\mathscr{P}^{{S_{\rm{3}}}}} = {C_{\rm{1}}} + {C_{\rm{3}}} + {C_4} + {C_{\rm{5}}} = {\rm{160}}{\rm{\mu s}}\end{align*}

\begin{align*}{\mathscr{B}}{{\rm{S}}^{{{\rm{S}}_{\rm{3}}}}} = {\rm{}}\left( {{\rm{1 + }}\left\lfloor {{{{{\rm{J}}^{{{\rm{S}}_{\rm{3}}}}}} \over {{{\rm{T}}_{\rm{1}}}}}} \right\rfloor } \right)\,{ \cdot T_{\rm{1}}} = {\rm{4000\mu s}}\end{align*}

where the arrival jitter ${J^{{S_{\rm{3}}}}} = 40{\rm{\mu s}}$ .

The length of the test interval is obtained as:

\[{\mathscr{B}^{{S_{\rm{3}}}}} = \mathscr{B}{\mathscr{P}^{{S_{\rm{3}}}}} + \mathscr{B}{S^{{S_{\rm{3}}}}} = {\rm{160 + 4000 = 4160\mu s}}\]

The maximum backlog encountered by flow v ₁ at node S ₃ is obtained at time t = 0, and $Bklg_{\rm{1}}^{{S_{\rm{3}}}} = {\rm{120\mu s}}$ .

The worst-case end-to-end transmission delay of flow v ₁ is

(27) \begin{align} R1= S{\rm{max}}_{\rm{1}}^{{S_{\rm{3}}}} + Bklg_{\rm{1}}^{{S_{\rm{3}}}} + {C_{\rm{1}}}{\rm{}} = {\rm{152}} + {\rm{120}} + {\rm{40}} = {\rm{312\mu s}} \end{align}

The corresponding scenario of the worst-case transmission is depicted in Fig. 12.

Figure 12. Worst-case scenario corresponding to the Forward Approach without considering serialisation effect.

When considering the serialisation effect, at the source node, the cumulative function ${W^h}(t)$ is computed according to Theorem 6 and 7. As there is only one flow at node e ₁, the cumulative transmission time is computed by

(28) \begin{align}{{\rm{W}}^{{{\rm{e}}_{\rm{1}}}}}\!\left( {\rm{t}} \right){ = \rm{ ma}}{{\rm{x}}_{0 \le {\rm{t}} \le {{\rm{B}}^{{{\rm{e}}_1}}}}}\left\{ {\left( {{\rm{1 + }}\left\lfloor {{{\rm{t}} \over {{{\rm{T}}_{\rm{1}}}}}} \right\rfloor } \right) \cdot {{\rm{C}}_{\rm{1}}}{\rm{ - t}}} \right\} \end{align}

The maximum value is 40 which is obtained at time t = 0. The maximum backlog

(29) \begin{align} Bklg_{\rm{1}}^{{e_{\rm{1}}}} = {W^{{e_{\rm{1}}}}}(0) - {C_{\rm{1}}}{ = \rm{ 40}} - {\rm{40 = 0}} \end{align}

We have

\begin{align*} S{\rm{max}}_{\rm{1}}^{{S_1}}{ = \rm{ 40 + 16 = 56\mu s}}\end{align*}

and

\begin{align*} S{\rm{min}}_{\rm{1}}^{{S_1}}{ = \rm{ 40 + 16 = 56\mu s}} \end{align*}

The arrival jitter is zero. Similarly, we also have $S{\rm{max}}_{\rm{2}}^{{S_1}} = S{\rm{max}}_{\rm{2}}^{{S_1}} = {\rm{56\mu s}}$ , and the arrival jitter is zero too.

Then v ₁ and v ₂ meet at node S ₁. The maximum cumulative transmission time is computed as

(30) \begin{align}{{\rm{W}}^{{{\rm{S}}_1}}}\!\left( {\rm{t}} \right){\rm{}} = {\rm{mi}}{{\rm{n}}_{}}\left\{ {\left( {{\rm{1 + }}\left\lfloor {{{\rm{t}} \over {{{\rm{T}}_{\rm{1}}}}}} \right\rfloor } \right) \cdot {{\rm{C}}_{\rm{1}}}{\rm{,t}} + {{\rm{C}}_{\rm{1}}}} \right\} + {\min _{}}\left\{ {\left( {{\rm{1 + }}\left\lfloor {{{\rm{t}} \over {{{\rm{T}}_{\rm{2}}}}}} \right\rfloor } \right) \cdot {{\rm{C}}_{\rm{2}}}{\rm{,t}} + {{\rm{C}}_{\rm{2}}}} \right\} \end{align}

The maximum value is obtained at time t = 0 and the maximum backlog

(31) \begin{align} Bklg_{\rm{1}}^{{S_1}} = {W^{{S_1}}}(0) - {C_{\rm{1}}}{ = \rm{ 80}} - {\rm{40 = 40\mu s}} \end{align}

We have

\begin{align*} S{\rm{max}}_{\rm{1}}^{{S_{\rm{3}}}} &= S{\rm{max}}_{\rm{1}}^{{S_1}} + Bklg_{\rm{1}}^{{S_1}} + {C_{\rm{1}}} + L \\ & = 56 + 40 + 40 + 16 \\ &= 152\mu \rm{s} \end{align*}

Then v ₁ meets v ₃, v ₄ and v ₅ at node S ₃. Flows v ₃ and v ₄ are coming through the same physical link, they are serialised. The maximum cumulative transmission time is computed as

(32) \begin{align}{W^{{S_{\rm{3}}}}}(t) &= \mathop {\min }\limits_{} \left\{ {\left( {{\rm{1 + }}\left\lfloor {{t \over {{T_{\rm{1}}}}}} \right\rfloor } \right) \cdot {C_{\rm{1}}}{\rm{,}}t + {C_{\rm{1}}}} \right\}\mathop { + \min }\limits_{} \left\{ \left( {{\rm{1 + }}\left\lfloor {{t \over {{T_{\rm{4}}}}}} \right\rfloor } \right) \cdot {C_{\rm{4}}} + \left( {{\rm{1 + }}\left\lfloor {{t \over {{T_{\rm{3}}}}}} \right\rfloor } \right) \cdot {C_{\rm{3}}}{\rm{,}}t \right.\nonumber\\ &\quad\left.+ \mathop {\max }\limits_{} \left\{ {{C_{\rm{3}}},{C_{\rm{4}}}} \right\} \right\} + {\min _{}}\left\{ {\left( {{\rm{1 + }}\left\lfloor {{{\rm{t}} \over {{{\rm{T}}_{\rm{5}}}}}} \right\rfloor } \right) \cdot {{\rm{C}}_{\rm{5}}}{\rm{,t}} + {{\rm{C}}_{\rm{5}}}} \right\}\end{align}

The maximum value is 120µs which is obtained at time t = 0. The corresponding maximum backlog is

(33) \begin{align}Bklg_{\rm{1}}^{{S_{\rm{3}}}} = {W^{{S_{\rm{3}}}}}(0) - {C_{\rm{1}}}{ = \rm{ 120}} - {\rm{40 = 80\mu s}}\end{align}

The worst-case end-to-end transmission delay is:

(34) \begin{align}{\rm{}}{R_1} = S{\rm{max}}_{\rm{1}}^{{S_{\rm{3}}}} + {\rm{}}Bklg_{\rm{1}}^{{S_{\rm{3}}}} + {C_1} = 152{\rm{ }} + {\rm{ }}80{\rm{ }} + {\rm{ }}40{\rm{ }} = {\rm{ }}272{\rm{ \mu s}}\end{align}

The corresponding scenario of the worst-case transmission is depicted in Fig. 13.

Compared with the transmission scenario depicted in Fig. 12, packets 3 and 4 cannot arrive at S ₃ simultaneously with packet 1 in Fig. 13. One of them arrives 40µs earlier and does not cause delay to packet 1 (packet 3 arrives 40µs earlier in Fig. 13, it is similar if packet 4 arrives earlier). That is why the worst-case end-to-end transmission delay computed in Eq. (27) has 40µs pessimism compared with Eq. (34).

Similarly, the worst-case end-to-end transmission delays of the other flows in Table 1 are given in Table 3. We can see, compared with the Forward Approach without considering the serialisation effect, the Forward Approach integrated with the serialisation effect can compute a tighter upper bound for flows v ₁, v ₃, v ₄ and v ₅. Flow v ₂ has no gains as there is no serialisation effect during its transmission. Compared with the Network Calculus approach and the Trajectory Approach, the Forward Approach can obtain a tighter upper bound than the Network Calculus approach, it and can keep the same tightness as the Trajectory Approach no matter with or without the serialisation effect. Moreover, when compared with the Model Checking approach, we find that the Forward Approach integrated with the serialisation effect can obtain the exact delay upper bounds for all the flows in the network configuration of this experiment. The simulation method usually obtains an optimistic upper bound due to missing some rare scenarios, and it is not necessary to do the comparative analysis between the simulation method and the Forward Approach integrated with the serialisation effect.

Table 3 Worst-case end-to-end transmission delays

Basic: computed without the serialisation effect.

Serial: computed with the serialisation effect.

Gain: improvement when considering the serialisation effect.

Figure 13. Worst-case scenario corresponding to the Forward Approach integrated with the serialisation effect.

5.2 Optimism in serialisation computation

In this section, we present the potential optimism existing in the computation of the serialisation effect, which is given in Section 4.6. An illustrative AFDX network is depicted in Fig. 14. The only difference with Fig. 11 is that flow v ₂ is transmitted to e ₆ in this figure. The flow characteristics are given in Table 4. It is the same as Table 1.

Table 4 Characteristics of flows

Figure 14. An illustrative AFDX network.

We still focus on flow v ₁ which is transmitted from e ₁ to e ₆. Flow v ₁ meets v ₂ at node S ₁, meets v ₃, v ₄ and v ₅ at node S ₃.

At the source node e ₁, there is only flow v ₁, and the backlog at node e ₁ is zero. The corresponding computation is similar to Eq. (28) and (29). We have:

\begin{align*}S{\rm{min}}_{\rm{1}}^{{S_1}} = S{\rm{max}}_{\rm{1}}^{{S_1}}{ = \rm{ 40 + 16 = 56\mu s}}\end{align*}

At node S ₁, $Bklg_{\rm{1}}^{{S_1}} = 40$ µs, the corresponding computation is similar to Eq. (30) and (31). We can obtain:

\begin{align*}S{\rm{max}}_{\rm{1}}^{{S_{\rm{3}}}} = S{\rm{max}}_{\rm{1}}^{{S_1}} + Bklg_{\rm{1}}^{{S_1}} + {C_1} + L = {\rm{56 + 40 + 40 + 16 = 152\mu s}}\end{align*}

At node S ₃, $Bklg_{\rm{1}}^{{S_{\rm{3}}}} = {\rm{80}}$ µs, the corresponding computation is similar to Eq. (32) and (33).

The worst-case end-to-end transmission delay is 272µs as computed in Eq. (34).

In fact, for flow v ₁, the exact worst-case end-to-end transmission delay is 312µs as depicted in Fig. 15.

In this configuration, both packets 3 and 4 can cause delay to packet 1 even they do not arrive at node S ₃ simultaneously with packet 1 (in fact, packet 3 arrives 40µs earlier than packet 1, packet 4 arrives simultaneously with packet 1). The actual backlog at node S ₃ is

\begin{align*}Bklg_1^{{S_3}} = {{\rm{C}}_{\rm{3}}} + {{\rm{C}}_{\rm{4}}} + {{\rm{C}}_{\rm{5}}}{ = \rm{ 120 \mu s}}\end{align*}

However, the backlog computed by Eq. (32) and (33) is only 80µs. It has 40µs optimism. This demonstrates that the computation of the serialisation effect presented in Section 4.6 computes an optimistic upper bound.

Figure 15. Actual worst-case end-to-end transmission delay.

5.3 New computation of serialisation effect

For the network configuration in Fig. 14, packet 3 can delay packet 1 even it does not arrive at node S ₃ simultaneously with packet 1 (in fact, it arrives 40µs earlier, it arrives at node S ₃ simultaneously with packet 2). In Fig. 15, we can see packets 3, 4 and 5 (all the packets coming from the other input physical inks) arrive at node S ₃ not earlier than packet 2.

As depicted in Fig. 8, node h has n input physical links. Packets at the input link $IP_x^{h}$ form a packet sequence $seq_x^{h}$ . Suppose flow v _i are transmitted through the first input physical link $IP_1^h$ .

Due to the FIFO scheduling policy, any packet arriving at node h later than packet i cannot delay it. In order to maximise the delay of packet i at node h, each competing packet sequence is postponed and it finishes at the same time as the packet sequence $seq_1^{h}$ . The finishing time is denoted as $\eta$ depicted in Fig. 16. $\delta _i^h$ is the maximum time difference between arrival of the first packet from $seq_x^{h}$ where $ x \in \{ 2, \ldots ,n\}$ and the arrival of the first packet from $seq_{\rm{1}}^{h}$ . Suppose p _i is the first packet in the sequence $seq_x^{h}$ . In Fig. 16, $\delta _i^h$ is the time difference between the arrivals of packets p ₂ and p ₁.

Figure 16. Illustration of ${\delta ^h}$ .

Packets in $\delta _i^h$ are transmitted before p ₁, and cannot cause any delay to packet i. The delay at node h suffered by packet i can be maximised when $\delta _i^h$ is minimised. For the computation of the worst-case end-to-end transmission delay, it comes to determine the minimum value of $\delta _i^h$ at each visited node h. it is obvious that the minimum value of $\delta _i^h$ is obtained by minimising $\alpha _{{p_{\rm{1}}}}^h$ and maximising $\alpha _{{p_x}}^h$ where $x{\rm{}} \in {\rm{\{ 2, \ldots ,}}n{\rm{\} }}$ .

Let us define $l_x^h\,\left( {{\rm{1}} \le {\rm{}}x{\rm{}} \le {\rm{}}n} \right)$ is the transmission time of the packet sequence $seq_x^{h}$ without its first packet, from Fig. 16, we can obtain:

(35) $$\matrix{{\alpha _{{p_x}}^h = {\rm{ }}\eta - l_x^h + L} \hfill \cr} $$

and when x = 1 we have

(36) $$\matrix{{\alpha _{{p_1}}^h = {\rm{}}\eta - l_1^h + L} \hfill \cr} $$

Consequently, minimising $\alpha _{{p_{\rm{1}}}}^h$ comes to maximise $l_1^h$ . It is obtained when the first packet is the smallest one in the packet sequence $seq_{\rm{1}}^{h}$ . Similarly, maximising $\alpha _{{p_x}}^h$ comes to minimise each $l_x^h$ for $\left( {{\rm{2}} \le {\rm{}}x{\rm{}} \le {\rm{}}n} \right)$ . It is obtained when the first packet is the largest one in the respective sequence.

To summarise, $\delta _i^h$ can be lower bounded by the maximum of 0 and Eq. (37).

(37) \begin{align}ma{x_{{\rm{2}} \le {{x}} \le {{n}}}}{\rm{(}}mi{n_{}}{\rm{(}}l_x^h{\rm{))}} - ma{x_{}}{\rm{(}}l_1^h{\rm{)}}\end{align}

The new computation of the serialisation effect can be expressed by Eq. (38).

For the cumulative transmission time caused by the packets from the same physical link as flow v _i, it is always limited to the transmission time of the maximum packet in this physical link as depicted in Fig. 17.

Figure 17. Maximum cumulative transmission time.

The pseudo-code for the Forward Approach integrated with the new computation of the serialisation effect is given by Algorithm 3. In fact, the computation can also be done in a discrete manner by testing time t only at arrival times of new packets.

(38) \begin{align}{\rm{\delta }}_{\rm{i}}^{\rm{h}} & = \left[ \mathop {{\rm{max}}}\limits_{{\rm{x}} \in \left[ {{\rm{2,n}}} \right]} \left( {\sum\nolimits_{{{\rm{v}}_{\rm{y}}} \in {\rm{IP}}_{\rm{x}}^{\rm{h}}} {\left( {1 + \left\lfloor {{{{\rm{t}} + {\rm{J}}_{\rm{y}}^{\rm{h}}} \over {{{\rm{T}}_{\rm{y}}}}}} \right\rfloor } \right)} \cdot {{\rm{C}}_{\rm{y}}} - {{\max }_{{{\rm{v}}_{\rm{y}}} \in {\rm{IP}}_{\rm{x}}^{\rm{h}}}}\left( {{{\rm{C}}_{\rm{y}}}} \right)} \right)\right. \nonumber\\ &\quad\left. - \left( {\sum\nolimits_{{{\rm{v}}_{\rm{y}}} \in {\rm{IP}}_{\rm{1}}^{\rm{h}}} {\left( {1 + \left\lfloor {{{{\rm{t}} + {\rm{J}}_{\rm{y}}^{\rm{h}}} \over {{{\rm{T}}_{\rm{y}}}}}} \right\rfloor } \right)} \cdot {{\rm{C}}_{\rm{y}}} - {{\min }_{{{\rm{v}}_{\rm{y}}} \in {\rm{IP}}_{\rm{1}}^{\rm{h}}}}\left( {{{\rm{C}}_{\rm{y}}}} \right)} \right) \right]^{+} \end{align}

Algorithm 3 Forward Approach integrated with the new computation of the serialisation effect

6.1 Delays on illustrative AFDX networks

For the network configuration depicted in Fig. 14, we still focus on flow v ₁.

At the source node e ₁, there is only flow v ₁, and the backlog at node e ₁ is zero. The corresponding computation is similar to Eq. (28) and (29). We also have

\begin{align*}S{\rm{min}}_{\rm{1}}^{{S_1}} = S{\rm{max}}_{\rm{1}}^{{S_1}}{ = \rm{ 40 + 16 = 56\mu s}}\end{align*}

At node S ₁, there are two input physical links, for $IP_{\rm{1}}^{{S_1}}$ , according to our new computation of the serialisation effect, the maximum cumulative transmission time is

\begin{align*}W_{\rm{1}}^{{S_1}}{ = \rm{ max\{ }}{C_{\rm{1}}}{\rm{\} = 40\mu s}}\end{align*}

For $IP_{\rm{2}}^{{S_1}}$ , the maximum cumulative transmission time

\[{\rm{}}W_{\rm{2}}^{{S_1}} = \left( {{\rm{1 + }}\left\lfloor {{t \over {{T_{\rm{2}}}}}} \right\rfloor } \right) \cdot {C_{\rm{2}}} = 40{\rm{\mu s}}\]

As C ₁ and C ₂ arrive at node S ₁ simultaneously, the serialisation effect at node S ₁ $\delta _{\rm{1}}^{{S_1}} = 0$ . The maximum cumulative transmission time

\[{{W}}_{}^{{{{S}}_1}}{ = { W}}_1^{{{{S}}_1}} + {{W}}_{{2}}^{{{{S}}_1}} - {\rm{\delta }}_{\rm{1}}^{{{{S}}_1}}{ = \rm{ 40 + 40}} - {\rm{0}} = \,80{\rm{\mu s}}\]

and the maximum backlog can be expressed:

\begin{align*}Bklg_1^{{S_1}} = {\rm{}}W_{}^{{S_1}} - {{\rm{C}}_1} - t{\rm{}} = {\rm{80}} - {\rm{40}} = {\rm{40\mu s}}\end{align*}

We can have:

\begin{align*}Smax_{\rm{1}}^{{S_{\rm{3}}}} = Smax_{\rm{1}}^{{S_1}} + Bklg_{\rm{1}}^{{S_1}} + {{\rm{C}}_1}{\rm{ + L}} = {\rm{56 + 40 + 40 + 16 = 152\mu s}}\end{align*}

At node S ₃, there are three input physical links, for $IP_{\rm{1}}^{{S_{\rm{3}}}}$ , the maximum cumulative transmission time

\[W_{\rm{1}}^{{S_1}}{ = \rm{ max\{ }}{C_{\rm{1}}}{\rm{\} = 40}}{\rm{\mu s}}\]

For $IP_{\rm{2}}^{{S_{\rm{3}}}}$ , the maximum cumulative transmission time is

\begin{align*}W_{\rm{2}}^{{S_1}} = \left( {{\rm{1 + }}\left\lfloor {{t \over {{T_3}}}} \right\rfloor } \right) \cdot {C_{\rm{3}}} + \left( {{\rm{1 + }}\left\lfloor {{t \over {{T_{\rm{4}}}}}} \right\rfloor } \right) \cdot {C_4}{ = \rm{ 80\mu s}}\end{align*}

For $IP_{\rm{3}}^{{S_{\rm{3}}}}$ , the maximum cumulative transmission time

\[W_{\rm{3}}^{{S_1}} = \max \{ {C_5}\} = 40{\rm{\mu s}}\]

Packet 3 arrives at node S ₃ earlier than packet 4, it arrives simultaneously with packet 2, so the serialisation effect at node S ₃ $\delta _{\rm{1}}^{{S_{\rm{3}}}} = {\rm{0}}$ . The maximum cumulative transmission time is

\[W_{}^{{S_{\rm{3}}}} = W_{\rm{1}}^{{S_{\rm{3}}}} + {\rm{}}W_{\rm{2}}^{{S_{\rm{3}}}} + {\rm{}}W_{\rm{3}}^{{S_{\rm{3}}}}{ = \rm{ 40 + 80 + 40 = 160\mu s}}\]

and the maximum backlog can be computed as below:

\[Bklg_i^h = W_{}^{{S_3}} - {C_1} - t = 160 - 40 = 120{\rm{\mu s}}\]

The worst-case end-to-end transmission delay is:

\[{R_{\rm{1}}} = S{\rm{max}}_{\rm{1}}^{{S_{\rm{3}}}} + {\rm{}}Bklg_{\rm{1}}^{{S_{\rm{3}}}} + {C_{\rm{1}}} = {\rm{152}} + {\rm{120}} + {\rm{40}} = {\rm{312\mu s}}\]

Compared with the original computation of the serialisation effect in Section 4.6, this new computation eliminates the optimism in Section 5.2. The obtained result is consistent with Fig. 15.

The upper bounds of other flows computed with different approaches are given in Table 5. The comparison histogram is depicted in Fig. 18.

Table 5 Upper bounds of end-to-end transmission delay

FA: Forward Approach without considering the serialisation effect.

FAOS: Forward Approach with original serialisation.

FANS: Forward Approach with new serialisation.

MC: Model Checking approach.

Figure 18. Comparison of delay upper bounds obtained with different approaches.

6.2 Delays on real avionics systems

This section evaluates the new computation of the serialisation effect on a small real avionics network depicted in Fig. 19. It includes 23 end systems, 7 switches and 23 virtual links corresponding to 26 paths due to the multicast characteristic. All the VLs have a common traffic feature (BAG = 4000μs, C = 40μs). The technological latency is 16μs.

Figure 19. A small real avionics network.

The worst-case end-to-end transmission delays computed with different approaches are summarised in Table 6, and the comparison is depicted in Fig. 20. The results obtained by the Model Checking approach are used as a reference.

Table 6 Delay upper bounds computed with different approaches

FA: Forward Approach. FAOS: Forward Approach with original serialisation.

MC: Model Checking approach. FANS: Forward Approach with new serialisation.

Figure 20. Comparison of delay upper bounds computed with different approaches.

For all these flows, FANS can compute the exact worst-case end-to-end delays.

For flows v ₁, v ₃, v ₄, v ₁₀, v ₂₃, FA, FAOS and FANS can compute the exact worst-case end-to-end delays, as no serialisation effect occurs during the whole transmission, and no pessimism is involved during the computation.

For flows v ₅, v ₆, v ₇, v ₁₃, v ₁₉, v ₂₀, v ₂₁ and v ₂₂, FA computes pessimistic upper bounds. FAOS and FANS can compute the exact worst-case end-to-end delays. It demonstrates that both the original computation of the serialisation effect and the new computation of the serialisation effect can eliminate the pessimism in the computation.

For flows v ₂, v ₈, v ₉, v ₁₁, v ₁₂, v ₁₄, v ₁₅, v ₁₆, v ₁₇ and v ₁₈, FA computes pessimistic upper bounds. FAOS computes optimistic upper bounds. It demonstrates FAOS involves optimism when eliminating the pessimism. However, FANS can compute the exact worst-case end-to-end transmission delays, it demonstrates that FANS eliminate the pessimism and does not involve the optimism in the original computation of the serialisation effect.

For example, flow v ₂ with path ${e_2} - {S_4} - {S_5}$ . The exact worst-case end-to-end transmission delay is 272μs as depicted in Fig. 21. FAOS eliminates the pessimism at S ₄, but involves optimism at S ₅ as it considers only one packet from e ₁₁ can delay packet 2. In fact, both packets 14 and 11 can delay packet 2.

Figure 21. Delay upper bound of flow v _2.

Both the experiments on the illustrative AFDX network and the real avionics system demonstrate that the new computation of the serialisation effect can eliminate the pessimism in the Forward Approach and does not induce the optimism existing in the original computation of the serialisation effect.