Optimizing the deployment of tree-shaped functional graphs of real-time system on distributed architectures

Abstract

Recent development methodologies from the industry and the academia for complex real-time systems define a stage in which system functions are deployed onto an execution platform. The deployment consists of the placement of functions on a distributed network of nodes, the partitioning of functions in tasks and the scheduling of tasks and messages. In this paper, we present two approaches towards the efficient deployment of realistic and complex real-time systems by considering tree-shaped functional models. A formal approach to compute optimal deployment and a heuristic approach to scale to industry-size systems. The approaches consider placement, partitioning and scheduling, and are based on mixed integer linear programming (MILP) technique. Furthermore, we present a deep evaluation of the proposed deployment approaches to show the benefits and limits of a MILP-based deployment approach. A set of synthetic use-cases as well as a real-life automotive system are used to assess the quality and scalability of our deployment approaches. Considering use-cases, we show an added value with respect to end-to-end latencies optimization when solving the three stages of the deployment problem at the same time. This is done by comparing the quality of the solutions obtained with our techniques to those returned by the existing approaches.

A Appendix: The complete MILP formulation for the PP and PS stages of the two-step optimization approach

This section details the MILP formulation for each sub problem of the two-step deployment approach. The objective function to optimize is expressed in the same way in the formulations of both sub-problems, it analogous to the one defined for the integrated MILP formulation, in Sect. 4:

$$\begin{aligned} Minimize: \frac{ \sum _{\rho _i \in \gamma } L_i}{\# \gamma } \end{aligned}$$

1.1 A.1 MILP formulation of the placement problem PP

The objective of this step is optimizing the placement of either an initial configuration or the output of the previous iteration. The parameters are the priorities assigned to functions and signals.

• 1. Placement of functions on nodes

  $$\begin{aligned} {{\left\{ \begin{array}{ll} \forall i \in F: \displaystyle \sum _{j\in C(i)} A_{i,j} =1 \\ \forall i \in F: \displaystyle \sum _{j \in C {\setminus } \{C(i)\}} A_{i,j}=0 \\ \forall i,j \in F, k \in C: 0 \le A_{i,k} + A_{j,k} - \left( 2 \cdot X_{i,j,k}\right) \le 1 \\ \forall j \in C: \displaystyle \sum _{i \in F} A_{i,j} \left( \frac{\omega _{i,j}}{ P_{i}} \right) \le \eta _{j} \\ \end{array}\right. }} \end{aligned}$$

• 2. Placement of signals on buses

  $$\begin{aligned} {{\left\{ \begin{array}{ll} \forall i \in \varPhi : \displaystyle \sum _{j\in \beta } AS_{i,j} = G_{i} \\ \forall i \in \varPhi , j \in rec(i): 1- \displaystyle \sum _{k\in C} X_{snd(i),j,k} = G_{i}\\ \forall i,j \in \varPhi , k \in \beta : 0 \le AS_{i,k} + AS_{j,k} - \left( 2 \cdot XS_{i,j,k}\right) \le 1 \\ \forall i \in \varPhi , j \in rec(i), b \in \beta : 0 \le G_{i}+\displaystyle \sum _{k \in NB(b)} A_{snd(i),k} + \displaystyle \sum _{k \in NB(b)} A_{j,k} - 3 \cdot AS_{i,b} \le 3 \\ \forall j \in \beta : \displaystyle \sum _{i \in \varPhi } AS_{i,j} \left( \frac{\omega _{i,j}}{ P_{i}} \right) \le \eta _{j}\\ \forall i,j \in \varPhi , \pi _i = \pi _j: \displaystyle \sum _{k \in \beta } XS_{i,j,k} \le \sum _{k \in C} X_{snd(i),snd(j),k}\\ \end{array}\right. }} \end{aligned}$$

• 3. Functions WCRT computation

  $$\begin{aligned} {{\left\{ \begin{array}{ll} \forall i \in F: R_{i} =W_{i} + J_{i}\\ \forall i,j \in F: 0 \le \sigma _{i,j}- \left( \frac{W_{i}+J_{j}}{P_{j}}\right) < 1 \\ \forall i,j \in F, \pi j> \pi _i: \sigma _{i,j}- M \left( 1- \displaystyle \sum _{k \in C} X_{i,j,k}\right) \le I_{i,j} \\ \forall i,j \in F, \pi j> \pi _i: I_{i,j} \le \sigma _{i,j} \\ \forall i,j \in F, \pi j> \pi _i: I_{i,j} \le M \cdot \displaystyle \sum _{k \in C} X_{i,j,k} \\ \forall i \in F: W_{i}= \displaystyle \sum _{k \in C} A_{i,k}\left( \omega _{i,k}+2 \cdot cs\right) + \displaystyle \sum _{\underset{\pi _{j}=\pi _{i}}{j \in F{\setminus } \{i\}:}} \displaystyle \sum _{k \in C} X_{i,j,k} \cdot \omega _{j,k} \\ \quad \quad \quad \quad \quad \quad \quad \quad + \displaystyle \sum _{ \begin{array}{c} {j \in F:} \\ { \xi _{i,j}=1}\\ {\pi _{j}>\pi _{i}} \end{array}} \sum _{k \in C} I_{i,j} \left( \omega _{j,k}+2 \cdot cs\right) \\ \forall i \in \gamma , j \in F, j=snk_{(i)}: L_i\ge R_j \\ \forall i \in \gamma : L_i\le D_i \\ \forall j \in \varPhi , i \in rec(j): \phi _{i} \ge RS_{j} \\ \forall i,j \in F , \pi _i = \pi _j : \phi _{j}- M \left( 1- \displaystyle \sum _{k \in C} X_{i,j,k}\right) \le \theta _{i,j} \\ \forall i,j \in F, \pi _i = \pi _j : \theta _{i,j} \le \phi _{j}\\ \forall i,j \in F, \pi _i = \pi _j :\theta _{i,j} \le M \cdot \displaystyle \sum _{k \in C} X_{i,j,k} \\ \forall i,j \in F, \pi _j = \pi _i: J_{i} \ge \theta _{i,j} \\ \end{array}\right. }} \end{aligned}$$

• 4. Signals WCRT computation

  $$\begin{aligned} {{\left\{ \begin{array}{ll} \forall i \in \varPhi : RS_{i} = WS_{i} + \displaystyle \sum _{k \in \beta } AS_{i,k} \cdot \omega _{i,k}+ \displaystyle \sum _{\begin{array}{c} {j \in \varPhi {\setminus } \{i\}:} \\ {\pi _i =\pi _j} \end{array}} \sum _{k \in \beta } XS_{i,j,k} \cdot \omega _{j,k} + JS_{i} \\ \forall i,j \in \varPhi : 0 \le \sigma {s_{i,j}}- \left( \frac{WS_{i}+JS_{j}}{P_{j}}\right)< 1 \\ \forall i,j \in \varPhi , \pi j> \pi _i : \sigma {s_{i,j}}- M\left( 1- \displaystyle \sum _{k \in \beta } XS_{i,j,k}\right) \le IS_{i,j} \\ \forall i,j \in \varPhi , \pi j> \pi _i : IS_{i,j} \le \sigma {s_{i,j}} \\ \forall i,j \in \varPhi , \pi j> \pi _i : IS_{i,j} \le M \cdot \displaystyle \sum _{k \in \beta }XS_{i,j,k} \\ \forall i,j \in \varPhi , \pi _j < \pi _i, k \in \beta : BS_{i,k} \ge XS_{i,j,k} \cdot \omega _{j,k} + \displaystyle \sum _{ \underset{\pi _j = \pi _l}{l \in \varPhi {\setminus } \{i,j\}\}:}} XS_{j,l,k} \cdot \omega _{l,k} \\ \forall i \in \varPhi , k \in \beta : BS_{i,k} \ge \displaystyle \sum _{ \underset{\pi _i = \pi _j}{j \in \varPhi :}} XS_{i,j,k} \cdot \omega _{j,k}\\ \forall i \in \varPhi : WS_{i}= \displaystyle \sum _{k \in \beta } BS_{i,k} + \displaystyle \sum _{ \begin{array}{c} {j \in \varPhi :} \\ {\pi _j > \pi _i} \\ {\xi S_{i,j}=1} \end{array}} \sum _{k \in \beta } IS_{i,j} \cdot \omega _{j,k} \\ \forall i \in \varPhi : R_{snd(i)}-M \left( 1- (1-G_i)\right) \le \alpha _i \\ \forall i \in \varPhi : \alpha _i \le R_{snd(i)}\\ \forall i \in \varPhi : \alpha _i \le M (1-G_i)\\ \forall i \in \varPhi : J_{snd(i)}-M\left( 1- (1-G_i)\right) \le \zeta _i \\ \forall i \in \varPhi : \zeta _i \le J_{snd(i)}\\ \forall i \in \varPhi : \zeta _i \le M\left( 1-G_i\right) \\ \forall i \in \varPhi , j \in rec(i): JS_i \ge \alpha _{i}\left( 1-SP_{snd(i),j}\right) + \zeta _i \cdot SP_{snd(i),j} \\ \forall i,j \in \varPhi : JS_i \ge \varLambda _{i,j}\\ \forall i,j \in \varPhi , \pi _j = \pi _i : R_{snd(j)} -M\left( 1-\displaystyle \sum _{k \in \beta }XS_{i,j,k}\right) \le \varLambda _{i,j}\\ \forall i,j \in \varPhi , \pi _j = \pi _i : \varLambda _{i,j} \le R_{snd(j)}\\ \forall i,j \in \varPhi , \pi _j = \pi _i : \varLambda _{i,j} \le M \cdot \displaystyle \sum _{k \in C}XS_{i,j,k} \\ \end{array}\right. }} \end{aligned}$$

The result of this PP MILP formulation consists of a placement of functions and signals which will be the input for the next stage.

1.2 A.2 MILP formulation for the partitioning and scheduling problem PS

At this stage, the placement of functions and signals is given and the following MILP formulation aims at improving (if possible) the partitioning and scheduling. The MILP formulation assigns priority orders to functions and signals within each node and bus, then tasks and messages are constructed based on priorities, i.e. a task (message) is the set of functions (signals) with the same priority order residing on the same node (bus).

• 1. Functions partitioning and scheduling

  $$\begin{aligned} {{\left\{ \begin{array}{ll} \forall i,j \in F: \psi _{i,j}+\psi _{j,i} \le 1 \\ \forall i,j,k \in F: \psi _{i,j}+\psi _{j,k} - 1 \le \psi _{i,k} \\ \forall i,j,k \in F: \psi _{i,j}- \left( \psi _{j,k}+ \psi _{k,j}\right) \le \psi _{i,k} \\ \forall i,j,k \in F: \psi _{j,k}- \left( \psi _{j,i}+ \psi _{i,j}\right) \le \psi _{i,k} \\ \forall i,j,k \in F: \psi _{i,j} + \psi _{j,i}+ \psi _{j,k}+ \psi _{k,j} \ge \psi _{i,k} \\ \forall i,j,k \in F: \psi _{i,j} + \psi _{j,i}+ \psi _{j,k}+ \psi _{k,j} \ge \psi _{k,i} \\ \forall i,j \in F: 1= SP_{i,j}+\psi _{j,i}+\psi _{i,j} \\ \text {A binary variable } SP_{i,j}\text { denotes whether}\\ f_i\text {/}s_i\text { and }f_j\text {/}s_j\text { have the same priority order.} \end{array}\right. }} \end{aligned}$$

• 2. Signals partitioning and scheduling

  $$\begin{aligned} {{\left\{ \begin{array}{ll} \forall i,j \in \varPhi : \psi S_{i,j}+\psi S_{j,i} \le 1 \\ \forall i,j,k \in \varPhi : \psi S_{i,j}+\psi S_{j,k} - 1 \le \psi S_{i,k} \\ \forall i,j,k \in \varPhi : \psi S_{i,j}- \left( \psi S_{j,k}+ \psi _{k,j}\right) \le \psi S_{i,k} \\ \forall i,j,k \in \varPhi : \psi S_{j,k}- \left( \psi S_{j,i}+ \psi _{i,j}\right) \le \psi S_{i,k} \\ \forall i,j,k \in \varPhi : \psi S_{i,j} + \psi S_{j,i}+ \psi _{j,k}+ \psi S_{k,j} \ge \psi S_{i,k} \\ \forall i,j,k \in \varPhi : \psi S_{i,j} + \psi S_{j,i}+ \psi S_{j,k}+ \psi S_{k,j} \ge \psi S_{k,i} \\ \forall i,j \in \varPhi : 1= SPS_{i,j}+\psi S_{j,i}+\psi S_{i,j} \\ \forall i \in \varPhi , j,k \in rec(i), SN_{j,k}=1 : SP_{snd(i),j}- SP_{snd(i),k}=0 \\ \forall i,j \in \varPhi , \varOmega _{i} = \varOmega _{j}, SN_{snd(i),snd(j)}=0 : SPS_{i,j} =0 \end{array}\right. }} \end{aligned}$$

• 3. Functions WCRT computation

  $$\begin{aligned} {{\left\{ \begin{array}{ll} \forall i \in F: R_{i} =W_{i} + J_{i}\\ \forall i,j \in F: 0 \le \sigma _{i,j}- \left( \frac{W_{i}+J_{j}}{P_{j}}\right) < 1 \\ \forall i,j \in F, SN_{i,j} = 1: \sigma _{i,j}- M \left( 1- \psi _{j,i}\right) \le I_{i,j} \\ \forall i,j \in F, SN_{i,j} = 1: I_{i,j} \le \sigma _{i,j} \\ \forall i,j \in F, SN_{i,j} = 1: I_{i,j} \le M \cdot \psi _{j,i} \\ \forall i \in F, node of i=k: W_{i}= \omega _{i,k}+ 2 \cdot cs + \displaystyle \sum _{\begin{array}{c} {j \in F{\setminus } \{i\}:}\\ {SN_{i,j}=1} \end{array}} SP_{i,j} \cdot \omega _{j,k} \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,+ \displaystyle \sum _{ \begin{array}{c} {j \in F:} { \xi _{i,j}=1}\\ {SN_{i,j}=1} \end{array}} \sum _{k \in C} I_{i,j} \left( \omega _{j,k}+2 \cdot cs\right) \\ \forall i \in \gamma , j \in F, j=snk_{(i)}: L_i\ge R_j \\ \forall i \in \gamma : L_i\le D_i \\ \forall j \in \varPhi , i \in rec(j): \phi _{i} \ge RS_{j} \\ \forall i,j \in F, SN_{i,j} = 1: \phi _{j}- M \left( 1- SP_{i,j}\right) \le \theta _{i,j} \\ \forall i,j \in F, SN_{i,j} = 1 : \theta _{i,j} \le \phi _{j}\\ \forall i,j \in F, SN_{i,j} = 1 :\theta _{i,j} \le M \cdot SP_{i,j} \\ \forall i,j \in F, SN_{i,j} = 1: J_{i} \ge \theta _{i,j} \\ \end{array}\right. }} \end{aligned}$$

• 4. Signals WCRT computation

  $$\begin{aligned} {{\left\{ \begin{array}{ll} \forall i \in \varPhi , \varOmega _i = k : RS_{i} = WS_{i} + \omega _{i,k} \cdot G_i +\displaystyle \sum _{\begin{array}{c} {j \in \varPhi {\setminus } \{i\}:} \\ \varOmega _j = k \end{array}} SPS_{i,j} \cdot \omega _{j,k} + JS_{i} \\ \forall i,j \in \varPhi : 0 \le \sigma {s_{i,j}}- \left( \frac{WS_{i}+JS_{j}}{P_{j}}\right) < 1 \\ \forall i,j \in \varPhi ,\varOmega _i = \varOmega _j : \sigma {s_{i,j}}- M\left( 1- \psi S_{j,i}\right) \le IS_{i,j} \\ \forall i,j \in \varPhi ,\varOmega _i = \varOmega _j : IS_{i,j} \le \sigma {s_{i,j}} \\ \forall i,j \in \varPhi ,\varOmega _i = \varOmega _j : IS_{i,j} \le M \cdot \psi S_{j,i}\\ \forall i \in \varPhi , G_i=0: BS_i =0 \\ \forall i,j \in \varPhi , \varOmega _i = \varOmega _j =k : BS_{i} \ge \psi S_{i,j} \cdot \omega _{j,k} + \displaystyle \sum _{ \underset{\varOmega _j = \varOmega _l}{l \in \varPhi {\setminus } \{i,j\}:}} SPS_{j,l} \cdot \omega _{l,k} \\ \forall i,j \in \varPhi : BS_{i} \ge \displaystyle \sum _{ \underset{\varOmega _i = \varOmega _j = k}{j \in \varPhi :}} SPS_{i,j} \cdot \omega _{j,k} \forall i \in \varPhi : WS_{i}= BS_{i} + \displaystyle \sum _{ \begin{array}{c} {j \in \varPhi :} \\ {\varOmega _j = \varOmega _i=k} \\ {\xi S_{i,j}=1} \end{array}} IS_{i,j} \cdot \omega _{j,k} \\ \forall i \in \varPhi ,j \in rec(i), G_i =0: R_{snd(i)}-M \left( 1- (1-SP_{snd(i),j})\right) \le \alpha _i \\ \forall i \in \varPhi : \alpha _i \le R_{snd(i)}\\ \forall i \in \varPhi : \alpha _i \le M \left( 1-SP_{snd(i),j}\right) \\ \forall i \in \varPhi : J_{snd(i)}-M \left( 1- SP_{snd(i),j} \right) \le \zeta _i \\ \forall i \in \varPhi : \zeta _i \le J_{snd(i)}\\ \forall i \in \varPhi : \zeta _i \le M \cdot SP_{snd(i),j}\\ \forall i \in \varPhi G_i = 0: JS_i = \alpha _{i} +\zeta _i \\ \forall i,j \in \varPhi , G_i=1 : JS_i \ge \varLambda _{i,j}\\ \forall i,j \in \varPhi , \varOmega _j = \varOmega _i : R_{snd(j)} -M\left( 1-SPS_{i,j}\right) \le \varLambda _{i,j}\\ \forall i,j \in \varPhi , \varOmega _j = \varOmega _i : \varLambda _{i,j} \le R_{snd(j)}\\ \forall i,j \in \varPhi , \varOmega _j = \varOmega _i : \varLambda _{i,j} \le M \cdot SPS_{i,j} \\ \end{array}\right. }} \end{aligned}$$

This MILP formulation returns as output a set of functions (signal) partitions as well as a priority order for each partition (task or message) on each node (bus).