End-to-end response time with fixed priority scheduling: trajectory approach versus holistic...

INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMSInt. J. Commun. Syst. 2005; 18:37–56Published online 26 November 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/dac.688

End-to-end response time with fixed priority scheduling:trajectory approach versus holistic approach

Steven Martin1, Pascale Minet2,n,y and Laurent George3

1Ecole Centrale d’Electronique, LACCSC, 53 rue de Grenelle, 75 007 Paris, France2 INRIA Rocquencourt, 78 153 Le Chesnay, France

3Universit !ee Paris 12, LIIA, 120 rue Paul Armangot, 94 400 Vitry, France

SUMMARY

In this paper, we are interested in providing deterministic end-to-end guarantees to real-time flows in adistributed system. We focus on the end-to-end response time, quality of service (QoS) parameter of theutmost importance for such flows. We assume that each node uses a Fixed Priority scheduling. Wedetermine a bound on the end-to-end response time of any real-time flow with a worst case analysis usingthe trajectory approach. We establish new results that we compare with those provided by the classicalholistic approach for flows visiting the same sequence of nodes. These results show that the trajectoryapproach is less pessimistic than the holistic one. Moreover, the bound provided by our worst-case analysisis reached in various configurations, as shown in the examples presented. Copyright # 2004 John Wiley &Sons, Ltd.

KEY WORDS: fixed priority scheduling; deterministic guarantee; quality of service; worst-case end-to-endresponse time; trajectory approach; holistic approach

1. INTRODUCTION

Fixed priority scheduling has been extensively studied in the last years. First, in the uniprocessorcontext (see Reference [1] for a survey). These results have been extended to the distributed caseby Reference [2], where the holistic approach is proposed to determine the end-to-end responsetime of any flow in a network. This approach considers the worst-case scenario on each nodevisited by a flow, accounting for the maximum possible release jitter introduced by the previousvisited nodes. The jitter increases throughout the visited nodes, then the minimum andmaximum response times on a node h induce a maximum release jitter on the next visited nodehþ 1 that leads to a worst-case response time and then a maximum release jitter on thefollowing node and so on. The holistic approach can be pessimistic as it considers worst-casescenarios on every node possibly leading to consider impossible scenarios.

Received 20 December 2003Revised 18 April 2004Accepted 21 July 2004Copyright # 2004 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: Pascale Minet, INRIA Rocquencourt, 78 153 Le Chesnay, France.

To compute the worst case end-to-end response time of any flow, another approach has beenintroduced. We call this approach the trajectory approach. This approach is less pessimistic thanthe holistic one, as it considers only possible scenarios but is somewhat more complex. Wepropose a solution to guarantee deterministic end-to-end response time to any real-time flow ina distributed system using the trajectory approach, when all flows are scheduled according toFixed Priority. We then compare the obtained results with those provided by the holisticapproach.

Fixed Priority scheduling exhibits interesting properties:

* No clock synchronization is required, unlike earliest deadline first [1].* The impact of a new flow ti with a priority Pi is limited to flows having priorities smaller

than or equal to Pi:* It is easy to implement.* It is well adapted for service differentiation: flows with high priorities have smaller

response times.

In this paper, we compute the end-to-end response time obtained with Fixed Priorityscheduling. Our results can be used in various configurations:

* In a Differentiated Services architecture [3], several classes are defined, each havingits own priority. The highest priority class, that is the expedited forwarding (EF)class, is scheduled Fixed Priority with the other classes. Moreover, if packets belongingto the EF class need to be differentiated, different priorities can be assigned to thesepackets. Hence, a Fixed Priority scheduling can be used to provide the requesteddifferentiation.

* In an Integrated Services architecture [4], a priority is assigned to each flow. Fixed Priorityscheduling is used to provide shorter response times to high priority flows.

* In an hybrid architecture, some flows are managed per class, whereas others are managedindividually.

The rest of the paper is organized as follows. Section 2 briefly discusses related work. InSection 3, we define the problem and the different models. Notations and definitions arepresented in Section 4. We show in Section 5 how to compute an upper bound on the end-to-endresponse time of any flow, based on a worst case analysis, when all the flows follow the samesequence of nodes (the same line). In Section 6, we compare our results, obtained by applyingthe trajectory approach, with those provided by the holistic approach. In Section 7, we considerdifferent configurations and compare for these configurations the bounds on the end-to-endresponse times obtained by the trajectory approach and the holistic approach with theexact worst case end-to-end response times. These exact values are obtained by a validationtool we have designed, tool that does an exhaustive analysis. Finally, we conclude the paper inSection 8.

2. RELATED WORK

In this section, we first examine the existing approaches to obtain deterministic end-to-endresponse time guarantees in a distributed system. Then we recall some classical resultsestablished in the uniprocessor case for the fixed priority scheduling.

Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Commun. Syst. 2005; 18:37–56

S. MARTIN, P. MINET AND L. GEORGE38

2.1. End-to-end response time

To determine the maximum end-to-end response time, several approaches can be used: astochastic or a deterministic one. A stochastic approach consists in determining the meanbehaviour of the considered network, leading to mean, statistical or probabilistic end-to-endresponse times [5, 6]. A deterministic approach is based on a worst case analysis of the networkbehaviour, leading to worst case end-to-end response times [7, 8].

In this paper, we are interested in the deterministic approach as we want to provide adeterministic guarantee of worst case end-to-end response times for any flow in the network. Inthis context, two different approaches can be used to determine the worst case end-to-end delay:the holistic approach and the trajectory approach.

* The holistic approach [2] considers the worst case scenario on each node visited by a flow,accounting for the maximum possible jitter introduced by the previous visited nodes. If nojitter control is done, the maximum jitter will increase throughout the visited nodes. In thiscase, the minimum and maximum response times on a node h induce a maximum jitter onthe next visited node hþ 1 that leads to a worst case response time and then a maximumjitter on the following node and so on. Otherwise, the jitter can be either cancelled orconstrained.* the Jitter Cancellation technique consists in cancelling, on each node, the jitter of a flow

before it is considered by the node scheduler [8]: a flow packet is held until its latestpossible reception time. Hence a flow packet arrives at node hþ 1 with a jitterdepending only on the jitter introduced by the previous node h and the link betweenthem. As soon as this jitter is cancelled, this packet is seen by the scheduler of nodehþ 1: The worst case end-to-end response time is obtained by adding the worst caseresponse time, without jitter (as cancelled) on every node;

* the Constrained Jitter technique consists in checking that the jitter of a flow remainsbounded by a maximum acceptable value before the flow is considered by the nodescheduler. If not, the jitter is reduced to the maximum acceptable value by means oftraffic shaping.

As a conclusion, the holistic approach can be pessimistic as it considers worst casescenarios on every node possibly leading to impossible scenarios.

* The trajectory approach [9] consists in examining the scheduling produced by all the visitednodes of a flow. In this approach, only possible scenarios are examined. For instance, thefluid model (see Reference [10] for GPS) is relevant to the trajectory approach. Thisapproach produces the best results as no impossible scenario is considered but is somewhatmore complex to use. This approach can also be used in conjunction with a jitter control(see Reference [11] for EDF, and Reference [10] for GPS). In this paper, we adopt thetrajectory approach without jitter control in a distributed system to determine themaximum end-to-end response time of a flow.

We can also distinguish two main traffic models: the sporadic model and the token bucketmodel. The sporadic model has been used in the holistic approach and in the trajectoryapproach, while the token bucket model has been used only in the trajectory approach.

* The sporadic model is classically defined by three parameters: the maximum processingtime, the minimum interarrival time and the maximum release jitter, (see Section 2.2). Thismodel is natural and well adapted for real-time applications.


END-TO-END RESPONSE TIME WITH FIXED PRIORITY SCHEDULING 39

* The token bucket [7, 10, 11] is defined by two parameters: s; the bucket size, and r; thetoken throughput. The token bucket can model a flow or a flow aggregate. In the first case,it requires to maintain per flow information on every visited node. This solution is notscalable. In the second case, the choice of good values for the token bucket parameters iscomplex when flows have different characteristics. With this model, arises the problem offixing the good values of these parameters for a given application. As shown in References[11, 12], the end-to-end response times strongly depend on the choice of the token bucketparameters. A bad choice can lead to bad response times. Furthermore, the token bucketparameters can be optimized for a given configuration, only valid at a given time. If theconfiguration evolves, the parameters of the token bucket should be recomputed on everynode to remain optimal. This is not generally done.

In this paper, we adopt the trajectory approach with the sporadic traffic model and weestablish new results that we compare with those provided by the classical holistic approach.

2.2. Fixed priority scheduling in a uniprocessor context

In this section, we recall some classical results used in hard real-time scheduling and we giveseveral properties and results established in the uniprocessor case for the non-preemptive FixedPriority scheduling algorithm.

We consider a set t ¼ ft1; . . . ; tng of n sporadic flows with ti ¼ fCi;Ti; Jinig; where:

* Ti denotes the minimum interarrival time (abusively called period) between two successivepackets of ti;

* Ci denotes the maximum processing time of a packet of ti;* Jini denotes the maximum jitter of packets of ti:

Due to the scheduling model, to any flow ti 2 t is assigned a fixed priority Pi: Then, wedenote:

* hpi ¼ fj 2 ½1; n�; j=i; such that Pj5Pig;* hpi ¼ fj 2 ½1; n� such that Pj5Pig:

In the following, we assume that time is discrete. Reference [13] shows that results obtainedwith a discrete scheduling are as general as those obtained with a continuous scheduling whenall flow parameters are multiples of the node clock tick. In such conditions, any set of flows isfeasible with a discrete scheduling if and only if it is feasible with a continuous scheduling.

As detailed in Section 3.1, packet scheduling is non-preemptive. Hence, despite the highpriority of any packet m; a packet with a lower priority can delay m processing due to non-preemption. Indeed, if a packet m of any flow ti is released while a packet m0 belonging to hpi isbeing processed, m has to wait until m0 completion. It is important to notice that the non-preemptive effect is not limited to this waiting time. Indeed, the delay incurred by packet mdirectly due to m0 may lead to consider packets with a higher priority than m released after m butbefore m starts its execution. The following lemma establishes the maximum delay incurred by apacket directly due to another packet with a lower priority.

Lemma 1The maximum delay incurred by any packet of any flow ti directly due to packets belonging tohpi is equal to: maxð0; max

j2hpifCjg � 1Þ; where max

j2hpifCjg ¼ 0 if hpi ¼ |:



ProofSee Reference [1]. &

We now focus on a flow ti 2 t: Let WiðtÞ denote the latest starting time of the packet of tireleased at time t: Let Rmaxi denote the worst case response time of flow ti:

Property 1In the uniprocessor case, when all flows are scheduled according to the non-preemptive fixedpriority algorithm, the worst case response time of any flow ti meets:

Rmaxi ¼ maxk¼0::K

fWiðk � Ti � Jini Þ � k � Tig þ Jini þ Ci

where

Wiðk � Ti � Jini Þ ¼Xj2hpi

1þWiðk � Ti � Jini Þ þ Jinj

Tj

� �� Cj

þ k � Ci þmax 0; maxj2hpi

fCjg � 1

!

with K the smallest integer value such that WiðK � Ti � Jini Þ þ Ci þ Jini4ðK þ 1Þ � Ti:

ProofSee Reference [1]. &

Lemma 2If for any flow ti; i 2 ½1; n�; the input jitter is increased or left unchanged, the quantityRmaxi � Jini is increased or left unchanged.

ProofLet R0

maxidenote the worst case response time of flow ti when the release jitter of ti is equal to

J 0ini; where J 0

ini5Jini : We want to show that: 8i 2 ½1; n�; R0

maxi� J 0

ini5Rmaxi � Jini : We have:

R0maxi

� J 0ini

¼ maxk¼0::KfWiðk � Ti � J 0iniÞ � k � Tig þ Ci: We consider the series:

Wð0Þi ðk � Ti � Jini Þ ¼

Xj2hpi

Cj þ k � Ci þmax 0;maxj2hpi

fCjg � 1

!

Wðpþ1Þi ðk � Ti � Jini Þ ¼

Xj2hpi

1þW

ðpÞi ðk � Ti � Jini Þ þ Jinj

Tj

$ % !� Cj

þ k � Ci þmax 0;maxj2hpi

fCjg � 1

!



At rank 0, we have Wð0Þi ðk � Ti � J 0

iniÞ5W

ð0Þi ðk � Ti � Jini Þ: Let us assume that

WðpÞi ðk � Ti � J 0

iniÞ5W

ðpÞi ðk � Ti � Jini Þ; we have W

ðpþ1Þi ðk � Ti � J 0

iniÞ equal to

Xj2hpi

1þW

ðpÞi ðk � Ti � J 0

iniÞ þ Jinj

Tj

$ % !� Cj þ k � Ci þmax 0;max

j2hpi

fCjg � 1

!

4Xj2hpi

1þW

ðpÞi ðk � Ti � Jini Þ þ Jinj

Tj

$ % !� Cj þ k � Ci þmax 0;max

j2hpi

fCjg � 1

!

4Wðpþ1Þi ðk � Ti � Jini Þ

Hence, we get Wiðk � Ti � J 0iniÞ5Wiðk � Ti � Jini Þ:

We deduce: R0maxi

� J 0ini5Rmaxi � Jini : &

3. THE PROBLEM

We investigate the problem of providing a deterministic end-to-end response time guarantee toany flow in a distributed system. The end-to-end response time of a flow is defined between itsingress node and its egress node. In this paper, we want to provide an upper bound on the end-to-end response time of any flow in the distributed system. As we make no particularassumption concerning the arrival times of packets in the distributed system, the feasibility of aset of flows is equivalent to meet the requirement, whatever the arrival times of the packets inthe distributed system. Moreover, we assume the following models.

3.1. Scheduling model

We consider that all the nodes in the distributed system schedule packets according to the non-preemptive fixed priority/highest priority first algorithm, denoted Fixed Priority in thefollowing. Moreover, we assume that packet scheduling is non-preemptive. Therefore, thenode scheduler waits for the completion of the current packet transmission (if any) beforeselecting the next packet.

3.2. Network model

The considered network is a distributed system where links interconnecting nodes are supposedto be FIFO and the network delay between two nodes has known lower and upper bounds: Lmin

and Lmax: Lmax is the maximum network delay between two nodes, one hop away. If Lmax doesnot exist in the network, it is then impossible to provide a deterministic guarantee on flow worstcase response times. The reader interested in the computation of Lmax is referred to Reference[14] for the CAN network and Reference [15] for point-to-point packet-switched networks.

In this paper, we consider neither network failures nor packet losses.

3.3. Traffic model

We consider a set t ¼ ft1; . . . ; tng of n sporadic flows. Each flow ti follows the same sequence ofnodes whose first node is the ingress node of the flow. In the following, we call line this sequence.



Moreover, a sporadic flow ti following a line L consisted of q nodes, numbered from 1 to q; isdefined by:

* Ti; the minimum interarrival time (abusively called period) between two successive packetsof ti;

* Chi ; the maximum processing time on node h of a packet of ti;

* J1ini; the maximum jitter of packets of ti arriving in the distributed system (i.e. on node 1,

the first node visited by ti: The ingress node is in charge of controlling the jitter ofany incoming flow. The maximum jitter accepted for flow ti after jitter control is equalto J1

ini:

This characterization is well adapted to real-time flows (e.g. process control, voice and video,sensor and actuator). Moreover, for any packet g; we denote tðgÞ the index number of the flowwhich g belongs to.

3.4. Admission control and worst case end-to-end response time

The worst case response time of a flow can be computed by the admission control, when a newflow ti requests its admission. The number of flows considered, n; is then the number of flowspresent at this time. The admission control decides to accept the new flow ti with a priority Pi ifand only if the two following conditions are met:

* The QoS requested by the new flow ti is met. In our case, the worst case end-to-endresponse time of ti is less than or equal to its end-to-end deadline;

* The QoS of already accepted flows with a priority Pi is always met. In our case, theadmission control has to check that the worst case end-to-end response times ofalready accepted flows with a priority Pi always meet their deadline. To avoid thatalready accepted flows with a priority strictly less than Pi miss their deadlines,the bandwidth granted to highest priority flows is limited. Indeed, to avoid flowstarvation, the bandwidth granted to the highest priority flows is limited to a smallpart. Hence flows with small priorities will still get bandwidth left by higher prioritiesflows. For instance, in the DiffServ model, the bandwidth granted to the EF class islimited, allowing assured forwarding and best effort classes to consume the bandwidth leftby the EF class.

4. NOTATIONS AND DEFINITIONS

In addition to the notations and definitions presented in our model description (see Section 3),we use the following ones in the rest of this paper. For any flow ti; i 2 ½1; n�; we denote:

* Jhini

(resp. Jhouti

) the worst case jitter of flow ti when entering (resp. when leaving) node h;* R1;q

maxi(resp. R

1;qmini

) the maximum (resp. the minimum) response time experienced by apacket of flow ti between its arrival time in node 1 and its departure time on node q;

* %RR1;qmaxi

(resp. %RR1;qmini

) the maximum (resp. the minimum) response time experienced by apacket of flow ti between its arrival time in node 1 and its arrival time in node q;

* H1;hi the maximum delay incurred by flow ti directly due to flows tj ; j 2 hpi; while visiting

nodes 1 to h:



Moreover, for any packet m of flow ti released at time t; we denote:

* Whi ðtÞ the time when packet m starts its execution in node h in the worst case. This time is

denoted latest starting time of packet m in node h:

Definition 1An idle time t is a time such that all packets arrived before t have been processed at time t:

Definition 2An idle time t of level i is a time such that all packets with a priority greater than or equal to iand arrived before t have been processed at time t:

Definition 3A busy period is defined by an interval ½t; t0Þ such that t and t0 are both idle times and there is noidle time 2 ðt; t0Þ:

Definition 4A busy period of level i is defined by an interval ½t; t0Þ such that t and t0 are both idle times oflevel i and there is no idle time of level i 2 ðt; t0Þ:

Definition 5For any node h; for any flow ti visiting h; the processor utilization factor for the flows belongingto hpðiÞ is denoted Uh

hpðiÞ: It is the fraction of processor time spent by node h to process packetsbelonging to hpðiÞ: It is equal to

Pj2hpðiÞ ðC

hj =TjÞ:

5. TRAJECTORY APPROACH

Let us consider any packet m belonging to any flow ti 2 t: Let t be the generation time of packetm on node 1: To compute the end-to-end response time of packet m; we identify the busy periodsof level Pi that affect the delay of m: For this, we consider the busy period of level Pi; denotedbp

qi ; in which m is processed on node q and we define f ðqÞ as the first packet processed in bp

qi

with a priority greater than or equal to Pi:The packet f ðqÞ has been processed in a busy period on node q� 1 at least of level Pi: Let

bpq�1i be this busy period. We then define f ðq� 1Þ as the first packet processed in bp

q�1i with a

priority greater than or equal to Pi: And so on until the busy period of node 1 in which thepacket f ð1Þ is processed (see Figure 1).

Figure 1. Starting time of packet m in node q:



Due to the non-preemptive effect, on any node h 2 ½1; q�; the execution of packet f ðhÞ can bedelayed once by a packet with a priority less than Pi:

5.1. Evaluation of the maximum delay directly due to the non-preemption

The following property gives a bound on the maximum delay incurred by flow ti directly due tothe non-preemption while visiting nodes 1 to q:

Property 2When all flows follow the same line L consisting of q nodes numbered from 1 to q; then themaximum delay incurred by any flow ti directly due to flows belonging to hpi meets:

H1;hþ1i 4H1;h

i þmax 0;maxj2hpi

fChþ1j g � min

j2hpi[figfCh

j g þ Lmax � Lmin

!

H1;1i ¼ max 0;max

j2hpi

fC1j g � 1

!

where 8h 2 ½1; q�; maxj2hpi

fChj g ¼ 0 if hpi ¼ |:

ProofBy recurrence on the number of nodes visited. On the first node visited, Property 2 is true.Assuming that Property 2 is true at rank h; we prove it at rank hþ 1:

Let us consider packet m of any flow ti generated at time t: Packet m arrives on node hþ 1 atthe earliest at time: Wh

i ðtÞ þ Chi þ Lmin: If a packet m0 2 hpi is being processed on node hþ 1;

its processing will end at the latest at time: Whi ðtÞ þ Lmax þ Chþ1

tðm0Þ: Hence, the maximumdelay incurred by packet m due to the non-preemptive scheduling is at most:Chþ1

tðm0Þ � ½ðWhi ðtÞ þ Ch

i þ LminÞ � ðWhi ðtÞ þ LmaxÞ�: This quantity can be bounded by

maxj2hpi

fChþ1j g �minj2hpi[figfCh

j g þ Lmax � Lmin: Hence the property (see Figure 2). &

Figure 2. Delay directly due to non-preemption on node hþ 1:



5.2. Computation of the latest starting time of packet m in node q

We focus again on packet m of flow ti generated at time t on node 1:We now expressWqi ðtÞ; that

is the latest starting time of packet m in node q: For the sake of simplicity, we numberconsecutively the packets of the considered busy periods of level Pi (see Figure 1). Hence, wedenote m0 � 1 (resp. m0 þ 1) the packet preceding (resp. succeeding) m0: Moreover, in thefollowing, we consider the arrival time of packet f ð1Þ as the time origin. By adding parts of theconsidered busy periods, we get W

qi ðtÞ equal to:

the processing time on node 1 of packets f ð1Þ to f ð2Þ þ Lmax

þ the processing time on node 2 of packets f ð2Þ to f ð3Þ þ Lmax

þ ...þ the processing time on node q of packets f ðqÞ to ðm� 1Þ:þ the maximum delay directly due to packets belonging to hpi while visiting nodes 1 to q:

By convention, f ðqþ 1Þ ¼ m: Then, the latest starting time of packet m in node q meets:

Wqi ðtÞ ¼

Xqh¼1

Xf ðhþ1Þ

g¼f ðhÞ

ChtðgÞ

!� C

qi þH

1;qi þ ðq� 1Þ � Lmax

We now consider the termPq

h¼1 ðPf ðhþ1Þ

g¼f ðhÞ ChtðgÞÞ: By definition, on any node h; packets f ðhÞ to

f ðhþ 1Þ belong to hpi since all the busy periods we consider are at least of level Pi: Let slow bethe slowest node of line L: This node is such that for any node h 2 ½1; q�; for any packet m0

visiting h; the processing time of m0 on node h is less than Cslowtðm0Þ: We then distinguish the nodes

visited by the flows before slow and those visited after. Thus, we getPq

h¼1 ðPf ðhþ1Þ

g¼f ðhÞ ChtðgÞÞ less

than or equal to:

Xslow�1

h¼1

Xf ðhþ1Þ�1

g¼f ðhÞ

ChtðgÞ þ Ch

tðf ðhþ1ÞÞ

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

nodes visited before slow

þXf ðslowþ1Þ

g¼f ðslowÞ

CslowtðgÞ|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

node slow

þXq

h¼slowþ1

Xf ðhþ1Þ

g¼f ðhÞþ1

ChtðgÞ þ Ch

tðf ðhÞÞ

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

nodes visited after slow

For any node h 2 ½1; q�; for any packet m0 visiting h: Chtðm0Þ4Cslow

tðm0Þ: Then, as packets arenumbered consecutively from f ð1Þ to f ðqþ 1Þ ¼ m; we get inequation (1). By considering that onany node h; the processing time of a packet with a priority greater than or equal to Pi is less thanor equal to maxj2hpi[figfCh

j g; we get inequation (2).

Xslow�1

h¼1

Xf ðhþ1Þ�1

g¼f ðhÞ

ChtðgÞ

!þ

Xf ðslowþ1Þ

g¼f ðslowÞ

CslowtðgÞ þ

Xqh¼slowþ1

Xf ðhþ1Þ

g¼f ðhÞþ1

ChtðgÞ

!4Xmg¼f ð1Þ

CslowtðgÞ ð1Þ

Xslow�1

h¼1

Chtðf ðhþ1ÞÞ þ

Xqh¼slowþ1

Chtðf ðhÞÞ4

Xqh¼1

h=slow

maxj2hpi[fig

fChj g ð2Þ



By (1) and (2), we get

Wqi ðtÞ ¼

Xmg¼f ð1Þ

CslowtðgÞ þ

Xqh¼1

h=slow

maxj2hpi[fig

fChj g � C

qi þH


The termPm

g¼f ð1Þ CslowtðgÞ is bounded by the maximum workload generated by flows tj ; j 2 hpi; in

the interval ½�J1inj;maxð0;Wq

i ðtÞ � %RR1;qminj

Þ� plus the maximum workload generated by packets of

ti in the interval ½�J1ini; t�: Indeed, any packet released before f ð1Þ does not interfere with the

considered packets processed in the selected busy periods. It is the same for any packet arrivingin node q after m starts its execution, that is W

qi ðtÞ: Then, a packet of any flow tj ; j 2 hpi does

not delay m if it arrives in node 1 after the time: Wqi ðtÞ � %RR

1;qminj

: By definition, themaximum workload generated by any flow tj in the interval ½t1; t2� on node slow is equal toð1þ bðt2 � t1Þ=Tj cÞ � Cslow

j : Then, we get

Xmg¼f ð1Þ

CslowtðgÞ 4

Xj2hpi

1þmaxð0;Wq


Þ þ J1inj

Tj

$ % !� Cslow

j þ 1þtþ J1

ini

Ti

$ % !� Cslow

i

Hence, we establish the following property.

Property 3When all flows follow the same line L consisting of q nodes numbered from 1 to q; then for anypacket of any flow ti 2 t; arrived at time t on node 1; its latest starting time on node q; is given by

Wqi ðtÞ ¼

Xj2hpi

1þmaxð0;Wq


Þ þ J1inj

Tj

$ % !� Cslow

j þ 1þtþ J1

ini

Ti

$ % !� Cslow

i

þXqh¼1

h=slow

maxj2hpi[fig

fChj g � C

qi þH


5.3. Analysis of the latest starting time of packet m in node q

In this section, we first show that the solution of the equation given in Property 3 exists for anytime t: This solution is used to compute the response time of a packet entering the distributedsystem at time t: We then show how to reduce the set of times to be tested to get the worst caseend-to-end response time, that can be reached by a packet of the flow considered.

5.3.1. Existence of a solution. To prove the existence of Wqi ðtÞ; solution of the equation given in

Property 3, we use the following series and show that it is convergent.

Wqð0Þi ðtÞ ¼

Xj2hpi

Cslowj þ 1þ

tþ J1ini

Ti

$ % !� Cslow

i þA1;qi



Wqðpþ1Þi ðtÞ ¼

Xj2hpi

1þmaxð0;WqðpÞ


Þ þ J1inj

Tj

$ % !� Cslow

j

þ 1þtþ J1

ini

Ti

$ % !� Cslow

i þA1;qi

with A1;qi ¼

Pqh¼1

h=slow

maxj2hpi[fig fChj g � C

qi þH

1;qi þ ðq� 1Þ � Lmax:

Lemma 3For any flow ti 2 t; if Uslow

hpi51; where Uslow

hpidenotes the utilization factor on node slow for the

flows belonging to hpi; then the series Wq ðpÞi ðtÞ is convergent.

ProofThe series W

qðpÞi ðtÞ is a non-decreasing series as the floor function is non-decreasing. Moreover,

by recurrence, we prove that Wq ðpÞi ðtÞ is upper bounded by: X=ð1�Uslow

hpiÞ; where:

X ¼Xj2hpi

1þJ1inj

Tj

!� Cslow

j þ 1þtþ J1

ini

Ti

$ % !� Cslow

i þA1;qi

Indeed, we have: Wqð0Þi ðtÞ ¼

Pj2hpi C

slowj þ ð1þ bðtþ J1

iniÞ=Ti cÞ � Cslow

i þA1;qi 4X ; that is less

than or equal to: X=ð1�Uslowhpi

Þ; assuming Uslowhpi

51: We now assume that the recurrence is trueat rank p and show that it is true at rank pþ 1: Indeed, we get

Wqðpþ1Þi ðtÞ ¼

Xj2hpi

1þmaxð0;WqðpÞ


Þ þ J1inj

Tj

$ % !� Cslow

j

þ 1þtþ J1

ini

Ti

$ % !� Cslow

i þA1;qi

4Xj2hpi

1þW

qðpÞi ðtÞ þ J1

inj

Tj

!� Cslow

j þ 1þtþ J1

ini

Ti

$ % !� Cslow

i þA1;qi

4WqðpÞi ðtÞ �Uslow

hpiþXj2hpi

1þJ1inj

Tj

!� Cslow

j þ 1þtþ J1

ini

Ti

$ % !� Cslow

i þA1;qi

4WqðpÞi ðtÞ �Uslow

hpiþ X4X �Uslow

hpi=ð1�Uslow

hpiÞ þ X ¼ X=ð1�Uslow

hpiÞ

The series is non-decreasing and upper bounded by X=ð1�Uslowhpi

Þ: Hence, this series isconvergent. &



5.3.2. Reduction of the times to be tested. In this section, we give an upper bound for the valuesof t to be tested to obtain the worst case end-to-end response time of a flow. We first proveproperties concerning the latest starting time of a flow packet.

Lemma 4

For any time t5� J1ini; we have: W

qi ðtþ Bslow

i Þ4Wqi ðtÞ þ Bslow

i ; where Bslowi meets: Bslow

i ¼Pj2hpi[fig dB

slowi =Tj e � Cslow

j :

ProofWe consider the series W

qðpÞi ðtÞ and prove the lemma by recurrence. At rank 0; we have

Wqð0Þi ðtþ Bslow

i Þ ¼Xj2hpi

Cslowj þ 1þ

tþ Bslowi þ J1

ini

Ti

$ % !� Cslow

i þA1;qi

4Xj2hpi

Cslowj þ 1þ

tþ J1ini

Ti

$ % !� Cslow

i þBslowi

Ti

� �� Cslow

i þA1;qi

4Wqð0Þi ðtÞ þ Bslow

i

Assuming that the recurrence is true at rank p; we show that it is true at rank pþ 1: IndeedW

q ðpþ1Þi ðtþ Bslow

i Þ equals:

Xj2hpi

1þmaxð0;WqðpÞ

i ðtþ Bslowi Þ � %RR

1;qminj

Þ þ J1inj

Tj

$ % !� Cslow

j

þ 1þtþ Bslow

i þ J1ini

Ti

$ % !� Cslow

i þA1;qi

4Xj2hpi

1þmaxð0;WqðpÞ


Þ þ Bslowi þ J1

inj

Tj

$ % !� Cslow

j

þ 1þtþ Bslow

i þ J1ini

Ti

$ % !� Cslow

i þA1;qi

4Xj2hpi

1þmaxð0;WqðpÞ


Þ þ J1inj

Tj

$ % !� Cslow

j þ 1þtþ J1

ini

Ti

$ % !� Cslow

i

þX

j2hpi[fig

Bslowi

Tj

� �� Cslow

j þA1;qi

4WqðpÞi ðtÞ þ Bslow

i &

We now show that only times t ¼ k � Ti � J1ini; where k 2 N; have to be tested.



Lemma 5For any k 2 N; we have: W

qi ðk � Ti � J1

iniÞ ¼ W

qi ðk � Ti þ D� J1

iniÞ; 8D 2 ½0;TiÞ:

ProofWe consider the series W

qðpÞi ðtÞ and prove the lemma by recurrence. Let t ¼ k � Ti � J1

iniand

t0 ¼ k � Ti þ D� J1ini; with k 2 N and D 2 ½0;TiÞ: The lemma is met at rank 0. Indeed

Wq ð0Þi ðtÞ ¼ W

q ð0Þi ðt0Þ: Assuming that the recurrence is true at rank p; we show that it is true at

rank pþ 1: Indeed, we have

Wqðpþ1Þi ðt0Þ ¼

Xj2hpi

1þmaxð0;WqðpÞ

i ðt0Þ � %RR1;qminj

Þ þ J1inj

Tj

$ % !� Cslow

j

þ 1þt0 þ J1

ini

Ti

$ % !� Cslow

i þA1;qi

¼Xj2hpi

1þmaxð0; W

qðpÞi ðtÞ � %RR

1;qminj

Þ þ J1inj

Tj

$ % !� Cslow

j

þ 1þk � Ti þ D

Ti

� �� Cslow

i þA1;qi

¼Xj2hpi

1þmaxð0; W

qðpÞi ðtÞ � %RR

1;qminj

Þ þ J1inj

Tj

$ % !� Cslow

j

þ ðkþ 1Þ � Cslowi þA

1;qi ¼ W

q ðpþ1Þi ðtÞ

5.4. Worst case end-to-end response time

The worst case end-to-end response time of any packet m of flow ti is equal to the latest startingtime of m in node q; plus Cq

i ; minus t; the arrival time of packet m in the distributed system.More precisely, it is equal to: W

qi ðtÞ þ C

qi � t: By Lemma 4, the worst case response time of a

packet of ti; entered the distributed system at time tþ Bslowi ; is less than the worst case response

time of m:Moreover, from Lemma 5 the worst case end-to-end response time of a packet of flowti is reached when this packet enters the distributed system at time t ¼ k � Ti � J1

ini; k 2 N:

The worst case end-to-end response time of flow ti is equal to the maximum of the worst caseend-to-end response times of its packets. Hence, to compute this worst case response time, wehave only to consider times t equal to: k � Ti � J1

ini5Bslow

i :

Property 4When all the flows follow the same line L consisting of q nodes numbered from 1 to q and arescheduled according to the Fixed Priority algorithm, the worst case end-to-end response time ofany flow ti meets:

R1;qmaxi

¼ maxk¼0::K

fWqi ðk � Ti � J1

iniÞ þ C

qi � k � Ti þ J1

inig



where

Wqi ðtÞ ¼

Xj2hpi

1þmaxð0;Wq


Þ þ J1inj

Tj

$ % !� Cslow

j þ 1þtþ J1

ini

Ti

$ % !� Cslow

i

þXqh¼1

h=slow

maxj2hpi[fig

fChj g � C

qi þH


and K is the smallest integer value such that Wqi ðK � Ti � J1

iniÞ þ C

qi � K � Ti þ J1

ini5Bslow

i ; withBslowi ¼

Pj2hpi[fig dB

slowi =Tj e � Cslow

j :

We can notice that in the single node case, this bound is exact.

Lemma 6In the single node case, the bound given by Property 4 is this of Property 1 given in the related work.

6. HOLISTIC APPROACH

We now apply the holistic approach to compute the worst case end-to-end response time of anyflow ti; when all flows follow the same line L:

6.1. Principle

The holistic approach proceeds iteratively and starts with node 1. Knowing the value of J1inj

for

any j 2 ½1; n�; we compute R1maxj

using Property 1 and R1minj

¼ C1j ; 8j 2 ½1; n�: We proceed in the

same way for any node h; h 2 ð1; q�: Knowing the value of Jhinj

¼P

k¼1::h�1 ðRkmaxj

� Rkminj

Þ þ

ðh� 1Þ � ðLmax � LminÞ for any j 2 ½1; n�; we compute Rhmaxj

using Property 1. Moreover,

Rhminj

¼ Chj :

A bound on the end-to-end response time of flow ti is given byPq

h¼1 Rhmaxi

�Pq

h¼2 Jhini

þðq� 1Þ � Lmax:

6.2. Comparative evaluation

In this section, we compare the bound on the end-to-end response time provided by the holisticapproach with this provided by the trajectory approach.

Property 5For any flow ti; i 2 ½1; n�; visiting line L; the bound on ti end-to-end response time provided bythe trajectory approach is less than or equal to this provided by the holistic approach.

ProofLet us denote R1;q

maxi;Tand R1;q

maxi;Hthe bounds obtained, respectively, by the trajectory approach

and the holistic approach on the end-to-end response time of flow ti visiting nodes 1 to q: Weshow that the bound provided by the trajectory approach is less than or equal to this providedby the holistic approach. We proceed by recurrence on the number of nodes visited. This



property is true for one node. Indeed, we can notice that when the number of visited nodes isequal to 1; both bounds are identical to the formula given in Property 1.

We consider that it is true for q nodes and consider a sequence of qþ 1 nodes. We can writewith the trajectory approach that: W

qþ1i ðk � Ti � J1

iniÞ ¼ W

qi ðk � Ti � J1

iniÞ þ C

qi þ Lmax þ X

qþ1i;T ;

where Xqþ1i;T denotes the waiting time on node qþ 1 of packet entered the distributed system at

time k � Ti � J1ini: Then, we have

R1;qþ1maxi;T

¼ maxk

fWqþ1i ðk � Ti � J1

iniÞ � k � Tig þ J1

iniþ C

qþ1i 4R1;q

maxi;Tþ C

qþ1i þ Lmax þ X

qþ1i;T

We now show that: Cqþ1i þ X

qþ1i;T 4Rqþ1

maxi;H� J

qþ1ini;H

: As for any flow, the input jitter on node

qþ 1 considered with the holistic approach is higher than this considered with the trajectory

approach, we apply Lemma 2 and get Cqþ1i þ X

qþ1i;T 4Rqþ1

maxi;H� J

qþ1ini;H

: We finally have: R1;qþ1maxi;T

4

R1;qmaxi;T

þ Rqþ1maxi;H

� Jqþ1ini;H

þ Lmax:

As R1;qmaxi;T

4R1;qmaxi;H

; we obtain R1;qþ1maxi;T

4R1;qmaxi;H

þ Rqþ1maxi;H

� Jqþ1ini;H

þ Lmax ¼ R1;qþ1maxi;H

: HenceProperty 5. &

7. EXAMPLES

In this section, we give examples of bounds on the end-to-end response times of flows in anetwork, when all flows follow the same line consisting of five nodes. We consider a sett ¼ ft1; t2; t3; t4g; of sporadic flows. Moreover, we assume that all flows enter the networkwithout jitter, and Lmax ¼ Lmin ¼ 1:

We have developed for this paper a validation tool providing the exhaustive solution of a real-time scheduling problem in a network. Indeed, once the different parameters have beenspecified, all the possible scenarios are generated and feasibility of the flow set is checked foreach of them. To do this, we consider the first packet generation times vector, V ; where VðiÞdenotes the generation time of the first packet of ti; i 2 ½1; n�: A scenario is then characterized byV : Notice that VðiÞ is in the interval ½�J1

in1;�J1

in1þ TiÞ: The number of scenarios to consider is

bounded byQn

j¼1 Tj : The validation result is a file containing the exact worst case end-to-endresponse time of each flow.

We now present three examples and compare the bounds on the end-to-end response times offlows, provided by the holistic and the trajectory approaches with the exact worst case values,provided by our validation tool. In the first two examples, we assume that 8i 2 ½1; 4�; Pi ¼ i:Flow t4 has thus the highest priority. All flows have a period equal to 30.

Example 1We first consider that all packets have a maximum processing time equal to 6 on each visitednode, that is: 8i 2 ½1; 4�; 8h 2 ½1; 5�; Ch

i ¼ 6: Figure 3 gives for any flow ti the exact value of itsworst case end-to-end response time and the value computed according to the trajectoryapproach. To show the improvement of our results compared with those obtained by theclassical technique, we also include the value computed according to the holistic approach.

We can see on Figure 3 that the values provided by the trajectory approach are exact for allflows, whereas those provided by the holistic approach are up to eight times the exact values.Notice that the accuracy of the holistic approach decreases with the priority of flows.



Example 2We now consider a more general case, by assuming that the maximum processing time of a flowis not the same on each visited node. More precisely, we consider ten configurations numberedfrom 1 to 10 and presented in Table I.

Figure 4 gives for any flow ti; i 2 ½1; 4�; the exact value of its worst case end-to-end responsetime and the value computed according to the trajectory approach in each configuration. Thevalue computed according to the holistic approach is also given.

We observe on Figure 4 that the values provided by the trajectory approach are exact or veryclose to the exact values. Concerning the bounds provided by the holistic approach, they arevery pessimistic. Indeed, these values are up to five times the exact ones. Moreover, we cannotice that our values are exact in two cases:

* For flow t1; the flow with the smallest priority, whatever the configuration. This can beexplained because the overestimation done with the trajectory approach concerns the term

H1;qi ; the delay due to the non-preemptive effect. Notice that this non-preemptive effect

does not exist for flow t1;* For any flow when the first visited node is the slowest node. This configuration is

frequently encountered in a network where the ingress node has to perform specificprocessing such that flow marking or flow reshaping.

Figure 3. Worst case end-to-end response times.

Table I. Processing time of any flow ti on each node.

1 2 3 4 5 6 7 8 9 10

C1i 6 2 5 4 3 4 5 5 2 6

C2i 2 6 3 2 5 6 4 3 6 2

C3i 5 4 6 6 2 2 3 4 3 3

C4i 3 3 2 3 6 5 2 6 5 4

C5i 4 5 4 5 4 3 6 2 4 5



Example 3In the last example, we study the influence of the priority choice on the worst case end-to-endresponse time. We consider 4! ¼ 24 configurations corresponding to all possible priorityassignments (see Table III). We assume that the processing time of a packet of flow ti; 8i 2 ½1; 4�does not depend on the node considered and increases with the minimum interarrival time Ti:These times are given in Table II.

Among all possible configurations, configuration 24 (see Table III) is the optimal onewhen flow deadlines are equal to flow interarrival times and for any flows ti and tj suchthat Ci4Cj ; we have Di4Dj. Indeed, in that case, IDM (inverse deadline monotonic) thatassigns flow priority inversely proportional to flow deadline is optimal (see Reference [1]). Wecan notice that for this configuration, the trajectory approach provides the exact values (seeFigure 5).

Moreover, we can see in Figure 5 that for all configurations, bounds provided by the holisticapproach are pessimistic compared to those provided by the trajectory approach: theoverestimation can reach þ356%:

Table II. Processing and interarrival times of flows.

Chi Ti

t1 2 10t2 3 15t3 4 20t4 5 25

Figure 4. End-to-end response time of each flow ti; i 2 ½1; 4�:



8. CONCLUSION

In this paper, we have established new results for Fixed Priority scheduling in a distributedsystem. We have determined the worst case end-to-end response time of any real-time flow, usinga trajectory approach. We have compared these results with those provided by the classicalholistic approach. The trajectory approach considers only possible scenarios, whereas the holisticapproach considers worst-case scenarios on every node, that generally do not occursimultaneously on all nodes. We have shown that the bound given by the trajectory approachis reached in various configurations, whereas the holistic approach provides only a bound that canbe very pessimistic. The accuracy of the holistic approach decreases with the priority of flows.

REFERENCES

1. George L, Rivierre N, Spuri M. Preemptive and non-preemptive scheduling real-time uniprocessor scheduling.INRIA Research Report No 2966, September 1996.

Figure 5. End-to-end response time of each flow ti; i 2 ½1; 4�:

Table III. Configurations considered.

P1 P2 P3 P4

config. 1 4 3 2 1config. 2 4 3 1 2config. 3 4 2 3 1config. 4 4 2 1 3config. 5 4 1 3 2config. 6 4 1 2 3config. 7 3 4 2 1config. 8 3 4 1 2... ... ... ... ...config. 24 1 2 3 4



2. Tindell K, Clark J. Holistic schedulability analysis for distributed hard real-time systems. Microprocessors andMicroprogramming, Euromicro Journal 1994; 40:117–134.

3. Blake S, Black D, Carlson M, Davies E, Wang Z, Weiss W. An architecture for differentiated services. RFC 2475,December 1998.

4. Braden R, Clark D, Shenker S. Integrated services in the internet architecture: an overview. RFC 1633, June 1994.5. Sivaraman V, Chiussi F, Gerla M. End-to-end statistical delay service under GPS and EDF scheduling: a

comparison study. INFOCOM’2001, Anchorage, Alaska, April 2001.6. Vojnovic M, Le Boudec J. Stochastic analysis of some expedited forwarding networks. INFOCOM’2002,New York,

June 2002.7. Chiussi F, Sivaraman V. Achieving high utilization in guaranteed services networks using early-deadline-first

scheduling. IWQoS’98, Napo, California, May 1998.8. George L, Marinca D, Minet P. A solution for a deterministic QoS in multimedia systems. International Journal on

Computer and Information Science 2001; 1(3):106–119.9. Le Boudec J, Thiran P. Network Calculus: A Theory of Deterministic Queuing Systems for the Internet. Lecture Notes

in Computer Science, vol. 2050. Springer: Berlin, 2003.10. Parekh A, Gallager R. A generalized processor sharing approach to flow control in integrated services networks: the

multiple node case. IEEE ACM Transactions on Networking 1994; 2(2):137–150.11. Georgiadis L, Gu!eerin R, Peris V, Sivarajan K. Efficient network QoS provisioning based on per node traffic shaping.

IEEE/ACM Transactions on Networking 1996; 4(4):482–501.12. Sivaraman V, Chiussi F, Gerla M. Traffic shaping for end-to-end delay guarantees with EDF scheduling.

IWQoS’2000, Pittsburgh, June 2000.13. Baruah S, Howell R, Rosier L. Algorithms and complexity concerning the preemptive scheduling of periodic real-

time tasks on one processor. Real-Time Systems 1990; 2:301–324.14. Tindell K, Burns A, Wellings A. Calculating controller area network (CAN) message response times. IFAC

DCCS’94, Toledo, Spain, 1994.15. Zheng Q, Shin KG. On the ability of establishing real-time channels in point-to-point packet-switched networks.

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AUTHORS’ BIOGRAPHIES

Steven Martin graduated from ESIEE engineering school in 2000 and received his PhDdegree from Paris 12 University in 2004. His research interests include quality of servicenetworks and real-time scheduling. He focuses on deterministic quantitative guarantees ofquality of service for real-time applications.

Pascale Minet is Senior Researcher with project Hipercom at INRIA. She qualified fromVersailles University (France) in advising PhD students in 1998. She received her PhDdegree from Toulouse University in 1982. She is co-author of the OLSR (Optimized LinkState Routing) protocol, that is an RFC for mobile ad-hoc networks. She is also interestedin quality of service networks, real-time scheduling and more generally distributed real-time systems.

Laurent George is a teacher in Networks and Telecommunications at the University ofParis XII. He prepared at INRIA, his PhD thesis on distributed real-time scheduling. Hereceived in 1998 his PhD degree from Versailles University, France. His research interestsare distributed real-time scheduling and real-time communication, fault-tolerance indistributed systems and quality of service in distributed multimedia systems.



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