Feyaerts, Bart

Wittevrongel, Sabine

English

Ghent University Academic Bibliography

Delay Analysis of a Place Reservation QueueBart FeyaertsSupervisor(s): Sabine WittevrongelI. INTRODUCTIONModern communication networks have tosupport an increasingly diverse range of appli-cations, each bearing their own particular set ofQuality of Service (QoS) requirements. Real-time applications (e.g. multimedia streaming,telephony, gaming, ...) are characterized byhard delay requirements: the mean delay andthe delay jitter of the data packets should typ-ically be kept minimal. For non-real-time ap-plications (e.g. www, ftp, e-mail, ...) on theother hand, delay requirements are typicallyless stringent, but packet loss should be (very)low. In short, the network should differenti-ate between the various types of traffic, in or-der to meet up with the traffic types’ QoS re-quirements. Over the last decades, many so-lutions have been proposed, varying in com-plexity and effect. A common method of sup-porting differentiated QoS, is the use of prior-ity scheduling in the network nodes, opposedto standard First-In First-Out (FIFO) schedul-ing. In what follows, we will divide all networktraffic in two separate classes: class 1 will coverdelay-sensitive traffic, class 2 consists of delay-tolerant traffic.In FIFO, all packets are treated equally, re-gardless of their requirements. The other ex-treme is posed by the Absolute Priority (AP)scheduling mechanism, where different traffictypes are mapped to different priority levels,and transmission priority is always given topackets of the highest priority level. In thisparticular case, we will consider class-1 traf-fic to be high-priority and class-2 traffic to below-priority. When the server of an AP sys-tem becomes available, the first class-1 packet,present in the queue, if any, is served first.Class-2 packets can then only enter the serverin the absence of class-1 packets. AP has beenstudied extensively for various cases. It hasbeen shown that AP does indeed decrease high-priority packet delay at the expense of low-priority traffic. This may lead to excessive low-priority packet delays or so-called starvation.To counter the problem of packet starvation,a new priority scheduling mechanism was pro-posed in [1], called the Reservation Discipline(RD). With RD, a dummy packet, the reservedplace R, is inserted in the queue, to act as aplaceholder for future class-1 packets. As aclass-1 packet arrives at the system, it is in-serted at R’s position and R gets reinserted atthe end of the queue. Class-2 packets are sim-ply inserted at the end of the queue, just as theywould have been under FIFO scheduling. Ifmultiple packets arrive at the system during thesame time slot, first the class-1 packets are in-serted one by one and then the class-2 packetsare appended in arrival order. This way, class-1packets are favoured twice: not only are theyinserted before the class-2 packets that arrivedduring the same slot; the first class-1 packet ofeach slot is inserted at R’s position, allowingthe packet to jump over some class-2 packets.Since any class-2 packet can be jumped overonly once, there is a limit to the disadvantagesustained. Research has been done before forservice times of exactly one slot, see [2]. Wehave studied RD in case of geometric and gen-eral service times, see [3] and [4] respectively.II. METHODOLOGYAlthough we would eventually like to findvalues for delay probabilities, we try to avoidthe direct use of probabilities. Fact is thatcalculations that involve probabilities easilybecome complicated. As an example, weintroduce two independent random variablesX and Y , of which we want to determineProb[X + Y = n]. Even the direct calcula-tion of this simple probability requires the useof a convolution. A PGF (probability generat-ing function) approach keeps the calculationsconcise. The PGF of a random variable R(in what follows shortened to R(z)) is definedas E[zR], where E[.] denotes the expectedvalue of the quantity between brackets. Thus,the PGF of X + Y can easily be obtained asE[zX]E[zY], which only involves a productbut no convolution.PGFs possess some useful properties. One ofthese is the moment generating property, whichallows us to recursively calculate the consecu-tive moments of a random variable, usingdnR(z)dzn∣∣∣∣z=1= E[n−1∏i=0(X − i)].In practice this implies that the mean E[R] of arandom variable R is given by R′(1), the firstderivative of the PGF at z = 1. The variance ofthat random variable R can then be calculatedas R′′(1) + R′(1) − R′(1)2. Moreover, if weare interested in the probabilities itself, we canapply various inversion techniques for PGFs.The first step in the analysis was to create anappropriate mathematical model to fit the sys-tem. The model involves a discrete-time single-server queueing system and two packet streamsof which the numbers of arrivals from slot toslot are independent and identically distributed(iid). The transmission times of the packets areiid from packet to packet and have a generaldistribution. Based on this model, we com-posed a vector of random variables that allowsus to adequately monitor the system state andto derive the desired performance measures.These variables were:• the number of slots until service completionof the packet in the server (if any);• the number of information packets in the sys-tem;• the position of R in the queue.This so-called system state vector was found tobe Markovian, meaning that the distribution ofthe system state in a random slot can be cal-culated solely from the system state in the pre-ceding slot. The relation between the systemstates in any two consecutive slots is given bythe system equations. We then established a re-lationship between the joint PGFs of the systemstates in two consecutive slots by means of thesystem equations. From this relation, we thenderived the joint PGF of the system state duringa random slot in the steady state.By means of the system state, expressionscould also be found for the delays of a randomclass-1 and class-2 packet arriving during a slotin equilibrium. We then calculated the PGFs ofthe delays, allowing us to easily construct ex-pressions for the mean values and the variancesof the delays of both packet classes.III. CONCLUSIONSNumerical tests and simulations have proventhat RD effectively solves the problem of star-vation. There is, however, an important pay-off: the class-1 delay performance decreasessignificantly, in relation to the class-1 delayperformance under AP.REFERENCES[1] W. Burakowski and H. Tarasiuk, On new strat-egy for prioritising the selected flow in queuingsystem, Proceedings of the COST 257 11th Man-agement Committee Meeting (Barcelona, January2000), COST-257 TD(00)03[2] S. De Vuyst, S. Wittevrongel and H. Bruneel,Place reservation: delay analysis of a novelscheduling mechanism, Computers & OperationsResearch, vol. 35, no. 8, pp. 2447–2462 (2008)[3] B. Feyaerts, S. De Vuyst, S. Wittevrongel andH. Bruneel, Analysis of a discrete-time prior-ity queue with place reservations and geometricservice times, Proceedings of the Sixth Confer-ence on Design, Analysis, and Simulation of Dis-tributed Systems, DASD 2008 (Edinburgh, June2008)[4] B. Feyaerts and S. Wittevrongel, Performanceanalysis of a priority queue with place reserva-tion and general transmission times, Proceed-ings of the Fifth European Performance Engineer-ing Workshop, EPEW 2008 (Mallorca, September2008)

Delay analysis of a place reservation queue

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Delay analysis of a place reservation queue

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