With count-based scheduling the maximum amount of service that is granted during one visit of the server at a particular queue is based on the number of customers served in the polling p
Trang 1Chapter 9
T HE principle of polling is well-known in many branches of computer science applica- tions In early timed-sharing computers, terminals were polled in order to investigate whether they had any processing to be done These days, intelligent workstations access file or computing servers via a shared communication medium that grants access using a polling scheme Also in other fields, e.g., manufacturing, logistics and maintenance, the principle of polling is often encountered
When trying to analyse systems that operate along some polling scheme, so-called polling models are needed In this chapter we provide a concise overview of the theory and application of polling models Although we do provide some mathematical derivations, our main aim is to show how relatively simple models can be used, albeit sometimes approximately, for the analysis of fairly complex systems
This chapter is further organised as follows In Section 9.1 we characterise polling models and introduce notation and terminology In Section 9.2 we address some important general results for polling models Symmetric and asymmetric count-based polling models are addressed in Section 9.3 and 9.4 respectively Using these models, the IBM token ring system is analysed in Section 9.5 Time-based polling models, both symmetric and asymmetric, are finally discussed in Section 9.6
In polling models, there is a single server which visits (polls) a number of queues in some predefined order Customers arrive at the queues following some arrival process Upon visiting a particular queue, queued customers are being served according to some scheduling strategy After that, t&e server leaves the queue and visits the next queue Going from
Performance of Computer Communication Systems: A Model-Based Approach.
Boudewijn R Haverkort Copyright © 1998 John Wiley & Sons Ltd ISBNs: 0-471-97228-2 (Hardback); 0-470-84192-3 (Electronic)
Trang 2one queue to another takes some time which is generally called the switch-over time
In the above description a number of issues have deliberately been left unspecified It
is these issues that, once specified, characterise the polling model In particular, the visit ordering of the server to the queues and the strategy being used to decide how long a particular queue receives service before the server leaves, characterise the model These issues will be addressed in Section 9.1.2 and 9.1.3 respectively after some preliminary notation and terminology has been introduced in Section 9.1.1
We will assume that we deal with a polling model with N stations, modelled by queues
Qi through QN We use queue indices i, j E ( 1, , N) At queue i customers arrive according to a Poisson process with rate Xi The mean and second moment of the service requirement of customers arriving at queue i is E[Si] and E[ST] respectively The total offered load is given by p = Cz, pi, with pi = XJZ[Si] The mean and variance of the time needed by the server to switch from queue i to queue j are denoted &j and 62(i) respectively When the queues are assumed to be unbounded, under stability conditions, the through- put of each queue equals the arrival rate of customers at each queue The main performance measure of interest is then the customer waiting time for queue i, i.e., Wi Most analytic models only provide insight into the average waiting time E[ Wi] When the queues are bounded, the throughput and blocking probability at the stations are also of interest; we will come back to finite-buffer polling models in Chapter 16
We distinguish three different visit orders: a cyclic ordering, a Markovian ordering and an ordering via a polling table
l Cyclic polling In a cyclic visiting scheme, after having served queue i, the server continues to poll station i @ 1 where @ is the modulo-N addition operator such that
N @ 1 = 1 As a consequence of this deterministic visit ordering only “neighbouring” switch-over times and variances are possibly non-zero, i.e., S;,j = ;,j 6(2) = 0 whenever
j # i @ 1 For ease of notation we set Si = 6i,+@i and 6c2) = dj:Li a The mean and variance of the total switch-over time, defined as the total time spent switching during a cycle in which all stations are visited once, are given by n = CEi & and A(“) = CE, 8j2) respectively In Figure 9.1 we show a polling model with cyclic visit ordering
Trang 39.1 Characterisation of polling models 175
Ql / \ /
I \
Figure 9.1: Basic polling model with cyclic visit ordering
Due to the fact that most, and especially the earlier, results on polling models as- sumea the cyclic visit order, polling models are often called cyclic server models This, however, is a slight abuse of terminology: the class of polling models is larger,
as discussed below
l Markovian polling In a Markovian polling scheme, after having polled queue
i, the server switches to poll queue j with probability ~i,~ Since the probabilities pi,j are ind ependent of the state of the polling model this form of polling is called Markovian polling The probabilities are gathered in an N x N matrix P The mean and variance of the total switch-over time are now defined as n = CL, C~=ipi,~&j and AC21 = CL, Cy J=l pi,jb,(i’ respectively
Notice that the degenerate case of Markovian polling in which pi,iei = 1 and pi,j =
0 (j # i @ 1) is equivalent to a polling model with a cyclic visit ordering
l Tabular polling Finally, an ordering via a polling table T = (Z’i, Tz, , TM) establishes a cyclic visit ordering of the server along the queues; however, these cycles may contain multiple visits to the same queue The server starts with visiting queue QT1, then goes to QT2, etc After having visited QTM the server visits QT~ again and a new cycle starts
Typically, when the polling process is controlled in a centralised way, a polling table is used, e.g., when station 1 actively controls the system, we have 2’ = (1,2,1,3,1,4, e ,
1, N - 1, 1, N) ,Also the scan-polling order can be observed quite frequently:
Trang 4T = (1,2,3,.*., N-l,N,N,N-l,N-2,- ,2, l), e.g., when the queues model disk tracks that are visited by a moving disk head
The mean and variance of the total switch-over time are now defined as A =
~;,M,I b&j, and AC21 = Cr=, ~~~‘T~~l respectively > Note that the @-operator is now defined on {l, ,M}
Whenever T = (1,2,-m-,N), i.e., M = N, a polling model with polling table T is equivalent to a polling model with a cyclic visit ordering
The scheduling strategy defines how long or how many customers are served by the server once it visits a particular queue Two main streams in scheduling strategies can be distin- guished: count-based scheduling and time-based scheduling
Count-based scheduling With count-based scheduling the maximum amount of service that is granted during one visit of the server at a particular queue is based on the number
of customers served in the polling period Among the well-known scheduling disciplines are the following (for a more complete survey, see [286, Chapter 11):
l Exhaustive (E) : the server continuously serves a queue until it is empty;
l Gated (G) : the server only serves those customers that were already in the queue
at the time the service started (the polling instant);
l k-limited (k-L): each queue is served until it is emptied, or until Ic customers have been served, whichever occurs first The case where Ic = 1 is often mentioned separately as it results in simpler models;
l Decrementing or semi-exhaustive (D) : when the server finds at least one cus- tomer at the queue it starts serving the queue until there is one customer less in the queue than at the polling instant;
l Bernoulli (B) : when the server finds at least one customer in the queue it serves that customer; with probability bi E (0, l] an extra customer is served, after which, again with probability bi another one is served, etc.;
l Binomial (Bi): when the server finds ki customers in queue i at the polling instant, the number of customers served in the current service period is binomially distributed with parameters ki and bi E (0, 11
Trang 59.2 Cyclic polling: cycle time and conservation law 177
All of the above strategies are local, i.e., they are determined per queue One can also imagine global count-based strategies For instance, the global-gated strategy marks all jobs present at the beginning of a polling cycle During that cycle all of those jobs are served exhaustively Jobs arriving during the current cycle are saved for the next cycle Time-based scheduling With time-based scheduling the maximum amount of service that is granted during one visit of the server at a particular queue is based on the time already spent at that queue Two basic variants exist:
l Local time-based: the server continues to serve a particular queue until either all customers have been served or until some local timer, which has been started at the polling instant, expires;
l Global time-based: the server continues to serve a particular queue until either all customers have been served or until some global timer, which might have been started when the server last left the queue, expires
In fact, the first mechanism can be found in the IBM token ring (IEEE P802.5) [35] whereas the second one can be found in FDDI [35, 277, 2841
In this section we restrict ourselves to polling models with a cyclic visit order and a mixture
of count-based scheduling strategies (exhaustive, l-limited, gated and/or decrementing) The mean cycle time E[C] ’ d fi is e ne as the average time between two successive polling d instants at a particular queue E[C] is independent of i, also for asymmetric systems This can easily be shown using the following conservation argument Assuming that the service discipline is exhaustive, one cycle consists, on average, of the servicing of all jobs plus the total switch-over time The latter component equals a = xi & In one cycle all the jobs that arrive at station i, i.e., X$[C] jobs, have to be served This requires, for the i-th station X$[C]E[Si] t ime units Thus we have:
E[C] = 5 & + 5 X+?qC]E[SJ = a + E[C]p =+ E[C] = & (94
i=l i=l This result is also valid when the service discipline is other than exhaustive The average service period E[Pi] for queue i can be derived as
E[ly = X~E[C]E[S~] = f$ (9.2)
Trang 6This equation follows from the fact that for stability reasons, on average, everything that arrives in one cycle at station i, must be servable in one cycle For the average time between the departure of the server from station i and the next arrival at station i, the so-called inter-visit time Ii, we have
E[IJ = E[C] - E[P,] = (’ lv~~” e (9.3)
Important to note is that upon the arrival of a job at station i, the average time until the server reaches that station is not E[Ii]/2 Th is is due to the fact that Ii is a random variable, and we thus have an example of the waiting time paradox The average time until the next server visit therefore equals the residual inter-visit time E[1:]/2E[Ii] Notice that, in general, an explicit expression for E[1!] is not available
The (cyclic) polling models we address are generally not work conserving, that is, there are situations in which there is work to be done (the queues are non empty) but in which the server does no real work since it is switching from one queue to another When the switching times are zero, the polling model would have been work conserving and Kleinrock’s conservation law would apply ([ 1601; see also Chapters 5 and 6 on M]G] 1 queues) :
When the model is not work conserving, that is, when the switch-over times are positive, Kleinrock’s conservation law does not hold anymore It has, however, been shown by Boxma et al [28, 27, 1121 that a so-called pseudo-conservation law still does hold This pseudo-conservation law is based on the principle of work decomposition:
where v is a random variable indicating the steady-state amount of work in the model with positive switch-over times, V is a random variable indicating the steady-state amount
Trang 79.2 Cyclic polling: cycle time and conservation law 179
of work in the model when the switch-over times are set to 0, and Y is a random vari- able indicating the steady-state amount of work in the model at an arbitrary switch-over instance The principle of work decomposition is valid for cyclic polling models as well
as for polling models with Markovian routing or a polling table V is totally independent
of the scheduling discipline, whereas Y and therefore ? are dependent on the scheduling Intuitively, one expects Y and ? to decrease if the switch-over times decrease, if the visit order becomes more efficient or if the scheduling becomes “more exhaustive” In particular, for polling models with non-zero switch-over times (with cyclic, tabular or Markovian visit ordering) a pseudo-conservation law of the following form applies:
When we are dealing with a cyclic polling order, one has:
C ;E[wi] + c ; (I- -=) E[Wi] + c ; (1 - A$ I;‘“) E[Wi] =
where E, G, L, and D are the index sets of the queues with an exhaustive, a gated,
a l-limited and a decrementing scheduling discipline, respectively Clearly, the pseudo- conservation law expresses that the sum of the waiting times at the queues, weighted by their relative utilisations (for E and G directly and with more ccmplex factors for L and D) equals a constant
The pseudo-conservation law does not give explicit expressions for the individual mean waiting times since it is only one equation with as many unknowns as there are stations Nevertheless, it does provide insight into system operation and in the efficiency of schedul- ing strategies Also, it can be used as a basis for approximations or to verify simulation results (see below)
It is interesting to study the stability conditions for cyclic server models For cyclic server models with an exhaustive or a gated service discipline a necessary and sufficient condition is p < 1 For models with a l-limited service strategy, a necessary condition can be derived as follows The mean number of customers arriving at station i per cycle equals XiE[C] Th is number must be smaller than 1, as there is only 1 customer served per cycle Using the fact that E[C] = n/(1 - p), th e necessary stability condition equals
p + xin < 1, for all i For models with a decrementing scheduling strategy a necessary stability condition of the form p + Xi(l - pi)A < 1, for all i, can be derived
Trang 8Example 9.1 A 2-station asymmetric polling model (I)
Consider an asymmetric polling model with 2 stations: station 1 has exhaustive scheduling and station 2 l-limited scheduling Furthermore, the following parameters apply: E[Si] = 0.4, E[$] = 0.32, Xi = 1, Si = 0.05 and Ji2) = 0 for i = 1,2 Clearly, the stability conditions are satisfied so finite average waiting times do exist for both stations We are not in the position to compute E[IVi] and E[W2] directly; however, we can apply the pseudo-conservation law, yielding the following relation between E[W,] and E[W2]:
This linear relation can be drawn in the E[IVi]-E[W,] pl ane; the exact solutions for E[Wi]
When we address models in which all the scheduling disciplines and parameters are station independent, we can obtain closed-form results for the average waiting times by using the pseudo-conservation law, since in a fully symmetric system all the average waiting times are equal to one another so that we are left with only one unknown in (9.7) We will not use this approach here, since it does not provide us much insight into the actual system operation Instead, we will derive the expected waiting time in a fully symmetric exhaustive scheduling polling model in an operational way, following the lines of the proofs for the expected waiting times in the MIG]l and related queues as presented in Chapters 5 and 6 For an exhaustive count-based symmetric polling model, the expected waiting time for
an arriving customer can be thought to consist of 4 components:
where the 4 components can be understood as follows:
1 An arriving customer will, due to the PASTA property, find another customer (at some queue) in service with probability p The remaining service time of this cus- tomer equals E[S”]/2E[S]
2 Similarly, with probability 1 - p an arriving customer will find the server switching from one queue to another The remaining switch-over time equals S2/2S (notice that Sc2) denotes the variance in switch-over times, whereas 62 denotes the second moment of the switch-over time here)
Trang 99.3 Count-based symmetric cyclic polling models 181
3 An arriving job arrives at any queue with equal probability, so on average (N - 1)/2 switch-overs, each of expected length 6, are needed for the server to arrive at the particular queue (since the number of queues N is a constant, the waiting time paradox does not apply)
4 Finally, upon arrival of a customer, the steady-state amount of work in the system equals NE[N,]E[S] I n e ui i q 1 b rium, this amount of work should be handled before the randomly arriving customer is served
Adding these 4 components, we have:
Using Little’s law to rewrite E[N,] = XE[W], we obtain
(l-p)E[W] = (I-~)$+~c~+P~
s2 + E[W] = -+ - N - ’ 6 + NXE[S] w21
26 2(1-p) w - Puwl ’ (9.11)
We can rewrite the first two additive terms as follows:
(N - p)6 2(1 - P> ’
so that we have
h(2) E[WT] = 26 + NXE[S2] + 6(N - p)
(9.12)
(9.13) The subscript ‘E’ is added to indicate that the formula is valid for exhaustive scheduling Along the above lines, one can also derive the mean waiting time when all scheduling strategies are of gated type:
2(1-p- NM) ’ Finally, for the decrementing scheduling discipline we have:
(9.14)
(9.15)
dc2) E[~D] = 26 + NXE[S2](1 - XS) + (N - p)(6 + xhC2))
2(1 - p - M(N - p)) ’ (9.16)
Trang 10P E[WE] E[WG] E[WL] J?@%] p -@[WE] E[WG] B[WL] E[b]
0.05 0.6000 0.6053 0.6085 0.6031 0.55 1.711 1.833 2.089 1.931 0.15 0.7176 0.7353 0.7485 0.7302 0.65 2.314 2.500 3.070 2.793 0.25 0.8667 0.9000 0.9310 0.8953 0.75 3.400 3.700 5.286 4.690 0.35 1.062 1.115 1.179 1.119 0.85 5.933 6.500 15.00 12.27 0.45 1.327 1.409 1.535 1.428 0.95 18.60 20.50 - -
Table 9.1: E[W] in symmetric polling models with exhaustive, gated, l-limited, and decre- menting scheduling
From these explicit formulae, the earlier derived stability conditions can also easily be seen; they correspond to those traffic conditions where the right-hand denominator becomes zero From these expressions, it can also be observed that
and
E[WG] > E[WD] (at low load), and E[WG] < E[WD] (at high load) (9.18)
Example 9.2 Symmetric polling models: influence of scheduling
In Table 9.1 we have tabulated the average waiting times for symmetric polling models with exhaustive, gated, l-limited, and decrementing scheduling strategies for increasing utilisation (established by increasing the arrival rate) The other parameters are: N = 10,
s = 0.1, JC2) = 0.01, E[S] = 1.0, and E[S2] = 1.0 The above inequalities can easily
be observed Also notice that for p = 0.95, the l-limited and decrementing systems are
The fact that the exhaustive scheduling discipline is the most efficient can easily be understood It simply does not spoil its time for switching purposes when there is still work to do, i.e., when the queue it is serving still is not empty The gated and decrement- ing discipline, however, sometimes take time to switch when there are still customers in the current queue This counts even more for the l-limited case where every service is effectively lengthened with the succeeding switch-over time The fact that the amount of switching overhead per customer is smallest with an exhaustive scheduling strategy does not necessarily imply that it is also the best From a fairness point of view, the other
Trang 119.4 Count-based asymmetric cyclic polling models 183
disciplines might
the system
be considered better since they prevent one station from totally hogging
Example 9.3 A 2-station asymmetric polling model (II)
Reconsider the asymmetric polling model addressed before Since station 2 uses a l-limited scheduling strategy, station 1 profits from this as it receives more opportunities to serve customers In fact, E[IVi] should be smaller in the mixed scheduling case than when both stations would have exhaustive scheduling In this latter symmetric case, however, we can exactly compute the average waiting times: E[VVr,J = 1.75 So, in the asymmetric case station 1 is expected to perform better, i.e., E[IVr] 5 1.75 This implies, by the pseudo- conservation law derived for this example, that in the asymmetric case station 2 will suffer more, i.e., EIIVZ] 2 3.9
We can improve on the above bounds by considering the case where the arrival rate of station 2 is set to zero In that case, the model reduces to an M]G]l queue with exhaustive service and multiple vacations (as seen from station 1) because at station 2 no jobs arrive Thus, after the queue in station 1 empties, the server switches to queue 2 and directly back
to queue 1 This switching can be interpreted as a vacation with average length 0.1 (two switches of length 0.05) The variance of the switching (vacation) time is 0, so that we can compute E[IVi] using (5.38) as follows:
We thus have: 0.317 < E[IVi] 5 1.75 and, using the pseudo-conservation law again,
The analysis of asymmetric cyclic server models is much more complicated than the analysis
of symmetric models We will present the exact analysis of an asymmetric cyclic polling model with exhaustive service in Section 9.4.1 followed by a number of approximate results derived by using the pseudo-conservation law in Section 9.4.2
9.4.1 Exhaustive service: exact analysis
In the exhaustive asymmetric case the average waiting time E[IVi] perceived at station
i has the following form (see also (5.38) for the M]GJl queue with server vacations in