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To be able to do so, however, we need load-dependent queueing stations, that is, queueing nodes in which the service rate depends on the number of customers present.. We then continue wi

Chapter 12

I N the previous chapter it has become clear that the evaluation of large closed queueing networks can be quite unattractive from a computational point of view; this was also the reason for addressing approximation schemes and bounding methods In this chapter

we go a different way to attack large queueing network models: hierarchical modelling and evaluation We address a modelling and evaluation approach where large submodels are solved in isolation and where the results of such an isolated evaluation are used in other models To be able to do so, however, we need load-dependent queueing stations, that

is, queueing nodes in which the service rate depends on the number of customers present

In Section 12.1 we introduce load-dependent servers and show the corresponding product- form results for closed queueing networks including such servers We then continue with the extension of the convolution algorithm to include load-dependent service stations in Section 12.2 and discuss two important special cases, namely infinite-server systems and multi-server systems, in Section 12.3 In Section 12.4 we extend the mean-value analysis method to the load-dependent case We then outline an exact hierarchical decomposition approach using load-dependent service centers in Section 12.5 The hierarchical decompo- sition method can also be used in an approximate fashion; an example of that is discussed

in Section 12.6, where we study memory management issues in time-sharing computer systems

Up till now we have assumed that the service rate at the nodes in a queueing network is constant and independent of the state of the queue or the state of the queueing network

It is, however, also possible to deal with load-dependent service rates (or load-dependent

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)

servers) In fact we have already encountered two special cases of load-dependent servers

in Chapter 4:

l multiple server stations in which the service rate grows linearly with the number

of customers present until there are more customers in the stations than there are servers;

l infinite server stations in which the service rate grows linearly (without bound) with the number of customers present

Observe that in both these cases the load-dependency is “local” to a single queueing station: the service rate in a certain station only depends on the number of customers in that station One can also imagine similar dependencies among queueing stations Although they can have practical significance, we do not address them here because their analysis is more difficult; in general such dependencies spoil the product-form properties of a queueing network so that mean-value and convolution solution approaches cannot be employed any more When using stochastic Petri nets, however, such dependencies can be modelled with relative ease, albeit at the cost of a more expensive solution (see Chapter 14)

Having load-dependent service rates, it becomes difficult to specify the service time distribution of a single job since this distribution depends on the number of customers present during the service process It is therefore easier to specify the service rate of node

i as a function of the number of customers present: pi(ni) The value E[Si(ni)] = l/pi(ni) can then be interpreted as the average time between service completions at station i, given that during the whole service period there are exactly ni customers present In principle, pi(ni) can be any non-negative function of ni

The load-independent case and the above two special cases can easily be expressed in the above formalism:

l load-independent nodes: pi(ni) = pi, for all ni;

l infinite server nodes (delay centers): pi (ni) = nipi, for all n;;

l K-server nodes: pi(ni) = min{nipi, Kpi} (see also Chapter 4)

Load-dependency as introduced above does not change the product-form structure of queueing networks of the Jackson (JQN) and Gordon-Newell (GNQN) type introduced

in Chapters 10 and 11 Having M queueing stations and population K we still have a product-form solution for the steady-state probabilities:

(12.1)

In both cases, the above result reduces to the simpler expressions for the load-independent case whenever pi (ni) = pi, for all i

Before we proceed with the analysis of QN including load-dependent service stations,

we give two examples of such stations

Example 12.1 Non-ideal multi-processing

Consider a K-processor system where, due to multiprocessing-overhead, not the full ca- pacity of all the processors can be used To be more precise, whenever there is only one customer present, this customer can be served in a single processor at full speed, i.e., p( 1) = p However, if two customers are present, each will be processed in a separate processor with speed ~(1 - e), where E is the fraction of the processing capacity “lost” due

to overhead, i.e., we have ~(2) = 2~( 1 - 6) Th is continues until the number of jobs present equals K, i.e.,p(lc) = Icp(l - e), Ic 2 K For Ic > K, we have p(Jc) = Kp(l - E) Clearly, the service rate of this multi-server system is load-dependent 0

Example 12.2 Modelling of CSMA/CD networks

For the modelling of CSMA/CD network access mechanisms like Ethernet [194, 277, 2841, the queueing analysis methods we have discussed so far do not suffice (we cannot model the carrier sensing, the collisions and the binary-exponential backoff period in all their details, to name a few examples) Instead of modelling the exact system operation in all its details, we can also try to incorporate in a model just their net effect In particular, measurement studies have revealed that the effective throughput at the network boundary

in CSMA/CD systems strongly depends on the length of the network, the used packet length and the number of users simultaneously trying to use the network In modelling studies, one therefore may include these three aspects in an expression for the effective capacity that a CSMA/CD network offers, as follows If the number of customers increases, more collisions will occur If a collision occurs, it will take longer to be resolved when the network length is larger due to the longer propagation time A good approximation of the network efficiency E(n) when n users are trying to access the network is therefore given

by:

EPI

C(n)tR collision resolution time plus the actual packet transmission time E[S] Of these

EM + cc >t n R, only the transmission time for E[S] is effectively used, yielding the above expression

E(n) = 1

1 - A(n)

'b> = A(n) 7

n-l

12.2 The convolution algorithm 269

We now proceed to the solution of closed queueing networks with load-dependent servers using the convolution scheme Consider a GNQN consisting of A4 MIMI1 stations with

K customers As before, the routing probabilities are denoted ri,j and the state space Z(A4, K) The traffic equations are solved as usual, yielding the visit counts L$ Since

we allow for load-dependent service rates now, we have to define pi(j), the service rate

at station i, given that station i is currently being visited by j customers We define the service demand (per passage) of station i given that there are j customers present as Di(j) = K/pi(j) For the steady-state customer distribution, we now have the following expression:

Pr{& = rz} = with pi(ni) = fi Di(j)

j=l

As we have seen before, the latter product replaces the ni-th power of Di we have seen in the load-independent case A direct calculation of the state probabilities and the normalising constant therefore does not change so much; however, a more practical way to calculate performance measures of interest is via a recursive algorithm This recursive solution of G(A4, K) does change by the introduction of load-dependency, as we now need to take care

of all different populations in each station In particular, we have:

In the first term, we recognize PM(k), the (unnormalised) probability of having Ic customers

in queue A4, and in the second term we recognize the normalising constant with 1 station (namely, the A4-th) and Ic customers less Hence, we can write:

This summation explains the name convolution method: the normalising constant G(M, K)

is computed as the convolution of the sequences PM(O), ,pM(K) and G(M - 1, K), , G(M - 1,O) As initial values for the recursion, we have G(m,O) = 1, for m = 1, a, M (there is only one way to divide 0 customers over m nodes), and G( 1, k) = pi (Ic) = l-& &(k), for Ic = 1, , K

Although this recursion scheme is slightly more involved than the load-independent case, we can easily represent it in a two-dimensional scheme as before (see Figure 12.1)

We can still work through the scheme column-wise, however, we need to remember the complete left-neighbouring column for the calculation of the entries in the current column

We therefore need to store one column more than in the load-independent case; the memory requirements are therefore of order O( 2K) If all the nodes are load-dependent, we need

to store the precomputed values Di(j) which costs O(MK) In summary, the memory requirements are of order O(MK) The time complexity can be bounded as follows To compute the Ic-th entry in a column, we have to add Ic products Since Ic can at most be equal to K, we need at most O(K) operations per element in the table Since the table contains MK elements, the overall computational complexity is O(MK2)

TO compute Pr{Ni = ni} we now proceed in a similar way as for load-independent nodes As we have seen there, it turns out to be convenient to first address the case

i = ikf:

Pr(NM =nM) = C

12.2 The convolution algorithm 271

= PA&M) G(M-l,K-nM)

As we have seen before, this expression contains normalising constants of columns M and

M - 1; hence, the ordering of columns (stations) is important In the load-independent case we were in the position to write G(M - 1, K - nM> as the difference of two normalising constants of the form G(M, ) by using the simple recursion (11.20) Due to the convolution- based expression in the load-dependent case (12.9) we cannot do so now, so we cannot generalise the above expression for all stations i If we want to compute this measure for more than one station, the only thing we can do is to repeat the convolution scheme with all of the stations of interest appearing once as station M Notice that the nodes for which

no such detailed measures are necessary can be numbered from 1 onwards and the part of the convolution for these nodes does not have to be repeated Using the above result, the utilisation of station M can be calculated as

k=l

(12.13) Here we are fortunate to find again an expression based on only the M-th column in the computational scheme, and hence it is valid not only for station M, but for all stations:

Xi(K) = v; G(M, K - 1)

As we have seen in the load-independent case, the throughput through the reference node (with visit-count 1) is the quotient of the last two normalising constants; all other node throughputs depend on that value via their visit ratio Vi For the average population of station AL! we find

k=l

(12.15)

Notice that this expression is again only valid for station M If one would be interested

in E[Ni(K)] (i # M), th e convolution algorithm should be run with a different ordering of stations so that station i is the last one to be added

We finally comment on the difficulty in computing Pr{NM 2 nM> in the load-dependent case Similar to the load-independent case, we can express this probability as follows:

12.3 Special cases of the convolution algorithm

There are two special cases when using the convolution algorithm for GNQN with load- dependent service rates: the case of having multiple servers per queue (Section 12.3.1) and the case of having an infinite server station (Section 12.3.2)

12.3.1 Convolution with multi-server queueing stations

Consider the case where we deal with a GNQN with M stations as we have seen before, however, the number of identical servers at station i is mi The service rate at station i

12.3 Special cases of the convolution algorithm 273

12.3.2 Convolution with an infinite-server station

rI G(WEo nl! iz2 0”“ (12.21)

The normalising constant is defined as usual to obtain probabilities that sum to one The following recursion then holds for the normalising constant:

Wf, K> = G(M - 1, K) + DMG(M, K - I), (12.22) with as initial conditions G(m,O) = 1, for m = 1, a, M, and G(l, Ic) = @/lc!, for

k = o, , K The only “irregularity” in the queueing network is brought into the compu- tational scheme directly at the initialisation; the rest of the computations do not change

In Chapter 11 we have developed an MVA recursion for the load-independent case and for the special load-dependent case of the infinite servers In this section we develop an MVA recursion scheme for general load-dependency

As before, the throughput X(K) is simply expressed as the fraction of the number of customers K present and the overall response time per passage:

(12.23)

In order to calculate the value of E[.&(K)] we again use the arrival theorem for closed queueing networks However, since the service times depend on the exact number of customers in the queue, the average number E[Ni(K)] in the queue does not provide

us with enough information for the calculation of E[&(K)] Instead, we need to know the probability r;(j]k) for j customers to be present at queue i, given overall network population Ic Let us for the time being assume we know these probabilities

In a more detailed version, the arrival theorem for closed queueing networks states that an arriving customer at queue i will find j (j = 0, e a , K - 1) customers already in that queue with probability x,(j]Ic - 1)) i.e., with the steady-state probability of having

j customers in queue i, given in total one customer less in the queueing network Since

As before, we have E[R(K)] = C& E[&(K)] and X(K) = K/E[R(K)], Xi(K) =

ViX(K), and the average population at station i can be can be computed using Little’s

12.4 Mean-value analysis 275

law or can be expressed as:

(12.25)

j=o

Example 12.3 Readdressing the load-independent case

E[&(K)] = &?r,(j - 1IK - l)Di

(12.27)

j=l

We then use (12.10) as follows:

l E[ki (k)] = Di, if node i is of IS type;

l E[&(k)] = cg=1j~i(j-l/k-l)Di(j), 1 no d e i is load-dependent, ‘f thereby using

12.5 Exact hierarchical decomposition 277 12.5 Exact hierarchical decomposition

Load-dependent servers are especially useful in hierarchical model decomposition, which not only saves computation time, but also keeps the modelling process structured and allows for reuse of subsystem models In this sect8ion, we restrict ourselves to the case where hierarchical model decomposition can be applied exactly We describe hierarchical model decomposition informally in Section 12.5.1, after which we formalize it in Section 12.5.2

12.5.1 Informal description of the decomposition method

Consider the case where one has to model a large computer or communication system involving many queueing stations (M) and customers (K) We have seen that the required computational effort is at least proportional to the product MK (in the load-independent case), so that we might try to decrease either K or M Therefore, instead of constructing a monolithic model and analysing that, we proceed to analyse subsystems first The results

of these detailed analyses, each with a smaller number of nodes, are “summarized” in a load-dependent server modelling the behaviour of the subsystem, which can subsequently

be used in a higher-level model of the whole system (again with a smaller number of nodes) The hierarchical decomposition approach as sketched here is also often referred

to as Norton’s approach by its similarity with the well-known decomposition approach

in electrical circuit analysis By the fact that a subsystem model, possibly consisting of multiple queueing stations, is replaced by a single load-dependent queueing station, this queueing station is often referred to as a fZow-equivalent service center (FESC) or as an aggregate service center The former name indicates that the load-dependency is chosen such that the single station acts in such a way that its customer flow is equivalent to that

of the original queueing network

Let us illustrate this approach for dealing with a GNQN with A4 stations numbered l; , b!l*,111* + l;-, A4 and K customers Stations 1 through M* are the nodes to be aggregated in a single FESC Stations M* + 1, , A4 are the queueing stations that will not

be affected; sometimes these are called the high-level nodes Note that the node numbering scheme does not affect generality Furthermore, we assume that the queueing network is structured in such a way that there is only a single customer stream from the high-level model stations to the stations to be aggregated and back This is visualized in Figure 12.2;

we come back to the interpretation of the probabilities ~1: and p later

Since the total number of customers in the GNQN equals K, the number of customers in the stations to be aggregated varies between 0 and K as well Given a certain population in

Figure 12.2: High-level view of a GNQN that is to be decomposed

o!

8

stations 1,***,A4*

population Ic = l, ,K

Figure 12.3: Decomposition approach using a FESC

the group of stations to be aggregated, the high-level stations perceive a fixed average delay for customers passing through the stations to be aggregated, namely the average response time (per passage!) for this subnetwork with Ic customers in it (denoted E[fi*(k)]) From this, we can compute the perceived rate X*(lc) = k/E[fi*(k)] at which customers are served in the subnetwork We compute X*(lc) by studying the subnetwork in isolation

by connecting the in- and out-going flows to the high-level model That is, the outgoing branch (labelled with probability al) is looped back to the queueing station at which the flow labelled 1 - p ends The throughput along this shorted circuit can then be used

as service rate for the aggregated subnetwork (the FESC) that can be embedded in the high-level nodes, as visualized in Figure 12.3 Notice that we have added an immediate feedback loop around the FESC (with probability 1 - Q); this is to ensure that the visit counts in the original non-decomposed GNQN of the non-aggregated stations, relative to the visit-count of the FESC, remain the same In many textbooks on queueing networks this immediate feedback loop is not explicitly mentioned

Note that since the overall network is a GNQN, the subnetwork is so as well Therefore, one can employ MVA or the convolution approach to solve it Since an MVA is recursive