The joint moments of the departure process

7.4 Departure process analysis of priority queues

7.4.3 The joint moments of the departure process

As shown in Section5.3.3, the departure process of a MAP/MAP/1 queue is an infinite state MAP. Similarly, the departure process of the MMAP[K]/PH[K]/1 priority queue is an infinite MMAPas well. ThisMMAPgenerates a high priority (low priority) arrival when a high pri-ority (low pripri-ority) departure occurs in the system. The computation of the joint moments of the departure process,η_i,j⁽^L⁾,η_i,j⁽^H⁾, is rather difficult based on this infiniteMMAP representa-tion. Instead of using this representation directly, we construct a finiteMMAP(considering the 6 cases listed above in the beginning of Section7.4), such that the joint distribution of the first two arrivals of thisMMAP– starting from the appropriate initial distribution – is identi-cal with the one of two consecutive stationary departures of the MMAP[K]/PH[K]/1 priority queue.

The blocks of the matrices of this finiteMMAPare constructed from the block matrices of theQBDdescribing the queue length process ((280)-(283)), such that the transitions between the 6 listed cases are taken into consideration. The initial probability distribution is computed according to the stationary distribution of the queue length process just after a departure considering the 6 listed cases.

The matrices of the resultingMMAPrepresentation are as follows:

H₀ =

118 analysis of the mmap[k]/ph[k]/1 priority qeue

where the diagonal blocks ofH₀are:

M₁= E₀,

MatricesH₀,HL andHH can be interpreted as follows. The first set of states correspond to case0, 0. From this state, a high priority arrival moves the Markov chain to state1, 0, and a low priority arrival to state0, 1. No service events can occur in0, 0. From state1, 0(second set of states) there are transitions to1, 1+due to a low priority arrival (EL), to2+, 0due to a high priority arrival (EH), and to0, 0due to a high priority service (J₁⁽^H⁾). The latter one is accompanied by a high priority departure event, hence the corresponding matrix block is located inH_H. The rest of the blocks can be interpreted similarly.

The steady-state distribution of the MMAP[K]/PH[K]/1 queue just after a departure can be calculated from the stationary distribution as

x_i,j = are computed based on (307) and (296) as

x_0,0= ^y^1,0^J

7.4 departure process analysis of priority qeues 119

x_0,1= ^y^1,1^J

(H)

1 +y_0,2J⁽₁^L⁾ λ⁽^L⁾+λ⁽^H⁾

= (y_0,0R⁽₁^L⁾+y_0,1R⁽₀^L⁾)J₁⁽^H⁾+y_0,2J₁⁽^L⁾

λ⁽^L⁾+λ⁽^H⁾ , (312)

x_0,2₊= ^∑

∞j=₃y_0,jJ⁽₁^L⁾+_∑^∞_j₌₂y_1,jJ⁽₁^H⁾ λ⁽^L⁾+λ⁽^H⁾

= (y⁽₀^H⁾−y_0,0−y_0,1−y_0,2)J⁽₁^L⁾+ (_∑^∞_j₌₀y_1,j−y_0,0R⁽₀^L⁾−y_0,0R⁽₁^L⁾−y_0,1R⁽₀^L⁾)J⁽₁^H⁾ λ⁽^L⁾+λ⁽^H⁾

= (y⁽₀^H⁾−y_0,0−y_0,1−y_0,2)J⁽₁^L⁾ λ⁽^L⁾+λ⁽^H⁾

+ (_∑^∞_j₌₀_∑_k^j₌₀y_0,kR⁽_j₋^L⁾_k−y_0,0R₀⁽^L⁾−y_0,0R⁽₁^L⁾−y_0,1R⁽₀^L⁾)J⁽₁^H⁾ λ⁽^L⁾+λ⁽^H⁾

= (y⁽₀^H⁾−y_0,0−y_0,1−y_0,2)J⁽₁^L⁾+(y⁽₀^H⁾Rˆ⁽^L⁾−y_0,0R⁽₀^L⁾−y_0,0R⁽₁^L⁾−y_0,1R⁽₀^L⁾)J⁽₁^H⁾ λ⁽^L⁾+λ⁽^H⁾

, (313) where vectory⁽₀^H⁾denotes the probability vector that there are no high priority customers in the system, thusy⁽₀^H⁾= _∑^∞_i₌₀y_0,i.

Having computed the vectorvand the matricesH₀,H_H andH_L, the joint moments of the departure process are obtained according to (64), hence

η⁽_i,j^c⁾=i!j!v(−H₀)⁻ⁱ⁻¹H_c(−H₀)⁻^j₁, i,j≥0,c={H,L}. (314) 7.4.4 An efficient, truncation-free procedure

Section7.4.2describes how to calculate the steady state joint distribution of the number of customers, based on which Section7.4.3 derives the joint moments of the inter-departure times. These results, however, are not straight forward to implement in an efficient way. The potential numerical pitfalls are:

• To obtain matrixGH₀ the matrix polynomial equation (300) has to be solved, but this matrix equation has infinitely many terms, and relies on infinitely many elements of matrix seriesG_i⁽^L⁾.

• To obtain vectorsy_0,j,j ≥ 0, matricesTi are required (see (301)), that are defined by infinite summations (302).

• To obtain vectorsy_i,j,i > 0,j ≥ 0, a convolution like formula needs to be evaluated that gets slower and slower with increasingi(see (296)).

• To obtain vectorv, vectors likex₂₊_,0₊are necessary to compute, which involves infinite summations of vectorsy_0,j.

The naive numerical implementation of such a procedure evaluates the infinite summa-tions by truncation. In order to avoid the loss of accuracy, the truncation threshold must be high, depending on the parameters (in particular, the load) of the system, which makes the procedure slow.

120 analysis of the mmap[k]/ph[k]/1 priority qeue

Fortunately, it is possible to develop a numerically efficient procedure that addresses all the above listed critical steps without using any truncation. The efficient solution of (300) is addressed in Section7.4.4.1, while the accurate calculation of the vectorsy_0,i and further related quantities is discussed in Section7.4.4.2.

7.4.4.1 Two fundamental matrices and their relations

There are two matrices that play key roles in the efficient analysis of the system. One of them isG_H₀, that is the solution of (300), and the other one is matrixZ, defined by

∑

∞ i=0

G_i⁽^L⁾G_H₀ⁱ. (315)

Theorem 33. If the algebraic multiplicities of the eigenvalues of matrices G_H₀ andZare one, G_H₀Z=ZG_H₀ holds.

To prove this theorem we first need the following Lemma:

Lemma 7. If the algebraic multiplicities of the eigenvalues of matrixG⁽^L⁾(z)_{and matrix}E₀+ zEL+EHG⁽^L⁾(z)are one, the eigenvectors of G⁽^L⁾(z)and E0+zEL+EHG⁽^L⁾(z)are the same.

Proof of the Lemma. The proof uses the same techniques as in [47].

Let ν_i andu_i be the eigenvalue and the corresponding right eigenvector of G⁽^L⁾(z)(for simplicity we assume thatG⁽^L⁾(z)has distinct eigenvalues). AsG⁽^L⁾(z)satisfies the matrix-quadratic equation of (295),ν_isatisfies

deth

J₁⁽^H⁾+ (E0+zEL+J₀⁽^H⁾)ν_i+EHν_i² i

=0, (316)

and the associated right eigenvectoru_iis the solution of h

J₁⁽^H⁾+ (E₀+zE_L+J₀⁽^H⁾)ν_i+E_Hν_i²

i·u_i =0. (317)

Note that bothν_iand vectorsu_iare functions ofz.

By substituting (284) into (316) and by some basic manipulations we get det

ν_iD₀+ν_izD_L+ν_i²D_H

| {z }

⊕ s_Hσ_H+ν_iS_H

| {z }

M₂

⊗I

=0, (318)

from which it follows thatM₁⊕M₂has a zero eigenvalue. Letδ_jandσ_kdenote the eigenval-ues ofM₁andM₂, respectively. Since the eigenvalues ofM₁⊕M₂areδ_j+σ_k, to have a zero eigenvalue there must existj⁰ andk⁰such thatδ_j⁰ =−σ_k⁰. The eigenvector ofM₁belonging toδ_j⁰ is denoted byθ_j⁰, the one of M₂ belonging toσ_k⁰ is denoted by ψ_k⁰. Let us introduce φ_i =θ_j⁰⊗ψ_k⁰⊗₁.

Now we show thatφ_iis an eigenvector ofG⁽^L⁾(z)associated withν_i, thus it satisfies (317):

J₁⁽^H⁾+ (E₀+zEL+J₀⁽^H⁾)ν_i+EHν_i² i·φ_i

= [I⊗M₂⊗I+M₁⊗I⊗I]·(θ_j0⊗ψ_k0⊗₁)

=θ_j0⊗(σ_k⁰ψ_k⁰)⊗₁+ (δ_j⁰θ_j0)⊗ψ_k0⊗₁=_0.

(319)

7.4 departure process analysis of priority qeues 121 since the service process of the low priority class is stopped during the busy period of the high priority class.

The proof will be similar to the one of Lemma7, using the same techniques as in [47] again.

Letλ_k andv_k be the eigenvalue and the corresponding right eigenvector ofG_H₀ (for sim-plicity we assume thatG_H₀ has distinct eigenvalues). AsG_H₀ satisfies the matrix equation of (300),λ_ksatisfies

and the associated right eigenvectorv_kis the solution of

" By substituting (284) into (321) and by some basic manipulation we get

det

from which it follows thatN₁⊕N₂has a zero eigenvalue. Letα_jandβ_hdenote the eigenval-ues ofN₁andN₂, respectively. Since the eigenvalues ofN₁⊕N₂areα_j+β_h, to have a zero Next, we show thatµ_kis an eigenvector ofZ. Observe thatµ_kis an eigenvector ofZif and only if it is an eigenvector ofG⁽^L⁾(z)|_z₌_λ_k since

122 analysis of the mmap[k]/ph[k]/1 priority qeue

As Lemma7states that the eigenvectors of matrixG⁽^L⁾(z)and matrixE₀+zE_L+E_HG⁽^L⁾(z) are the same, it is enough to prove thatµ_kis an eigenvector of(E₀+zEL+EHG⁽^L⁾(z))|_z₌_λ_k:

(E₀+λ_kEL+EHG⁽^L⁾(λ_k))·µ_k

(D₀⊗I⊗I+λ_kDL⊗I⊗I+

∑

∞ i=0

λⁱ_k(DH⊗I⊗I)(G^˜⁽_i^L⁾⊗I)

·(ζ_j⁰⊗ξ_h⁰)

= [N₁/λ_k⊗I]·(ζ_j0⊗ξ_h0) =α_j⁰/λ_k(ζ_j⁰⊗ξ_h0).

(325)

AsG_H₀ andZhave the same eigenvectors, the same matrix diagonalizes them, consequently they commute.

Note that the theorem can be generalized to the case when the eigenvalues are not distinct, but it requires the detailed discussion of the combination of the multiplicities of the eigenval-ues ofM1andM2(N1andN2) that we neglect here.

Based on this commutative property, the next theorem enables the efficient computation ofG_H₀ andZ.

Theorem 34. MatricesG_H₀ andZsatisfy the following coupled matrix-quadratic equations:

0= J₁⁽^H⁾+ (E₀+J₀⁽^H⁾)Z+E_LG_H₀Z+E_HZ², 0= J₁⁽^L⁾+ (E₀+J₀⁽^L⁾)GH₀ +EHZGH₀+ELGH₀2.

(326)

Proof. To obtain equations forZ, we multiply (290) byG_H₀ⁱfrom the right, sum it fromi=1 to∞, and add (289) to it. By using (315) we get

0= J₁⁽^H⁾+ELZGH0+ (E0+J₀⁽^H⁾)Z+EH

∑

∞ i=0

∑

i k=0

G_k⁽^L⁾G_i⁽₋^L⁾_kGH0

i. (327)

The last term becomesE_HZ²by swapping the sums and exploiting thatG_H₀andZcommute, providing the first matrix quadratic equation. The second matrix quadratic equation can be obtained from (300), when the definition ofZis applied.

Interestingly, the two matrix equations show perfect symmetry. While the solution of cou-pled Sylvester equations has an extensive literature (e.g. [27, 60,89] are recent methods), there are no methods available for coupled matrix quadratic equations (according to our best knowledge). A very simple method is given by Algorithm6. (We are sure that more efficient solution methods can be developed as well, but even this simple method performs very well in our numerical examples.)

Note that Algorithm6needs only successive solution of matrix quadratic equations, that is much more efficient than the direct application of the results of Section7.4.2both in time and in space requirement, since the infinite series ofG_i⁽^L⁾matrices and the solution of (300) are not needed.

Although the results above have been derived in an algebraic way, matricesG_H₀ and Z have important probabilistic interpretations. Let us denote by(nH,nL)the set of states in which there aren_H high andn_L low priority customers in the queue. Then, entry (a,b) of matrixG_H₀ is the conditional probability that starting from state ain(0, 1)the first visit to (0, 0)occurs in stateb. Similarly, the entry (a,b) of matrixZis the conditional probability that

7.4 departure process analysis of priority qeues 123

Algorithm 6Solving the coupled matrix quadratic equations of (326) procedureGH₀,Z= SolveCoupled(E₀,EH,EL,J₀^(H),J₁^(H),J₀^(L),J₁^(L))

k←0 GH₀(0)←I repeat

Solve0=J₁^(H)+ (E0+J₀^(H)+ELGH₀(k))Z^(k+1)+EHZ^(k+1)²forZ^(k+1) Solve0=J₁^(L)+ (E0+J₀^(L)+EHZ^(k+1))GH₀(k+1)+ELGH₀(k+1)2

forGH₀(k+1)

k←k+1

untilkGH₀(k)−GH₀(k−1)k<eandkZ^(k)−Z^(k−1)k<e returnGH₀(k), Z^(k)

end procedure

starting from stateain(1, 0)the first visit to(0, 0)occurs in stateb. Using these probabilistic interpretations it is easy to see thatZandG_H₀ commute: let us investigate the busy period generated by a high and a low priority customer, thus the system is in(1, 1)initially. Since the probability that the first passage to(0, 0)occurs in statebis not affected by the order of service, we immediately have thatG_H₀Z=ZG_H₀.

7.4.4.2 The boundary distribution

MatricesT_i,i ≥0, that are necessary to compute boundary probabilitiesy_0,i, are defined by infinite summations (see (302)). To obtain matricesT_iefficiently we introduce closely related matricesAndefined by

A_n=

∑

∞ i=n

G_i⁽^L⁾G_H₀ⁱ⁻ⁿ, n≥0. (328)

Theorem 35. MatricesA_ncan be obtained recursively as the solutions of discrete Sylvester-type matrix equations

0= (−E0−J₀⁽^H⁾−EHG₀⁽^L⁾)⁻¹ ELAn−1+EH n−1 k

∑

G_k⁽^L⁾A_n−k

−An

+ (−E₀−J₀⁽^H⁾−E_HG₀⁽^L⁾)⁻¹E_HA_nZ

(329)

forn>0, and forn=0we have thatA₀= Z.

Proof. The case ofn=0is trivial (see (315) for the definition ofZ).

To derive equations forn > 0, let us multiply equations (290) byG_H₀ⁱ⁻ⁿfrom the right and sum them up fromi=nto∞. We get

0=EL

∑

∞ i=n

G_i⁽₋^L⁾₁GH₀i−n

| {z }

An−1

+(E₀+J₀⁽^H⁾)

∑

∞ i=n

G_i⁽^L⁾GH₀i−n

| {z }

+EH

∑

∞ i=n

∑

i k=0

G_k⁽^L⁾G_i⁽₋^L⁾_kGH₀i−n.

124 analysis of the mmap[k]/ph[k]/1 priority qeue

The last term can be transformed further by swapping the order of the two summations, lead-ing to

where the commuting property of matricesZ andG_H₀ was utilized. Collecting all parts of the equation gives Sylvester equations, establishing the theorem.

Observe that the Sylvester equation for Andepends only on matrices Ai,i < n, that are already available in stepn.

A consequence of Theorem35is that matricesT_ican be expressed explicitly.

Corollary 18. MatricesT_i,i≥0,can be obtained as

Finally, it is possible to solve the sum of matricesAnexplicitly as well, that will be useful later.

Proof. Summing equations (329) from n = 1 to ∞, swapping the sums, and applying the definition ofAˆ provides the corollary after some simple algebraic manipulations.

Corollary 20. The sum of matricesTi, denoted byTˆ =_∑^∞_i₌₀Ti, is given by

Tˆ =E0+J₁⁽^H⁾+ELGH0+EL+EHA.ˆ (334) Proof. The corollary follows from Corollaries18and19.

7.4 departure process analysis of priority qeues 125

7.4.4.3 Calculating the joint moments of the departure process

The steps to obtain the quantities necessary for the departure process analysis are as follows.

1. First solve matricesG_H₀ andZwith Algorithm6.

2. Solve the matrix quadratic equations to obtain matricesGˆ⁽^L⁾andRˆ⁽^L⁾.

3. Calculate matricesGifori =0, 1, 2based on (289) and (290), matricesRi fori= 0, 1 based on (291) and (292), matrices A_i fori = 1, 2based on (329), and matricesT_i for i=1, 2based on Corollary18.

4. Compute matrixAˆ by solving (333) and matrixTˆ from (334).

After this preparation the required stationary probabilities are calculated.

• Vectorsy_0,0,y_0,1andy_0,2 are computed directly from (304) and (301) using theTi ma-trices computed above.

• To derive vectory⁽₀^H⁾=_∑^∞_i₌₀y_0,i, equations (301) are summed up fromi=1to∞, and (304) is added leading to

∑

∞ i=1

∑

i k=0

y_0,kT_i−k

| {z }

from (301)

+y_0,0T₀−y_0,0J₀⁽^L⁾

| {z }

from (304)

∑

∞ k=0

y_0,k

| {z }

y⁽₀^H⁾

∑

∞ i=k

T_i−k

| {z }

Tˆ

−y_0,0J₀⁽^L⁾,

(335)

implyingy₀⁽^H⁾ =y_0,0J₀⁽^L⁾Tˆ⁻¹.

Using the quantities calculated so far, it is possible to obtain all the six components of vector v, namelyx_0,0,x_1,0,x_1,1₊,x₂₊_,0₊,x_0,1andx_0,2₊. The joint moments of the departure process are given by (314).

Part III

Q U E U E I N G N E T W O R K S

8

Q U E U E I N G N E T W O R K A N A LY S I S B A S E D O N T H E J O I N T M O M E N T S

8.1 traffic based decomposition for the analysis of qeueing networks At the beginning the queueing network models were restricted to have Poisson arrival pro-cesses withiid.service times in the nodes. More general and multi-class queueing network models are also available for a while [7], but the inter-dependency of the traffic classes is not captured in these classical models. Apart of the exact analytical results which are based on the product form of the stationary distribution of the number of customers at the queueing nodes, a set of approximate analysis methods were developed. One of the most commonly applied approximate analysis methods for queueing networks is thetraffic based decomposi-tion, where the nodes of the queueing network are evaluated sequentially, in isolation [42]. A node analysis is composed by four main steps (also depicted in Figure35),

1. the aggregation of the input streams of the node,

2. the queueing analysis of node (computing the performance measures), 3. the characterization of the departure process,

4. and splitting the departure process according to the routing of the traffic.

The evolution ofMAPandMMAPbased traffic models allowed the extension of the traffic based queueing network analysis to cope with dependent inter-arrival and service times [42, 43]. An important benefit ofMAPsandMMAPsis that from the four node analysis steps listed above, three can be performed in an exact and efficient way. These traffic models are closed for the aggregation and splitting operations (see Sections3.1.1and3.1.2), and the related queueing models can be solved by matrix-analytic methods. The only step where an approximation is needed is the modeling of the departure process. The accuracy of the queueing network analysis depends on who well the departure process of the queues is characterized.

This chapter puts the different parts of the dissertation together. TheMAPandMMAP char-acterization results described in Chapter3will be used for the internal traffic representation and basic traffic operations (the first and the fourth step in the list above), while Chapters5,

Node

Traffic aggregation

Traffic splitting Performance

analysis

Departure process approximation

Figure 35.: Steps of the node analysis in traffic-based decomposition

130 qeueing network analysis based on the joint moments

6and7provide the apparatus for the performance analysis of the nodes (step 2) and the char-acterization of the departure process (step 3).

In the rest of the chapter we restrict our attention to feed-forward queueing networks with-out loops, where the stability condition is satisfied for all nodes.

8.2 single-type qeueing networks

Several approaches are available in the literature to model the departure process of MAP/MAP/1 queues by a MAP. This section provides a numerical study to compare the noteworthy algorithms published in the past with the lag-1 joint moment based approach presented by this dissertation.

8.2.1 MAP based approximations for the departure process

AMAPis a suitable approximation for the departure process of a node if it satisfies the fol-lowing properties:

• Obviously, it must capture the behavior of the departure process reasonably well.

• The approximatingMAPmust have a Markovian representation, otherwise it is impos-sible to decide if it is a valid process or not. Remember that all performance measures obtained with an invalid arrival process are invalid.

• TheMAPrepresentation for the departure process should not be considerably larger than the one for the input process. Otherwise theMAPdescribing the traffic gets larger and larger every time it passes through a node, making the analysis of large queueing networks problematic.

The methods involved in the comparison study are introduced below.

p o i s s o n m o d e l

According to the simplest possible method the network traffic is approximated by a Poisson process, hence the representation of theMAPdescribing the departure process of nodeiis given by

H₀⁽ⁱ⁾=−λ⁽ⁱ⁾, H₁⁽ⁱ⁾ =λ⁽ⁱ⁾, (336)

whereλ⁽ⁱ⁾ is the traffic rate of the node. The Poisson approximation is included in the com-parison only to demonstrate the effect of ignoring the burstiness and the correlation of the traffic to the accuracy of the approximation.

This approximation is always Markovian (λ⁽ⁱ⁾>0), and the representation of the departure process is not larger than the one of the arrival process (both are of size1).

s c a l e d s e r v i c e p r o c e s s m o d e l l

According to this model (introduced in [22]), the departure process is obtained from the service process with appropriate scaling, thus

H₀⁽ⁱ⁾=ρ⁽ⁱ⁾S₀⁽ⁱ⁾, H₁⁽ⁱ⁾ =ρ⁽ⁱ⁾S₁⁽ⁱ⁾, (337)

8.2 single-type qeueing networks 131 whereρ⁽ⁱ⁾is the utilization of nodei. This approximation is accurate whenρ⁽ⁱ⁾ →1, since in

this case there are customers in the queue in almost all of the time, implying that the inter-departure times are given by the service times. At lower load, however, the accuracy of the approximation is expected to be worse.

This departure model is always Markovian, and its size always equals the size of the service process, hence the traffic model does not grow when passing through many nodes after each other.

b u s y p e r i o d a n a l y s i s b a s e d m e t h o d

The departure process is modeled in a unique way in [42]. It is based on the observation that the inter-departure times are equal to the service times as long as the queue is busy, and they are equal to a remaining arrival time plus a service time if the last customer left the system idle. According to the procedure in [42] a 2-phase discretePHdistribution is created to fit the first three moments of the number of departures in a busy period, and aMAPis constructed for the departure process along the aforementioned interpretation.

The busy period based method always gives a Markovian representation, but the model for the departure process is larger (has more phases) than the one of the arrival process, hence the MAPcharacterizing the traffic grows every time when it goes through a queue. Consequently, only queueing networks with a low number of nodes can be analyzed by this method.

t r u n c at i o n b a s e d m e t h o d s

As mentioned in Section5.3the exact departure process of a MAP/MAP/1 queue is aMAP with infinitely many phases. There were several efforts to truncate this infinite process (in-cluding [10],[74]), with perhaps the most sophisticated one being the ETAQA truncation ([44]).

The ETAQA truncation preserves the joint densities of the inter-departure time up to the truncation level exactly. The construction of theMAPmatrices of the departure process are provided in Section5.3.2.

The ETAQA truncation model has two important drawbacks. First, it gives a non-Markovian representation in almost all cases. The other issue is that the resulting repre-sentation is huge, making the analysis of even the simplest tandem networks numerically challenging.

l a g - 1 j o i n t m o m e n t b a s e d a p p r o a c h

As proven in Section 3.2, size N MAPs can be characterized by N² marginal moments and joint moments. According to this approach a number of such moments are calculated from the infinite, but exactMAP representation of the departure process and a finite, but approximateMAPrepresentation is created based on these moments. The construction of the matrices characterizing thisMAPis detailed in Section5.3.3.

The representation created this way is, however, typically non-Markovian. The transforma-tion method introduced in Sectransforma-tion3.2.4solves this issue in many cases. In case this trans-formation fails, the procedures presented in Section3.3 can be applied to find the closest possibleMAP according to the joint moment distance (Section3.3.3) and according to the distance of the joint densities (Section3.3.4), respectively.

As a result, the representation is always Markovian, and the size is very compact, indepen-dent on the size of the input and service processes of the queue.

132 qeueing network analysis based on the joint moments

Node1 Node2

(D0(1),D1(1))

Figure 36.: A tandem queueing network with two nodes

8.2.2 Numerical results with a tandem network

For the first numerical study let us consider a simple2-node tandem queueing network as shown by Figure36. Traffic entering the network is directed to node1, after getting served it is forwarded to node2.

The parameters of the system are taken from real traffic measurements. For the first sce-nario the inputMAPhas been created from the inter-arrival times of the LBL trace based on 5marginal and2×2joint moments using the moment matching procedure described in Sec-tion3.2. The matrices of thisMAPare

D₀⁽^LBL⁾= With these parameters the arrival rate of the packets isλ = 188.29, the squared coefficient of variation of the inter-arrival times isc²_A=2.183, and the lag-1auto-correlation is0.159.

The same LBL trace file contains information on the packet sizes as well. Matching the first two moments of the packet sizes we got the followingPHdistribution:

σ=^h0.19937 0.80063

by which the mean packet size is138.86and the squared coefficient of variation is2.508. The packet size distribution and the speed of the transmission lineC_iat nodeitogether determine the service process of the packets as

S₀⁽ⁱ⁾ =C_i·S,

S₁⁽ⁱ⁾ =C_i·(−S)₁·σ.

The values ofC_iare set such that utilization of both queues are equal to the desired valueρ.

In document E F F I C I E N T M AT R I X-A N A LY T I C S O L U T I O N O F M U L T I -T Y P E Q U E U E I N G S Y S T E M S W I T H C O R R E L AT E D T R A F F I C gábor horváth Dissertation submitted to the Hungarian Academy of Sciences for the degree of Doctor of (Pldal 129-0)