Open access peer-reviewed chapter

Performance Analysis of Multi-Server Queueing System Operating under Control of a Random Environment

By Che Soong Kim, Alexander Dudin, Valentina Klimenok, and Valentina Khramova

Published: March 1st 2010

DOI: 10.5772/8475

Downloaded: 1336

1. Introduction

Since the early 1900th the Erlang multi-server queueing systems with losses (B-model orM/M/N/0system) and with an infinite size buffer (C-model orM/M/Nsystem) provided good mathematical tools for capacity planning and performance evaluation in the classic telephone networks for many years. Good quality of the loss probability forecasting in real world networks based on the formulas obtained for theM/M/N/0system and the delay prediction based on the formula obtained for theM/M/Nsystem was a rather surprising because the requirement that inter-arrival and service times have an exponential distribution, which is imposed in theM/M/N/0andM/M/Nmodels, seems to be too strict. The interest of mathematicians to the fact of good matching of the calculated under debatable assumptions characteristics to their measured value in real world systems have lead to the following two results.

By efforts of many mathematicians (A. Ya. Khinchin and B. I. Grigelionis first of all), it was proved that the superposition of a large number of independent flows having uniformly small intensity approaches to the stationary Poisson input when the number of the superposed inputs tends to infinity. It explains the fact that the flows in classic telephone networks (where flows are composed by small individual flows from independent subscribers) have the exponentially distributed inter-arrival times.

Concerning the service time distribution, situation was more complicated. The real-life measurements have shown that the service (conversation) time can not be well approximated by means of the exponentially distributed random variable. So, due to the good matching of results obtained for theM/M/N/0queue performance characteristics to characteristics of real systems modelled by such a queue, the hypothesis has arisen that the stationary state distribution in theM/M/N/0queue is the same as the one in theM/G/N/0queue conditional that the average service times in both models coincide. This property was called the invariant (or insensitivity) property of the model with respect to the service time distribution. The work [10] by B. A. Sevastjanov is the first one where this property was proven strictly.

So, the question why the Erlang's models give very good results for a practice was highlighted. The special books containing the tables for a loss probability under the given values of the numberNof channels and intensities of the input and service exist. Different design problems (for a fixed value of permissible loss probability, to find the maximal intensity of the flow, which can be served by the line consisting of a fixed number of channels under the fixed average service time, or to find the necessary number of channels sufficient for transmission of the flow with a fixed intensity, etc) are solved by means of these tables.

However, the flows in the modern telecommunication networks have lost the nice properties of their predecessors in the old classic networks. In opposite to the stationary Poisson input (stationary ordinary input with no aftereffect), the modern real life flows are non-stationary, group and correlated. TheBMAP(Batch Markovian Arrival Process) arrival process was introduced as a versatile Markovian point process (VMPP) by M.F. Neuts in the 70th. The original development ofVMPPcontained extensive notations; however these notations were simplified greatly in [7] and ever since this process bears the nameBMAP. The class ofBMAPs includes many input flows considered previously, such as stationary Poisson (M), Erlangian (Ek), Hyper-Markovian (HM), Phase-Type (PH), Interrupted Poisson Process (IPP), Markov Modulated Poisson Process (MMPP). Generally speaking, theBMAPis correlated, so it is ideal to model correlated and (or) bursty traffic in modern telecommunication networks.

As it was mentioned above, the question why the inter-arrival times in the classical networks have the exponential distribution was answered in literature. However, Erlang's assumption that the service time distribution has the exponential distribution is not supported by the real networks measurements. In the case of theM/M/N/0system, good fitting of performance measures of this system with the respective measures of real world systems is easy explained by Sevastjanov's result. But in the case of theM/M/Nsystem, it was necessary to generalize results by Erlang to the cases of another, than exponential, service time distributions. This work was started by Erlang who offered so called Erlang's distribution. He introduced Erlangian of orderkdistribution as a distribution of a sum ofkindependent identically exponentially distributed random variables (phases). Further, so called phase type (PH) distribution was introduced into consideration as the straightforward generalization of Erlangian distribution, see, e.g., [8].PHdistribution includes as the special cases the exponential, Erlangian, Hyper-exponential, Coxian distributions. In our chapter we assume that service times at the fixed operation mode of the system havePHdistribution.

It follows from discussion above that it is interesting to extend investigation of Erlang's models to the case of theBMAPinput andPHtype service process. This work was started by M. Combe in [1] and V. Klimenok in [5] where theBMAP/M/NandBMAP/M/N/0models, respectively, were investigated.

In paper [2], we investigated theBMAP/PH/N/0model having no buffer. It was shown there that the stationary distribution of the system states essentially depends on the shape of the service time distribution and so Sevastjanov's invariant property does not hold true in the case of the generalBMAParrival process. Here we analyze theBMAP/PH/N/Lsystem with a finite buffer and theBMAP/PH/Nsystem with infinite buffer. Simultaneously, we make one more essential generalization of the model under study. Motivation of this generalization is as follows.

Even if one will use such general models of the arrival and service process as theBMAPandPH, he may fail in application to practical systems. The reason is the following. Assumption that the input flow is described by theBMAPallows to take into consideration a burstiness, an effect of correlation in the arrival process and variation of inter-arrival times. Assumption that the service process is described by thePHdistribution allows to take into consideration variation of service times. But theBMAParrival process andPHservice process are assumed to be stationary and independent of each other within the borders of the models ofBMAP/PH/N/Ltype,0L.While in many real world systems the input and service processes are not absolutely stable and may be mutually dependent. They may be influenced by some external factors, e.g., the different level of the noise in the transmission channel, hardware degradation and recovering, change of the distance by a mobile user from the base station, parallel transmission of high priority information, etc. Information transmission channel modeled by means of theBMAP/PH/N/Lqueueing system can be a part of complex communication network. The rest of the network may essentially vary characteristics of the arrival and service process in this system by means of: (i) changing the bandwidth of the channel (due to reliability factors or the needs to provide good quality of service in another parts of the network when congestion occurs); (ii) changing the mean arrival rate due breakdowns, overflow or underflow of alternative information transmission channels. Thus, to get the mathematical tool for adequate modeling such information transmission channels, more complicated queues than theBMAP/PH/N/Lqueueing system should be analyzed. These queues, in addition to the account of complicated internal structure of the arrival and service processes by means of considering theBMAPandPHmust take into account the influence of random external factors. In some extent, it can be done by means of analyzing the models of queues operating in a random environment. Such an analysis is the topic of this chapter.

Importance of investigation of the queues operating in a random environment (RE) drastically increased in the last years due to the following reason. The flows of information in the modern communication networks are essentially heterogeneous. Some types of information are very sensitive with respect to a delay and an jitter but tolerant with respect to losses. Another ones are tolerant with respect to the delay but very sensitive with respect to the loss of the packets. So, different schemes of the dynamic bandwidth sharing among these types exist and are developing. They assume that, in the case of congestion, transmission of the delay tolerant flows is temporarily postponed to provide better conditions for transmission of the delay sensitive flows. Analysis of such schemes requires the probabilistic analysis of the multi-dimensional processes describing transmission process of the different flows. This analysis is often impossible due to the mathematical complexity. In such a case, it is reasonable to decompose a simultaneous consideration of all flows to separate analysis of the processes of transmission of the delay sensitive and the delay tolerant flows. To this end, we model transmission of the delay sensitive flows in terms of the queues with the controlled service or (and) arrival rate where the service or the arrival rate can be changed depending on the queue length or the waiting time. Redistribution of a bandwidth to avoid congestion for the delay sensitive flows causes a variation, at random moments, of an available bandwidth for the delay tolerant flows. Correspondingly, the queues operating in a random environment naturally arise as the mathematical model for the delay tolerant flows transmission.

Mention that theBMAP/PH/N/0model operating in theREwas recently investigated in [4]. Short overview of the recent research of queues operating in the RE can be found there. In this chapter, we consider the modelsBMAP/PH/N/LandBMAP/PH/Noperating in theRE

2. The Mathematical Model

We consider the queueing system havingNidentical servers. The system behavior depends on the state of the stochastic process (random environment)rtt0which is assumed to be an irreducible continuous time Markov chain with the state space{1R}R2and the infinitesimal generatorQ

The input flow into the system is the following modification of theBMAP. In this input flow, the arrival of batches is directed by the processνtt0(the underlying process) with the state space{0,1,W}Under the fixed staterof theREthis process behaves as an irreducible continuous time Markov chain. Intensities of transitions of the chainνtt0which are accompanied by arrival ofk-size batch, are described by the matricesDk(r)k0r=1R¯with the generating functionD(r)(z)=k=0Dk(r)zk|z|1The matrixD(r)(1)is an irreducible generator for allr=1R¯Under the fixed staterof the random environment, the average intensityλ(r)(fundamental rate) of theBMAPis defined asλ(r)=θ(r)(D(r)(z))|z=1eand the intensityλb(r)of batch arrivals is defined asλb(r)=θ(r)(D0(r))eHere the row vectorθ(r)is the solution to the equationsθ(r)D(r)(1)=0θ(r)e=1eis a column vector of appropriate size consisting of 1's. The variation coefficientcvar(r)of intervals between batch arrivals is given by

(cvar(r))2=2λb(r)θ(r)(D0(r))1e1E1

while the correlation coefficientccor(r)of intervals between successive batch arrivals is calculated as

ccor(r)=(λb(r)θ(r)(D0(r))1(D(r)(1)D0(r))(D0(r))1e-1)/(cvar(r))2E2

At the epochs of the processrtt0transitions, the state of the processνtt0is not changed, but the intensities of its transitions are immediately changed.

The service process is defined by the modification of thePH-type service time distribution. Service time is interpreted as the time until the irreducible continuous time Markov chainmtt0with the state space{0,1,M+1}reaches the absorbing stateM+1Under the fixed valuerof the random environment, transitions of the chainmtt0within the state space{1M}are defined by an irreducible sub-generatorS(r)while the intensities of transition into the absorbing state are defined by the vectorS0(r)=S(r)eAt the service beginning epoch, the state of the processmtt0is chosen according to the probabilistic row vectorβ(r)r=1R¯It is assumed that the state of the processmtt0is not changed at the epoch of the processrtt0transitions. Just the exponentially distributed sojourn time of the processmtt0in the current state is re-started with a new intensity defined by the sub-generator corresponding to the new state of the random environmentrtt0

The system under consideration hasL0Lwaiting positions. In the case of an infinite buffer (L=) all customers are always admitted to the system. In the case of a finiteLthe system behaves as follows. If the system has all servers being busy at a batch arrival epoch, the batch looks for the available waiting position, and occupies it in case of success. If the system has all servers and all waiting positions being busy, the batch leaves the system forever and is considered to be lost. Due to a possibility of the batch arrivals, it can occur that there are free servers or waiting positions in the system at an arrival epoch, however the number of these positions is less than the number of the customers in an arriving batch. In such situation the acceptance of the customers to the system is realized according to the partial admission (PA) discipline (only a part of the batch corresponding to the number of free servers is allowed to enter the system while the rest of the batch is lost), the complete rejection (CR) discipline (a whole batch leaves the system if the number of free servers is less than the number of customers in the batch), complete admission (CA) discipline (a part of the batch corresponding to a number of free servers starts the service immediately while the rest of the batch waits for a service in the system in some special waiting space). All these disciplines are popular in the real life systems and got a lot of attention in the literature. Here, we consider all these disciplines.

Our aim is to calculate the stationary state distribution and main performance measures of the described queueing model.

For the use in the sequel, let us introduce the following notation:

en(0n)is a column (row) vector of sizenconsisting of 1's (0's). Suffix may be omitted if the dimension of the vector is clear from context;I(O)is an identity (zero) matrix of appropriate dimension (when needed the dimension of this matrix is identified with a suffix);diag{akk=1K¯}is a diagonal matrix with diagonal entries or blocksak;andare symbols of the Kronecker product and sum of matrices;Ωl=ΩΩll1Ω0=1Ωl=m=0l1InmΩInlm1l1wherenis the dimension of square matrixΩ;
D(z)=k=0diag{Dk(r)r=1R¯}zkE3
Dk(n)=diag{Dk(r)IMnr=1R¯}n=0N¯k0E4
D(n)(z)=k=0Dk(n)zkn=0N¯;E5
l(n)=diag{IW¯IMn(β(r))lr=1R¯}n=1N¯W¯=W+1E6
S(n)=diag{IW¯(S(r))nr=1R¯}n=1N¯;E7
S=diag{S(r)r=1R¯}E8
S0(n)=diag{IW¯(S0(r))nr=1R¯}n=1N¯E9
S¯0(N)=diag{IW¯(S0(r)β(r))Nr=1R¯}E10
C(n)=QIW¯IMn+D0(n)+S(n)n=0N¯E11
C¯(n)=QIW¯IMmin{nN}+D0(min{nN})+k=N+Ln+1Dk(min{nN})+S(min{nN})n=0N¯E12
C¯(N+L)=QIW¯IMN+k=0Dk(N)+S(N)E13

3. Process of the System States

It is easy to see that operation of the considered queueing model is described in terms of the regular irreducible continuous-time Markov chain

ξt={ntrtνtmt(1)mt(min{ntN})}t0E14

where

ntis the number of customers in the system, wherent=0N+L¯in case ofPAandCRdisciplines andnt0in case ofCAdiscipline;rtis the state of the random environment,rt=1R¯νtis the state of theBMAPunderlying process,νt=0W¯mt(n)is the phase ofPHservice process in thenth busy server,mt(n)=1M¯nt=1N¯(we assume here that the busy servers are numerated in order of their occupying, i.e. the server, which begins the service, is appointed the maximal number among all busy servers; when some server finishes the service, the servers are correspondingly enumerated) at epochtt0

Let us enumerate the states of the chainξtt0in the lexicographic order and form the row vectorspnof probabilities corresponding to the statenof the first component of the processξtt0Denote alsop=(p0p1p2)

It is well known that the vectorpsatisfies the system of the linear algebraic equations (so called equilibrium equations or Chapman-Kolmogorov equations) of the form:

pA=0pe=1E15

whereAis the infinitesimal generator of the Markov chainξtt0

Structure of this generator and methods of system (1) solution vary depending on the admission discipline.

3.1. The Case of Partial Admission Discipline

Lemma 1. Infinitesimal generatorAof the Markov chainξtt0in the case of partial admission discipline has the following block structure:
A=(Ann)nn=0N+L¯=E16
=(C(0)Ψ1,1(0)ΨN1N1(0)ΨNN(0)ΨN+1N(0)ΨN+L1N(0)Ψ^N+LN(0)S0(1)C(1)ΨN2,N2(1)ΨN1N1(1)ΨNN1(1)ΨN+L2,N1(1)Ψ^N+L1N1(1)OS0(2)ΨN3,N3(2)ΨN2,N2(2)ΨN1N2(2)ΨN+L3,N2(2)Ψ^N+L2,N2(2)OOC(N1)Ψ1,1(N1)Ψ2,1(N1)ΨL,1(N1)Ψ^L+1,1(N1)OOS0(N)C(N)D1(N)DL1(N)D^L(N)OOOS¯0(N)C(N)DL2(N)D^L1(N)OOOOOS¯0(N)C¯(N+L))E17

where

Ψmm(k)=Dm(k)m(k)k=0N1¯m1m=1N¯E18
Ψ^m1m(k)=m=m1Ψmm(k)D^m1(k)=m=m1Dm(k)E19

Proof of the Lemma follows from analysis of Markov chainξtt0transitions during an infinitesimal interval. Block entries of the generator have the following meaning. The non-diagonal entries of the matrixC(k)define intensity of transition of the components{rtνtmt(1)mt(nt)}of the Markov chainξtt0which do not lead to the change of the numberkof busy servers. The diagonal entries of the matrixC(k)are negative and define, up to the sign, intensity of leaving the corresponding states of the Markov chainξtt0The entries of the matrixΨmm(k)=Dm(k)m(k)define intensity of transitions of the components{rtνtmt(1)mt(nt)}of the Markov chainξtt0which are accompanied by arrival ofmcustomers and occupyingmservers conditional the number of busy servers iskThe entries of the matrixS0(k)define intensity of transitions, which are accompanied by a departure of a customer, conditional the number of busy servers isk

To solve system (1) with the matrixAdefined by Lemma 1, we use the effective numerically stable procedure developed in [2] that exploits the special structure of the matrixA(it is upper block Hessenberg) and probabilistic meaning of the unknown vectorpThis procedure is given by the following statement.

Theorem 1. In case of partial admission, the stationary probability vectors

pii=0N+L¯are computed as follows:
pl=p0Fll=1N+L¯E20

where the matricesFlare calculated recurrently:

Fl=(A¯0l+i=1l1FiA¯il)(A¯ll)1l=1N+L1¯E21
FN+L=(A0N+L+i=1N+L1FiAiN+L)(AN+LN+L)1E22

the matricesA¯IN+Lare calculated from the backward recursions:

A¯IN+L=AiN+Li=0N+L¯E23
A¯il=Ail+A¯il+1Gli=0l¯l=N+L1N+L2,,0E24

the matricesGii=0N+L1¯are calculated from the backward recursion:

Gi=(Ai+1i+1l=1N+Li1Ai+1i+1+lGi+lGi+l1Gi+1)1Ai+1iE25
i=N+L1N+L2,,0E26

the vectorp0is calculated as the unique solution to the following system of linear algebraic equations:

p0A¯0,0=0p0(l=1N+LFle+e)=1E27

3.2. The Case of Complete Rejection Discipline

Lemma 2. Infinitesimal generator

Aof the Markov chainξtt0in the case of complete rejection discipline has the following block structure:
A=(Ann)nn=0N+L¯=E28
=(C¯(0)Ψ1,1(0)ΨN1N1(0)ΨNN(0)ΨN+1N(0)ΨN+L1N(0)ΨN+LN(0)S0(1)C¯(1)ΨN2,N2(1)ΨN1N1(1)ΨNN1(1)ΨN+L2,N1(1)ΨN+L1N1(1)OS0(2)ΨN3,N3(2)ΨN2,N2(2)ΨN1N2(2)ΨN+L3,N2(2)ΨN+L2,N2(2)OOC¯(N1)Ψ1,1(N1)Ψ2,1(N1)ΨL,1(N1)ΨL+1,1(N1)OOS0(N)C¯(N)D1(N)DL1(N)DL(N)OOOS¯0(N)C¯(N)DL2(N)DL1(N)OOOOOS¯0(N)C¯(N+L))E29

The proof of the Lemma is analogous to the proof of the previous Lemma and takes into account the fact that the number of customers in the system does not change when the number of customers in an arriving batch exceeds the number of free servers.

To solve system (1) with the matrixAdefined by Lemma 2, we also use the procedure described by Theorem 1.

3.3. The Case of Complete Admission Discipline

Lemma 3. Infinitesimal generatorAof the Markov chainξtt0in the case of complete admission discipline has the following block structure:
A=(Ann)nn0=E30
=(C(0)Ψ1,1(0)ΨNN(0)ΨN+1N(0)ΨN+L1N(0)ΨN+LN(0)ΨN+L+1N(0)S0(1)C(1)ΨN1N1(1)ΨNN1(1)ΨN+L2,N1(1)ΨN+L1N1(1)ΨN+LN1(1)OS0(2)ΨN2,N2(2)ΨN1N2(2)ΨN+L3,N2(2)ΨN+L2,N2(2)ΨN+L1N2(2)OOΨ1,1(N1)Ψ2,1(N1)ΨL,1(N1)ΨL+1,1(N1)ΨL+2,1(N1)OOC(N)D1(N)DL1(N)DL(N)DL+1(N)OOS¯0(N)C(N)DL2(N)DL1(N)DL(N)OOOOC(N)D1(N)D2(N)OOOOS¯0(N)C¯(N+L)OOOOOOS¯0(N)C¯(N+L))E31

Essential difference of complete admission discipline is that the state space of the Markov chainξtt0is infinite and this makes its analysis more complicated. However, the block rows, except the firstN+Lboundary block rows, have only two non-zero blocks and this Markov chain behaves as Quasi-Death process when the state of the first componentntof the Markov chainξtt0if greater thanN+LIt allows to construct effective stable algorithm for calculation of the stationary distribution of this Markov chain. Note, that although the state space of the Markov chainξtt0is infinite, this Markov chain is ergodic under the standard assumptions about the parameters of theBMAPinput, thePHtype service and the random environment. The algorithm for calculation of the stationary distribution is given in the following statement.

Theorem 2. In case of complete admission discipline, the stationary probability vectors

pll0are calculated as follows:
pl=p0Fll1E32

where the matricesFlare calculated recurrently

Fl=(A¯0l+i=1l1FiA¯il)(A¯ll)1l1E33

the matricesA¯iIare calculated as:

A¯il=Ail+k=1Ail+kGmax{0l+kNL}Gmin{N+Ll+k}1Gmin{N+Ll+k}2Gli=0ll1E34

the matrixGhas a form

G=(C¯(N+L))1S¯0(N)E35

the matricesGii=0N+L1¯are calculated from the backward recursion

Gi=(Ai+1i+1+l=i+2Ai+1lGmax{0lNL} Gmin{N+Ll}1Gmin{N+Ll}2Gi+1)1Ai+1iE36

the vectorp0is the unique solution of the system:

p0A¯0,0=0p0(l=1Fle+e)=1E37

The proof of the Theorem follows from the theory of multi-dimensional Markov chains with continuous time, see [6]. It is worth to note that Neuts' matrixGwhich is usually found numerically as solution to matrix equation, see [9], here is obtained in the explicit form.

3.4. The Case of an Infinite Size of a Buffer(L=)

The system under consideration in this section has an infinite waiting space. If an arriving batch of customers sees idle servers, a part of the batch corresponding to the number of free servers occupy these servers while the rest of the batch joins the queue. If the system has all servers being busy at a batch arrival epoch, all customer of the batch go to the queue.

Lemma 3. Infinitesimal generator

Aof the Markov chainξtt0has the following block structure:
A=(Ann)nn0=E38
=(C(0)Ψ1,1(0)Ψ2,2(0)ΨN1N1(0)ΨNN(0)ΨN+1N(0)ΨN+2,N(0)S0(1)C(1)Ψ1,1(1)ΨN2,N2(1)ΨN1N1(1)ΨNN1(1)ΨN+1N1(1)OS0(2)C(2)ΨN3,N3(2)ΨN2,N2(2)ΨN1N2(2)ΨNN2(2)OOOC(N1)Ψ1,1(N1)Ψ2,1(N1)Ψ3,1(N1)OOOS0(N)C(N)D1(N)D2(N)OOOOS¯0(N)C(N)D1(N)OOOOOS¯0(N)C(N))E39

In what follows we perform the steady state analysis of the Markov chain having generator of form (2). To this end, we use the results for continuous time multi-dimensional Markov chain (QTMC) presented in [6].

Theorem 3. The necessary and sufficient condition for existence of the Markov chain

ξtt0stationary distribution is the fulfillment of the inequality
ρ=λ/μ¯1E40

where

λ=x1D'(z)|z=1eE41
μ¯=x2diag{(S0(r))Nr=1R¯}eE42

the vectorsxnn=1,2are the unique solutions to the following systems of linear algebraic equations:

x1(QIW¯+D(1))=0x1e=1E43
x2(QIMN+diag{(S(r)+S0(r)β(r))Nr=1R¯})=0x2e=1E44

Proof. Using the results of [6], we directly obtain the desired condition in the form of inequality

x[C(N)z+zD(N)(z)]z=1e0E45

wherexis the unique solution to the system

x(QIW¯MN+D(N)(1)+S(N)+S¯0(N))=0xe=1E46

It is easy to show that inequality (7) is reduced to the following inequality:

x1D'(z)|z=1ex2diag{(S0(r))Nr=1R¯}eE47

wherex1=x(IRW¯eMN)

x2=x(IReW¯IMN)E48

To get the equations for the row vectorsx1andx2we multiply equation (8) by the matricesIRW¯eMNandIReW¯IMNrespectively. After multiplication and some algebra we obtain equations (5), (6) for the vectorsx1andx2So, inequality (9) is equivalent to inequality (3) and the theorem is proved.

The valueρhas a meaning of the system load. In what follows we assume inequality (3) be fulfilled.

To solve system (1) with the matrixAdefined by (2), we use the effective numerically stable procedure [6] based on the account special structure of the matrixAnotion of the censored Markov chain and probabilistic meaning of the unknown vectorpFor more detail see [6]. This procedure is given by the following statement.

Theorem 4. The stationary probability vectors

pll0are calculated as follows:
pl=p0Fll1E49

where the matricesFlare calculated recurrently:

Fl=(A¯0l+i=1l1FiA¯il)(A¯ll)1l1E50

the matricesA¯iIare calculated as:

A¯il=Ail+A¯il+1Gl0ilNE51
A¯il=Ail+A¯il+1Glmax{iN}i0E52

the matrixGis calculated from the equation

G=(l=1AN+1N+lGl1)1AN+1NE53

the matricesGii=0N1¯are calculated from the backward recursion:

Gi=(Ai+1i+1+l=i+2Ai+1lGmax{0lN}Gmin{Nl}1Gmin{Nl}2Gi+1)1Ai+1iE54

the vectorp0is calculated as the unique solution to the following system of linear algebraic equations:

p0A¯0,0=0p0(l=1Fle+e)=1E55

4. Performance Measures

Having the probability vectorpbeen computed, we are able to calculate performance measures of the considered model. The main performance measure in the case of a finite buffer is the probabilityPlossthat an arbitrary customer will be lost (the loss probability).

Theorem 5. The loss probability

Plossis calculated as follows

(i) in the case ofPAdiscipline

Ploss=11λi=0N+L1pik=0N+Li(k+iNL)Dk(i)eE56

(ii) in the case ofCRdiscipline

Ploss=11λi=0N+L1pik=0N+LikDk(i)eE57

(iii) in the case ofCAdiscipline

Ploss=11λi=0N+L1pik=0kDk(i)eE58

Proofs of formulae (10) - (12) are analogous. So, we will prove only formula (10). According to a formula of the total probability, the probabilityPlossis calculated as

Ploss=1i=0N+L1k=1PkPi(k)R(ik)E59

wherePkis a probability that an arbitrary customer arrives in a batch consisting ofkcustomers;Pi(k)is a probability to seeiservers being busy at the epoch of theksize batch arrival;R(ik)is a probability that an arbitrary customer will not be lost conditional it arrives in a batch consisting ofkcustomers andiservers are busy at the arrival epoch.

It can be shown that

Pi(k)=piDk(i)ex1Dk(0)ei=0N+L1¯k1E60
Pk=kx1Dk(0)ex1D(z)|z=1'e=kx1Dk(0)eλk1E61
R(ik)={1kN+LiN+LikkN+Lii=0N+L1¯E62

By substituting (14)-(16) into (13) after some algebra we get (10).

Some performance measures for the caseL=are presented below.

The probability to seeicustomers in the system

pi=piei0E63

The mean number of customers in the system

Lqueue=i=0ipieE64

The probability to seenbusy servers

pn=pnen=0N1¯pN=n=NpneE65

The mean number of busy servers

Nbusy=n=1Nnpne+Nn=NpneE66

The mean numberNidleof idle servers

Nidle=NNbusyE67

The vectorp^nwhose(W¯(r1)+ν+1)-th entry is the joint probability to seenbusy servers, the random environment in the staterand the processνtin the stateν

p^n=pn(IRW¯eMn)n=0N1¯p^N=n=Npn(IRW¯eMn)E68

The vector of conditional means of the number of busy servers under the fixed states of the random environment

n=n=1min{nN}p^nn(IReW¯)diag{qr1r=1R¯}E69

The vectorp(a)(n)whose(W¯(r1)+ν+1)-th entry is the joint probability that an arbitrary arriving call seesnbusy servers and the random environment in the staterand the state of the processνtbecomesνafter the arrival epoch

p(a)(n)=λ1p^nD'(z)|z=1n=0N¯E70

The probabilityp(a)(n)that an arbitrary arriving call seesnbusy servers

p(a)(n)=p(a)(n)en=0N¯E71

The vectorpb(a)(n)whose(W¯(r1)+ν+1)-th entry is the joint probability that an arbitrary arriving batch of sizekseesnbusy servers and the random environment in the staterand the state of the processνtbecomesνafter the arrival epoch

pb(a)(kn)=λb1p^ndiag{Dk(r)r=1R¯}n=0N¯k1E72

where

λb=x(D(1)D(0))eE73

The probabilitypb(a)(kn)that an arbitrary arriving batch of sizekseesnbusy servers

pb(a)(kn)=pb(a)(kn)en=0N¯k1E74

The vectorpb(a)(n)whose(W¯(r1)+ν+1)-th entry is the joint probability that an arbitrary arriving batch seesnbusy servers and the random environment in the staterand the state of the processνtbecomesνafter the arrival epoch

pb(a)(n)=λb1p^n(D(1)D(0))n=0N¯E75

The probabilitypb(a)(n)that an arbitrary arriving batch seesnbusy servers

pb(a)(n)=pb(a)(n)en=0N¯E76

The probabilityPimmthat an arbitrary customer will enter the service immediately upon arrival (without visiting a buffer)

Pimm=λ1i=0N1pik=0Ni(k+iN)Dk(i)eE77

5. Actual Sojourn Time

Letνa(s)Res0be the Laplace-Stieltjes transform (LST) of the sojourn time distribution andν¯abe the mean sojourn time of the arbitrary customer in the system.

Theorem 6. The Laplace-Stieltjes transform

νa(s)is calculated as follows
νa(s)=1λ{i=0N1pik=1min{kNi}Dk(i)(IReW¯mi)+E78
+i=0pik=max{1Ni+1}Dk(min{iN})(max{0Ni})l=max{1Ni+1}k((s))iN+l(IReMN)}(s)eRE79

where

(s)=diag{β(r)r=1R¯}(sI(QIM+S))1diag{S0(r)r=1R¯}E80
(s)=(sI(QIMN+diag{(S(r))Nr=1R¯}))1diag{(S0(r)β(r))Nr=1R¯}E81
(n)=diag{eW¯IMNn(β(r))nr=1R¯}n=0N¯E82
S=diag{S(r)r=1R¯}E83

Proof. We derive the expression for the

LSTνa(s)by means of the method of collective marks (method of additional event, method of catastrophes) for references see, e.g. [3], [11]. To this end, we interpret the variablesas the intensity of some virtual stationary Poisson flow of catastrophes. So,νa(s)has the meaning of probability that no one catastrophe arrives during the sojourn time of an arbitrary customer. Then, the proof of the theorem follows from the formula of total probability if we analyze the states of the system at an arbitrary customer arrival epoch and take into account the probabilistic meaning of the involved matrices. The matrix(s)is the matrixLSTof an arbitrary customer service time distribution. It is theR-size square matrix whose(rr)entry is a probability that during the service time of a customer a catastrophe does not arrive and the processrtt0transits from the staterto the staterrr=1R¯It is defined by the formula:
(s)=diag{β(r)r=1R¯}0este(QIM+diag{S(r)r=1R¯})tdtdiag{S0(r)r=1R¯}E84

Analogously, the entries of the matrixLST(s)are the probabilities of no catastrophe arrival and corresponding transitions of the process{rtmt(1)mt(N)}t0during the time interval from an arbitrary moment when allNservers are busy till the first epoch when one of these servers finishes the service of a customer. This matrix is defined by the formula:

(s)=0este(QIMN+diag{(S(r))Nr=1R¯})tdtdiag{(S0(r)β(r))Nr=1R¯}E85

Theorem 7. The mean sojourn time

ν¯aof an arbitrary customer in the system is calculated by
ν¯a=1λ{i=0pik=max{1Ni+1}Dk(min{iN})(max{0Ni})l=max{1Ni+1}km=0i+lN1((0))m(0)(IReMN)+E86
+i=0N1pik=1min{kNi}Dk(i)(IReW¯Mi)(0)+E87
+i=0pik=max{1Ni+1}Dk(min{iN})(max{0Ni})l=max{1Ni+1}k((0))iN+l(IReMN)(0)}eRE88

where

(0)=diag{β(r)r=1R¯}[QIM+S]2diag{S0(r)r=1R¯}E89
(0)=[QIMN+diag{(S(r))Nr=1R¯}]2diag{(S0(r)β(r))Nr=1R¯}E90

Proof. To get expression (18) forν¯awe differentiate (17) at the points=0and use the formulaν¯a=ν¯a'(0)

6. Numerical Examples

The goal of the numerical experiments is to demonstrate the feasibility of the proposed algorithms for computing the stationary distributions of the number of customers and the sojourn time in the system and to give some insight into behavior of the considered queueing systems. In particular, the following issues are addressed:

Comparison of the mean sojourn time of an arbitrary customer and the probability of immediate access to the servers in the systems with varying traffic intensities and different coefficients of correlation in theBMAPs (experiment#1);

Comparison of the mean sojourn time of an arbitrary customer and the probability of immediate access to the servers in the original system in aREand in more simple queueing systems for different system loads (experiments#2);

Demonstration of possible positive effect of redistribution of traffic between the peak traffic periods and normal traffic periods (experiment#3);

Comparison of the exact value of performance measures of the system in aREand their simple engineering approximations in cases of slowly and quickly varyingRE(experiment#4);

Investigation of the rate of convergence of the mean sojourn time and the probability of immediate access in the system with the finite buffer to corresponding performance measures of the system with an infinite buffer when the buffer size increases (experiment#5);

Demonstration of the possibility to apply the presented results for optimization of the number of servers in the system (experiment#6).

In numerical examples, we consider the systems operating in theREwhich has two states (R=2). The generator of the random environment isQ=(551515)The stationary distribution of theREstates is defined by the vectorq=(0.75,0.25)The number of servers isN=3

In the presented examples, we will use several differentMAPs andBMAPs for description of the arrival process and twoPHtype distributions for description of the service processes under the fixed value of theREFor the use in the sequel, let us define these processes.

We consider four arrival processesMAPrr=1,4¯MAPris defined by the matricesD0(r)

D1(r)r=1,4¯E91
where

D0(1)=(3.90.150.150.130.60.10.150.140.5)D1(1)=(3.50.080.020.030.30.040.020.060.13)E92
D0(2)=(6.40.10.10.040.60.10.070.10.44)D1(2)=(6.060.120.020.010.40.050.010.060.2)E93
D0(3)=(2.90.730.770.873.060.530.850.52)D1(3)=(0.680.450.270.481.080.10.350.050.25)E94
D0(4)=(1.30.210.170.162.040.210.010.161.3)D1(4)=(0.460.320.140.131.340.20.020.011.1)E95

All theseMAPs have fundamental rateλ(r)=1.25TheMAP1has the squared variation coefficient(cvar(1))2=4and the coefficient of correlation of the lengths of successive inter-arrival timesccor(1)=0.2For the rest of theMAPs, the corresponding parameters are:(cvar(2))2=4ccor(2)=0.3(cvar(3))2=1.097ccor(3)=0.0052(cvar(4))2=1.037

ccor(4)=0.0065E96

Based on theseMAPs, we construct batch flowsBMAPs as follows. If theMAPis defined by the matricesD0(r)andD1(r)r=1,4¯then theBMAPhaving the maximal size of a batch equal toKis defined by the matricesD0(r)Dk(r)=D1(r)qk1(1q)/(1qK)k=1K¯r=1R¯where

q=0.9E97

Following this way, we construct theBMAP1BMAP2BMAP3BMAP4flows based on theMAP1MAP2MAP3MAP4correspondingly, withK=5Note that the coefficients of variation and correlation of allBMAPs are the same as these coefficients for the correspondingMAPs. Fundamental rateλ(r)and the mean batch sizek¯(r)of theBMAPs are the following:λ(1)=λ(2)=λ(3)=λ(4)=3.488

k¯(1)=k¯(2)=k¯(3)=k¯(4)=1.989E98

ThePHrr=1,2¯service processes are defined by the vectorsβ(1)=(10)β(2)=(0.2,0.8)and the matrices
S(1)=(4404)S(2)=(102220)E99

The mean rates of service areμ(1)=2,μ(2)=14The coefficients of variation of the service time distribution are defined by(cvar(1))2=0.5

(cvar(2))2=1.24E100

In the first experiment, we compare the dependence ofν¯aandPimmon the system loadρfor theBMAPs with different correlations.

In the experiment we use service processes defined byPH1andPH2and four different input flows which are described byBMAP1BMAP2BMAP3andBMAP4having the same mean fundamental rate equal to 3.488 but different correlation coefficients.

We consider three queueing systems which have different combinations of theBMAPs under the first and second states of theRE

The input flow in the first system is defined byBMAP1andBMAP2TheseBMAPs have large coefficients of correlationccor(1)=0.2and

ccor(2)=0.3E101

The input flow in the second system is defined byBMAP3andBMAP4TheseBMAPs have small coefficients of correlationccor(3)=0.0052and

ccor(4)=0.0065E102

In the third system the input is defined byBMAP1andBMAP4The correlation coefficients of theseBMAPs differ significantly.

Figures 1 and 2 show the dependence of the mean sojourn timeν¯aand the probabilityPimmon the system loadρfor all these systems. Variation of the value ofρin all experiments is performed by means of multiplying the entries of the matrices, which define the correspondingBMAPby some varying factorγThis implies the increase of the fundamental rate of all theBMAPby a factorγService time distributions are not modified. It is clear from Figure 1 that correlation inBMAPhas a great impact on the sojourn time in the system. An increase of correlation at least in one of theBMAPs describing input in the system implies an increase of the sojourn time in the system in all range of the system load.

Figure 1.

Mean sojourn time in the system as a function of the system load for different correlations in the B M A P s

Figure 2.

Probability of immediate access to the servers in the system as a function of the system load for different correlations in the B M A P s

In the second experiment we compare the valuesν¯aandPimmin theBMAP/PH/Nsystem operating in theREand in more simple queueing systems which can be considered as its simplified analogs. The first type analog is theMX/PH/Nsystem in theREwhere, under the fixed value of theREthe input flow is a group stationary Poisson with the same batch size distribution and intensity equal to fundamental rate of the correspondingBMAPin the original system. The second type analog is the systemMX/M/Nwith parameters of arrival and service processes which are obtained by means of averaging, according to stationary distribution of theREparameters of the original system.

Input flow is described byBMAP1andBMAP2Service processes arePH1and

PH2E103

Figures 3 and 4 show the dependence of the the mean sojourn timeν¯aand the probabilityPimmon the value ofρ

Figure 3.

Mean sojourn time in original system and more simple queueing systems

Figure 4.

Probability of immediate access to the servers in original system and more simple queueing systems

It can be seen from Figures 3 and 4 that an approximation of the mean sojourn time and the probability that an arbitrary call reaches the server immediately by means of their values in some specially constructed more simple queueing system can be rather bad.

The idea of the third experiment is the following. Let us assume that theREhas two states. One state corresponds to the peak traffic periods, the second one corresponds to the normal traffic periods. Service times during these periods are defined byPH1andPH2distributions. Arrivals during these periods are defined by the stationary Poisson flow with the ratesλ1andλ2correspondingly and initially we assume thatλ1λ2It is intuitively clear that if it is possible to redistribute the arrival processes (i.e., to reduce the arrival rate during the peak periods and to increase it correspondingly during the normal traffic periods) without changing the total average arrival rate, the mean sojourn time in the system can be reduced. In real life system such a redistribution is sometimes possible, e.g., by means of controlling tariffs during the peak traffic periods. The goal of this experiment is to show that this intuitive consideration is correct and to illustrate the effect of the redistribution.

We assume that the averaged arrival rateλshould be 12.5 and consider four different situations: a huge difference of arrival ratesλ1=50λ2a very big differenceλ1=10λ2a big differenceλ1=3λ2and equal arrival ratesλ1=λ2The generator of the random environment isQ=(151555)

It can be seen from Figures 5 and 6 that the smoothing of the peak rates can cause essential decrease of the mean sojourn time and the increase of the probability that an arbitrary call reaches the server immediately upon arrival in the system.

Figure 5.

Mean sojourn time as a function of system load for different relations of arrival rates

Figure 6.

Probability of immediate access to the servers as a function of system load for different relations of arrival rates

In the second experiment, we have seen that an approximation of the system performance measures by means of their values in more simple queueing system can be bad. However, it is intuitively clear the following. If the random environment is "very slow" (the rate of theREis much less then the rates of the input flow and the service processes), an approximation called below as "mixed system" can be applied successfully. This approximation consists of calculation of the system characteristics under the fixed states of the RE and their averaging by the RE distribution. If the random environment is "very fast", approximation called below as "mixed parameters" can be successfully applied. This approximation consists of averaging parameters of the arrival and service processes by the distribution of theREand calculation of performance measures inBMAP/PH/Nsystem with the averaged arrival and service rates.

In the fourth experiment, we show numerically that sometimes the described approximations make sense. However, in situations when environment is neither "very slow" nor "very fast", these approximations can be very poor. We consider theREs with different rate which are characterized by the generators of the formQ(k)=(551515)10k

We vary the parameterkfrom -7 to 4 what corresponds to the variation of theRErate from "very slow" to "very fast". In this and further experiments, the input flow is described by theBMAP1andBMAP2and the service process is defined by thePH1andPH2The results are presented in Figures 7, 8 and 9. In application of "mixed system" approximation, the averaged arrival rates under both states of the RE are equal to 3.488. The averaged service rate is equal to 2 at the first state of theREand is equal to 14 at the second state. The mean sojourn times of an arbitrary customer at these states are equal to 3.5297 and 0.0998, respectively; the probabilities of immediate access to the servers are equal to 0.2021 and 0.81399; the mean numbers of customers in the system are equal to 12.311 and 0.3482. The averaged, according to the stationary distribution of theREmean sojourn time of an arbitrary customer is equal to 2.6722 and the probability that an arbitrary arriving customer sees an idle server is equal to 0.355. In application of "mixed parameters" approximation, the averaged, according to the stationary distribution of theREarrival rate is equal to 3.778 while the averaged service rate is equal to 4.0625. The value of the mean sojourn time of an arbitrary customer in the system with averaged arrival and service rates is equal to 0.7541, the probability of immediate access to the servers is equal to 0.4168, the mean number of customers in the system is equal to 2.849.

Figures 7, 8 and 9 confirm the hypothesis that the first type approximation ("mixed system") is good in case of "very slow"REand the second one ("mixed parameters") can be applied to case of "very fast"REBut sometimes the second type approximation is not very good (see Figures 8 ) because it is not quite clear how to make averaging of service intensity. Simple averaging of service rates under the different states of theREmay be not correct when the load of the system is not high because there are time intervals when the system is empty and no service is provided. It is worth to note also that there is an interval forRErate (intervalk[3,0]) where one should not use the values of the system performance measures calculated based on the considered approximating models. The use of these values can lead to the large relative error. Thus, Figures 7, 8 and 9 confirm the importance of investigation implemented in this chapter. Simple engineering approximations can lead to unsatisfactory performance evaluation and capacity planning in real world systems.

Figure 7.

Mean sojourn time of an arbitrary customer as a function of the R E rate

Figure 8.

Probability of immediate access to the servers as a function of the R E rate

Figure 9.

The mean number of customers in the system L q u e u e as a function of the R E rate

In the fifth experiment we compare the mean number of customers, probability of immediate access to the servers and loss probability in theBMAP/PH/NandBMAP/PH/N/Lsystems operating in theREfor different valuesLof the buffer capacity and different customers admission discipline.

Figure 10.

Mean number of customers in the system as a function of the buffer capacity L

Figure 11.

Probability of immediate access to the servers as a function of the buffer capacity L

Figure 12.

Loss probability as a function of the buffer capacity L

Looking at Figures 10-12, it should be noted that the rate of convergence of the curves corresponding to the disciplinesPAandCRto their limits defined by the system with an infinite buffer is not very high. When we further increase the valueL, we discover that even for the buffer capacityLabout 5000, the difference is not negligible. So, estimation of performance measures of the system with an infinite buffer by the respective measures of the system with a finite buffer can be not very good. This explains why we made the separate analysis of the system with an infinite buffer.

Finally, in the sixth experiment we consider the next optimization problem:

J(N)=c1ν¯a+c2NmaxNE104

where

ν¯ais the mean sojourn time in the system,Nis the number of servers,c1is the charge for an unit of customer sojourn time in the system,c2is the cost of a server maintenance per unit of time.

It is clear that this problem is not trivial. When the number of servers is small, the cost of servers maintenance is also small, but the mean sojourn time is large. If we increase the number of servers, the mean sojourn time decreases while the cost of servers maintenance increases.

Let us assume that the cost coefficients be fixed asc1=5andc2=3Service time distribution at both states of theREis exponential with intensitiesμ(1)=1

μ(2)=7E105

On Figure 13, dependence of the cost criterionJ(N)onNis presented along with the dependences of the summandsc1ν¯aand

c2NE106

Figure 13.

Criterion J ( N ) as a function of number of servers in the system

Based on Figure 13 , one can conclude that our analysis allows effectively solve the problems of the system design and that the optimal value of the cost criterion (in this example it is provided byN=4) can be significantly smaller than the values of the cost criterion for other values onN

7. Conclusion

TheBMAP/PH/N/Lsystem operating in a finite state space Markovian random environment is investigated for the finite and infinite buffer capacity. The joint stationary distribution of the number of the customers in the system, the state of the random environment, and the states of the underlying processes of arrival and service processes is calculated. The analytic formulas for performance measures of the system are derived. The Laplace-Stieltjes transform of sojourn time distribution is derived and the mean sojourn time is calculated. Selected results of numerical study are presented. They show an impact of the correlation in arrival process, illustrate the poor quality of the system characteristics approximation by means of more simple models, confirm the positive effect of the traffic redistribution between the peak and normal operation periods. The results can be used for the optimal design, capacity planning, and performance evaluation of real world systems in which operation of the system can be changed depending on some external factors.

Acknowledgments

This work was supported by the Korea Research Foundation Grant Funded by the Korean Government (MOEHRD)(KRF-2008-313-D01211).

© 2010 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited and derivative works building on this content are distributed under the same license.

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Che Soong Kim, Alexander Dudin, Valentina Klimenok, and Valentina Khramova (March 1st 2010). Performance Analysis of Multi-Server Queueing System Operating under Control of a Random Environment, Trends in Telecommunications Technologies, Christos J Bouras, IntechOpen, DOI: 10.5772/8475. Available from:

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