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Since the early 1900th the Erlang multi-server queueing systems with losses (-model orsystem) and with an infinite size buffer (-model orsystem) 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 thesystem and the delay prediction based on the formula obtained for thesystem was a rather surprising because the requirement that inter-arrival and service times have an exponential distribution, which is imposed in theandmodels, 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 thequeue performance characteristics to characteristics of real systems modelled by such a queue, the hypothesis has arisen that the stationary state distribution in thequeue is the same as the one in thequeue 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  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 numberof 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. The(Batch Markovian Arrival Process) arrival process was introduced as a versatile Markovian point process () by M.F. Neuts in the 70th. The original development ofcontained extensive notations; however these notations were simplified greatly in  and ever since this process bears the name. The class ofs includes many input flows considered previously, such as stationary Poisson (), Erlangian (), Hyper-Markovian (), Phase-Type (), Interrupted Poisson Process (), Markov Modulated Poisson Process (). Generally speaking, theis 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 thesystem, 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 thesystem, 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 orderdistribution as a distribution of a sum ofindependent identically exponentially distributed random variables (phases). Further, so called phase type () distribution was introduced into consideration as the straightforward generalization of Erlangian distribution, see, e.g., .distribution 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 havedistribution.
It follows from discussion above that it is interesting to extend investigation of Erlang's models to the case of theinput andtype service process. This work was started by M. Combe in  and V. Klimenok in  where theandmodels, respectively, were investigated.
In paper , we investigated themodel 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 generalarrival process. Here we analyze thesystem with a finite buffer and thesystem 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 theand, he may fail in application to practical systems. The reason is the following. Assumption that the input flow is described by theallows 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 thedistribution allows to take into consideration variation of service times. But thearrival process andservice process are assumed to be stationary and independent of each other within the borders of the models oftype,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 thequeueing 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 thequeueing 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 theandmust 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 () 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 themodel operating in thewas recently investigated in . Short overview of the recent research of queues operating in the RE can be found there. In this chapter, we consider the modelsandoperating in the
2. The Mathematical Model
We consider the queueing system havingidentical servers. The system behavior depends on the state of the stochastic process (random environment)which is assumed to be an irreducible continuous time Markov chain with the state spaceand the infinitesimal generator
The input flow into the system is the following modification of the. In this input flow, the arrival of batches is directed by the process(the underlying process) with the state spaceUnder the fixed stateof thethis process behaves as an irreducible continuous time Markov chain. Intensities of transitions of the chainwhich are accompanied by arrival of-size batch, are described by the matriceswith the generating functionThe matrixis an irreducible generator for allUnder the fixed stateof the random environment, the average intensity(fundamental rate) of theis defined asand the intensityof batch arrivals is defined asHere the row vectoris the solution to the equationsis a column vector of appropriate size consisting of 1's. The variation coefficientof intervals between batch arrivals is given by
while the correlation coefficientof intervals between successive batch arrivals is calculated as
At the epochs of the processtransitions, the state of the processis not changed, but the intensities of its transitions are immediately changed.
The service process is defined by the modification of the-type service time distribution. Service time is interpreted as the time until the irreducible continuous time Markov chainwith the state spacereaches the absorbing stateUnder the fixed valueof the random environment, transitions of the chainwithin the state spaceare defined by an irreducible sub-generatorwhile the intensities of transition into the absorbing state are defined by the vectorAt the service beginning epoch, the state of the processis chosen according to the probabilistic row vectorIt is assumed that the state of the processis not changed at the epoch of the processtransitions. Just the exponentially distributed sojourn time of the processin the current state is re-started with a new intensity defined by the sub-generator corresponding to the new state of the random environment
The system under consideration haswaiting positions. In the case of an infinite buffer () all customers are always admitted to the system. In the case of a finitethe 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 () 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 () 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 () 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:
is a column (row) vector of sizeconsisting of 1's (0's). Suffix may be omitted if the dimension of the vector is clear from context;is an identity (zero) matrix of appropriate dimension (when needed the dimension of this matrix is identified with a suffix);is a diagonal matrix with diagonal entries or blocksandare symbols of the Kronecker product and sum of matrices;whereis the dimension of square matrix;
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
is the number of customers in the system, wherein case ofanddisciplines andin case ofdiscipline;is the state of the random environment,is the state of theunderlying process,is the phase ofservice process in theth busy server,(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 epoch
Let us enumerate the states of the chainin the lexicographic order and form the row vectorsof probabilities corresponding to the stateof the first component of the processDenote also
It is well known that the vectorsatisfies the system of the linear algebraic equations (so called equilibrium equations or Chapman-Kolmogorov equations) of the form:
whereis the infinitesimal generator of the Markov chain
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 generatorof the Markov chainin the case of partial admission discipline has the following block structure:
Proof of the Lemma follows from analysis of Markov chaintransitions during an infinitesimal interval. Block entries of the generator have the following meaning. The non-diagonal entries of the matrixdefine intensity of transition of the componentsof the Markov chainwhich do not lead to the change of the numberof busy servers. The diagonal entries of the matrixare negative and define, up to the sign, intensity of leaving the corresponding states of the Markov chainThe entries of the matrixdefine intensity of transitions of the componentsof the Markov chainwhich are accompanied by arrival ofcustomers and occupyingservers conditional the number of busy servers isThe entries of the matrixdefine intensity of transitions, which are accompanied by a departure of a customer, conditional the number of busy servers is
To solve system (1) with the matrixdefined by Lemma 1, we use the effective numerically stable procedure developed in  that exploits the special structure of the matrix(it is upper block Hessenberg) and probabilistic meaning of the unknown vectorThis procedure is given by the following statement.
Theorem 1. In case of partial admission, the stationary probability vectors
are computed as follows:
where the matricesare calculated recurrently:
the matricesare calculated from the backward recursions:
the matricesare calculated from the backward recursion:
the vectoris calculated as the unique solution to the following system of linear algebraic equations:
3.2. The Case of Complete Rejection Discipline
Lemma 2. Infinitesimal generator
of the Markov chainin the case of complete rejection discipline has the following block structure:
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 matrixdefined by Lemma 2, we also use the procedure described by Theorem 1.
3.3. The Case of Complete Admission Discipline
Lemma 3. Infinitesimal generatorof the Markov chainin the case of complete admission discipline has the following block structure:
Essential difference of complete admission discipline is that the state space of the Markov chainis infinite and this makes its analysis more complicated. However, the block rows, except the firstboundary block rows, have only two non-zero blocks and this Markov chain behaves as Quasi-Death process when the state of the first componentof the Markov chainif greater thanIt 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 chainis infinite, this Markov chain is ergodic under the standard assumptions about the parameters of theinput, thetype 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
are calculated as follows:
where the matricesare calculated recurrently
the matricesare calculated as:
the matrixhas a form
the matricesare calculated from the backward recursion
the vectoris the unique solution of the system:
The proof of the Theorem follows from the theory of multi-dimensional Markov chains with continuous time, see . It is worth to note that Neuts' matrixwhich is usually found numerically as solution to matrix equation, see , here is obtained in the explicit form.
3.4. The Case of an Infinite Size of a Buffer
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
of the Markov chainhas the following block structure:
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 () presented in .
Theorem 3. The necessary and sufficient condition for existence of the Markov chain
stationary distribution is the fulfillment of the inequality
the vectorsare the unique solutions to the following systems of linear algebraic equations:
Proof. Using the results of , we directly obtain the desired condition in the form of inequality
whereis the unique solution to the system
It is easy to show that inequality (7) is reduced to the following inequality:
To get the equations for the row vectorsandwe multiply equation (8) by the matricesandrespectively. After multiplication and some algebra we obtain equations (5), (6) for the vectorsandSo, inequality (9) is equivalent to inequality (3) and the theorem is proved.
The valuehas a meaning of the system load. In what follows we assume inequality (3) be fulfilled.
To solve system (1) with the matrixdefined by (2), we use the effective numerically stable procedure  based on the account special structure of the matrixnotion of the censored Markov chain and probabilistic meaning of the unknown vectorFor more detail see . This procedure is given by the following statement.
Theorem 4. The stationary probability vectors
are calculated as follows:
where the matricesare calculated recurrently:
the matricesare calculated as:
the matrixis calculated from the equation
the matricesare calculated from the backward recursion:
the vectoris calculated as the unique solution to the following system of linear algebraic equations:
4. Performance Measures
Having the probability vectorbeen 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 probabilitythat an arbitrary customer will be lost (the loss probability).
Theorem 5. The loss probability
is calculated as follows
(i) in the case ofdiscipline
(ii) in the case ofdiscipline
(iii) in the case ofdiscipline
Proofs of formulae (10) - (12) are analogous. So, we will prove only formula (10). According to a formula of the total probability, the probabilityis calculated as
whereis a probability that an arbitrary customer arrives in a batch consisting ofcustomers;is a probability to seeservers being busy at the epoch of thesize batch arrival;is a probability that an arbitrary customer will not be lost conditional it arrives in a batch consisting ofcustomers andservers are busy at the arrival epoch.
It can be shown that
By substituting (14)-(16) into (13) after some algebra we get (10).
Some performance measures for the caseare presented below.
The probability to seecustomers in the system
The mean number of customers in the system
The probability to seebusy servers
The mean number of busy servers
The mean numberof idle servers
The vectorwhose-th entry is the joint probability to seebusy servers, the random environment in the stateand the processin the state
The vector of conditional means of the number of busy servers under the fixed states of the random environment
The vectorwhose-th entry is the joint probability that an arbitrary arriving call seesbusy servers and the random environment in the stateand the state of the processbecomesafter the arrival epoch
The probabilitythat an arbitrary arriving call seesbusy servers
The vectorwhose-th entry is the joint probability that an arbitrary arriving batch of sizeseesbusy servers and the random environment in the stateand the state of the processbecomesafter the arrival epoch
The probabilitythat an arbitrary arriving batch of sizeseesbusy servers
The vectorwhose-th entry is the joint probability that an arbitrary arriving batch seesbusy servers and the random environment in the stateand the state of the processbecomesafter the arrival epoch
The probabilitythat an arbitrary arriving batch seesbusy servers
The probabilitythat an arbitrary customer will enter the service immediately upon arrival (without visiting a buffer)
5. Actual Sojourn Time
Letbe the Laplace-Stieltjes transform () of the sojourn time distribution andbe the mean sojourn time of the arbitrary customer in the system.
Theorem 6. The Laplace-Stieltjes transform
is calculated as follows
Proof. We derive the expression for the
by means of the method of collective marks (method of additional event, method of catastrophes) for references see, e.g. , . To this end, we interpret the variableas the intensity of some virtual stationary Poisson flow of catastrophes. So,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 matrixis the matrixof an arbitrary customer service time distribution. It is the-size square matrix whoseentry is a probability that during the service time of a customer a catastrophe does not arrive and the processtransits from the stateto the stateIt is defined by the formula:
Analogously, the entries of the matrixare the probabilities of no catastrophe arrival and corresponding transitions of the processduring the time interval from an arbitrary moment when allservers are busy till the first epoch when one of these servers finishes the service of a customer. This matrix is defined by the formula:
Theorem 7. The mean sojourn time
of an arbitrary customer in the system is calculated by
Proof. To get expression (18) forwe differentiate (17) at the pointand use the formula
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 thes (experiment);
Comparison of the mean sojourn time of an arbitrary customer and the probability of immediate access to the servers in the original system in aand in more simple queueing systems for different system loads (experiments);
Demonstration of possible positive effect of redistribution of traffic between the peak traffic periods and normal traffic periods (experiment);
Comparison of the exact value of performance measures of the system in aand their simple engineering approximations in cases of slowly and quickly varying(experiment);
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);
Demonstration of the possibility to apply the presented results for optimization of the number of servers in the system (experiment).
In numerical examples, we consider the systems operating in thewhich has two states (). The generator of the random environment isThe stationary distribution of thestates is defined by the vectorThe number of servers is
In the presented examples, we will use several differents ands for description of the arrival process and twotype distributions for description of the service processes under the fixed value of theFor the use in the sequel, let us define these processes.
We consider four arrival processesis defined by the matrices
All theses have fundamental rateThehas the squared variation coefficientand the coefficient of correlation of the lengths of successive inter-arrival timesFor the rest of thes, the corresponding parameters are:
Based on theses, we construct batch flowss as follows. If theis defined by the matricesandthen thehaving the maximal size of a batch equal tois defined by the matriceswhere
Following this way, we construct theflows based on thecorrespondingly, withNote that the coefficients of variation and correlation of alls are the same as these coefficients for the correspondings. Fundamental rateand the mean batch sizeof thes are the following:
Theservice processes are defined by the vectorsand the matrices
The mean rates of service areThe coefficients of variation of the service time distribution are defined by
In the first experiment, we compare the dependence ofandon the system loadfor thes with different correlations.
In the experiment we use service processes defined byandand four different input flows which are described byandhaving the same mean fundamental rate equal to 3.488 but different correlation coefficients.
We consider three queueing systems which have different combinations of thes under the first and second states of the
The input flow in the first system is defined byandTheses have large coefficients of correlationand
The input flow in the second system is defined byandTheses have small coefficients of correlationand
In the third system the input is defined byandThe correlation coefficients of theses differ significantly.
Figures 1 and 2 show the dependence of the mean sojourn timeand the probabilityon the system loadfor all these systems. Variation of the value ofin all experiments is performed by means of multiplying the entries of the matrices, which define the correspondingby some varying factorThis implies the increase of the fundamental rate of all theby a factorService time distributions are not modified. It is clear from Figure 1 that correlation inhas a great impact on the sojourn time in the system. An increase of correlation at least in one of thes describing input in the system implies an increase of the sojourn time in the system in all range of the system load.
In the second experiment we compare the valuesandin thesystem operating in theand in more simple queueing systems which can be considered as its simplified analogs. The first type analog is thesystem in thewhere, under the fixed value of thethe input flow is a group stationary Poisson with the same batch size distribution and intensity equal to fundamental rate of the correspondingin the original system. The second type analog is the systemwith parameters of arrival and service processes which are obtained by means of averaging, according to stationary distribution of theparameters of the original system.
Input flow is described byandService processes areand
Figures 3 and 4 show the dependence of the the mean sojourn timeand the probabilityon the value of
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 thehas 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 byanddistributions. Arrivals during these periods are defined by the stationary Poisson flow with the ratesandcorrespondingly and initially we assume thatIt 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 rateshould be 12.5 and consider four different situations: a huge difference of arrival ratesa very big differencea big differenceand equal arrival ratesThe generator of the random environment is
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.
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 theis 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 theand calculation of performance measures insystem 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 thes with different rate which are characterized by the generators of the form
We vary the parameterfrom -7 to 4 what corresponds to the variation of therate from "very slow" to "very fast". In this and further experiments, the input flow is described by theandand the service process is defined by theandThe 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 theand 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 themean 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 thearrival 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"and the second one ("mixed parameters") can be applied to case of "very fast"But 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 themay 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 forrate (interval) 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.
In the fifth experiment we compare the mean number of customers, probability of immediate access to the servers and loss probability in theandsystems operating in thefor different valuesof the buffer capacity and different customers admission discipline.
Looking at Figures 10-12, it should be noted that the rate of convergence of the curves corresponding to the disciplinesandto their limits defined by the system with an infinite buffer is not very high. When we further increase the value, we discover that even for the buffer capacityabout 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:
is the mean sojourn time in the system,is the number of servers,is the charge for an unit of customer sojourn time in the system,is 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 asandService time distribution at both states of theis exponential with intensities
On Figure 13, dependence of the cost criteriononis presented along with the dependences of the summandsand
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 by) can be significantly smaller than the values of the cost criterion for other values on
Thesystem 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.
This work was supported by the Korea Research Foundation Grant Funded by the Korean Government (MOEHRD)(KRF-2008-313-D01211).
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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