1. Introduction
Since the early 1900th the Erlang multi-server queueing systems with losses (
B
-model or
M/M/N/0
system) and with an infinite size buffer (
C
-model or
M/M/N
system) 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 the
M/M/N/0
system and the delay prediction based on the formula obtained for the
M/M/N
system was a rather surprising because the requirement that inter-arrival and service times have an exponential distribution, which is imposed in the
M/M/N/0
and
M/M/N
models, 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 the
M/M/N/0
queue performance characteristics to characteristics of real systems modelled by such a queue, the hypothesis has arisen that the stationary state distribution in the
M/M/N/0
queue is the same as the one in the
M/G/N/0
queue 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 number
N
of 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
BMAP
(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 of
VMPP
contained extensive notations; however these notations were simplified greatly in [7] and ever since this process bears the name
BMAP
. The class of
BMAP
s 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, the
BMAP
is 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 the
M/M/N/0
system, 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 the
M/M/N
system, 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 order
k
distribution as a distribution of a sum of
k
independent 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].
PH
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 have
PH
distribution.
It follows from discussion above that it is interesting to extend investigation of Erlang's models to the case of the
BMAP
input and
PH
type service process. This work was started by M. Combe in [1] and V. Klimenok in [5] where the
BMAP/M/N
and
BMAP/M/N/0
models, respectively, were investigated.
In paper [2], we investigated the
BMAP/PH/N/0
model 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 general
BMAP
arrival process. Here we analyze the
BMAP/PH/N/L
system with a finite buffer and the
BMAP/PH/N
system 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 the
BMAP
and
PH
, he may fail in application to practical systems. The reason is the following. Assumption that the input flow is described by the
BMAP
allows 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 the
PH
distribution allows to take into consideration variation of service times. But the
BMAP
arrival process and
PH
service process are assumed to be stationary and independent of each other within the borders of the models of
BMAP/PH/N/L
type,
0≤L≤∞.
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 the
BMAP/PH/N/L
queueing 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 the
BMAP/PH/N/L
queueing 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 the
BMAP
and
PH
must 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 the
BMAP/PH/N/0
model operating in the
RE
was 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 models
BMAP/PH/N/L
and
BMAP/PH/N
operating in the
RE
2. The Mathematical Model
We consider the queueing system having
N
identical servers. The system behavior depends on the state of the stochastic process (random environment)
rtt≥0
which is assumed to be an irreducible continuous time Markov chain with the state space
{1…R}
R≥2
and the infinitesimal generator
Q
The input flow into the system is the following modification of the
BMAP
. In this input flow, the arrival of batches is directed by the process
νtt≥0
(the underlying process) with the state space
{0,1,…W}
Under the fixed state
r
of the
RE
this process behaves as an irreducible continuous time Markov chain. Intensities of transitions of the chain
νtt≥0
which are accompanied by arrival of
k
-size batch, are described by the matrices
Dk(r)k≥0
r=1R¯
with the generating function
D(r)(z)=∑k=0∞Dk(r)zk|z|≤1
The matrix
D(r)(1)
is an irreducible generator for all
r=1R¯
Under the fixed state
r
of the random environment, the average intensity
λ(r)
(fundamental rate) of the
BMAP
is defined as
λ(r)=θ(r)(D(r)(z))′|z=1e
and the intensity
λb(r)
of batch arrivals is defined as
λb(r)=θ(r)(−D0(r))e
Here the row vector
θ(r)
is the solution to the equations
θ(r)D(r)(1)=0θ(r)e=1
e
is a column vector of appropriate size consisting of 1's. The variation coefficient
cvar(r)
of intervals between batch arrivals is given by
while the correlation coefficient
ccor(r)
of intervals between successive batch arrivals is calculated as
At the epochs of the process
rtt≥0
transitions, the state of the process
νtt≥0
is not changed, but the intensities of its transitions are immediately changed.
The service process is defined by the modification of the
PH
-type service time distribution. Service time is interpreted as the time until the irreducible continuous time Markov chain
mtt≥0
with the state space
{0,1,…M+1}
reaches the absorbing state
M+1
Under the fixed value
r
of the random environment, transitions of the chain
mtt≥0
within the state space
{1…M}
are defined by an irreducible sub-generator
S(r)
while the intensities of transition into the absorbing state are defined by the vector
S0(r)=−S(r)e
At the service beginning epoch, the state of the process
mtt≥0
is chosen according to the probabilistic row vector
β(r)r=1R¯
It is assumed that the state of the process
mtt≥0
is not changed at the epoch of the process
rtt≥0
transitions. Just the exponentially distributed sojourn time of the process
mtt≥0
in the current state is re-started with a new intensity defined by the sub-generator corresponding to the new state of the random environment
rtt≥0
The system under consideration has
L0≤L≤∞
waiting positions. In the case of an infinite buffer (
L=∞
) all customers are always admitted to the system. In the case of a finite
L
the 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 size
n
consisting 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 blocks
ak;
⊗
and
⊕
are symbols of the Kronecker product and sum of matrices;
Ω⊗l=Ω⊗…⊗Ω︸ll≥1Ω⊗0=1
Ω⊕l=∑m=0l−1Inm⊗Ω⊗Inl−m−1l≥1
where
n
is the dimension of square matrix
Ω
;
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
where
nt
is the number of customers in the system, where
nt=0N+L¯
in case of
PA
and
CR
disciplines and
nt≥0
in case of
CA
discipline;
rt
is the state of the random environment,
rt=1R¯
νt
is the state of the
BMAP
underlying process,
νt=0W¯
mt(n)
is the phase of
PH
service process in the
n
th 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 epoch
tt≥0
Let us enumerate the states of the chain
ξtt≥0
in the lexicographic order and form the row vectors
pn
of probabilities corresponding to the state
n
of the first component of the process
ξtt≥0
Denote also
p=(p0p1p2…)
It is well known that the vector
p
satisfies the system of the linear algebraic equations (so called equilibrium equations or Chapman-Kolmogorov equations) of the form:
where
A
is the infinitesimal generator of the Markov chain
ξtt≥0
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 generator
A
of the Markov chain
ξtt≥0
in the case of partial admission discipline has the following block structure:
where
Proof of the Lemma follows from analysis of Markov chain
ξtt≥0
transitions during an infinitesimal interval. Block entries of the generator have the following meaning. The non-diagonal entries of the matrix
C(k)
define intensity of transition of the components
{rtνtmt(1)…mt(nt)}
of the Markov chain
ξtt≥0
which do not lead to the change of the number
k
of busy servers. The diagonal entries of the matrix
C(k)
are negative and define, up to the sign, intensity of leaving the corresponding states of the Markov chain
ξtt≥0
The entries of the matrix
Ψmm′(k)=Dm(k)ℬm′(k)
define intensity of transitions of the components
{rtνtmt(1)…mt(nt)}
of the Markov chain
ξtt≥0
which are accompanied by arrival of
m
customers and occupying
m′
servers conditional the number of busy servers is
k
The entries of the matrix
S0(k)
define intensity of transitions, which are accompanied by a departure of a customer, conditional the number of busy servers is
k
To solve system (1) with the matrix
A
defined by Lemma 1, we use the effective numerically stable procedure developed in [2] that exploits the special structure of the matrix
A
(it is upper block Hessenberg) and probabilistic meaning of the unknown vector
p
This 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:
where the matrices
Fl
are calculated recurrently:
the matrices
A¯IN+L
are calculated from the backward recursions:
the matrices
Gii=0N+L−1¯
are calculated from the backward recursion:
the vector
p0
is 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
A
of the Markov chain
ξtt≥0
in 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 matrix
A
defined by Lemma 2, we also use the procedure described by Theorem 1.
3.3. The Case of Complete Admission Discipline
Lemma 3. Infinitesimal generator
A
of the Markov chain
ξtt≥0
in 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 chain
ξtt≥0
is infinite and this makes its analysis more complicated. However, the block rows, except the first
N+L
boundary block rows, have only two non-zero blocks and this Markov chain behaves as Quasi-Death process when the state of the first component
nt
of the Markov chain
ξtt≥0
if greater than
N+L
It 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 chain
ξtt≥0
is infinite, this Markov chain is ergodic under the standard assumptions about the parameters of the
BMAP
input, the
PH
type 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
pll≥0
are calculated as follows:
where the matrices
Fl
are calculated recurrently
the matrices
A¯iI
are calculated as:
the matrix
G
has a form
the matrices
Gii=0N+L−1¯
are calculated from the backward recursion
the vector
p0
is the unique solution of the system:
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' matrix
G
which 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
A
of the Markov chain
ξtt≥0
has 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 (
QTMC
) presented in [6].
Theorem 3. The necessary and sufficient condition for existence of the Markov chain
ξtt≥0
stationary distribution is the fulfillment of the inequality
where
the vectors
xnn=1,2
are the unique solutions to the following systems of linear algebraic equations:
Proof. Using the results of [6], we directly obtain the desired condition in the form of inequality
where
x
is the unique solution to the system
It is easy to show that inequality (7) is reduced to the following inequality:
where
x1=x(IRW¯⊗eMN)
To get the equations for the row vectors
x1
and
x2
we multiply equation (8) by the matrices
IRW¯⊗eMN
and
IR⊗eW¯⊗IMN
respectively. After multiplication and some algebra we obtain equations (5), (6) for the vectors
x1
and
x2
So, inequality (9) is equivalent to inequality (3) and the theorem is proved.
The value
ρ
has a meaning of the system load. In what follows we assume inequality (3) be fulfilled.
To solve system (1) with the matrix
A
defined by (2), we use the effective numerically stable procedure [6] based on the account special structure of the matrix
A
notion of the censored Markov chain and probabilistic meaning of the unknown vector
p
For more detail see [6]. This procedure is given by the following statement.
Theorem 4. The stationary probability vectors
pll≥0
are calculated as follows:
where the matrices
Fl
are calculated recurrently:
the matrices
A¯iI
are calculated as:
the matrix
G
is calculated from the equation
the matrices
Gii=0N−1¯
are calculated from the backward recursion:
the vector
p0
is calculated as the unique solution to the following system of linear algebraic equations:
4. Performance Measures
Having the probability vector
p
been 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 probability
Ploss
that an arbitrary customer will be lost (the loss probability).
Theorem 5. The loss probability
Ploss
is calculated as follows
(i) in the case of
PA
discipline
(ii) in the case of
CR
discipline
(iii) in the case of
CA
discipline
Proofs of formulae (10) - (12) are analogous. So, we will prove only formula (10). According to a formula of the total probability, the probability
Ploss
is calculated as
where
Pk
is a probability that an arbitrary customer arrives in a batch consisting of
k
customers;
Pi(k)
is a probability to see
i
servers being busy at the epoch of the
k−
size batch arrival;
R(ik)
is a probability that an arbitrary customer will not be lost conditional it arrives in a batch consisting of
k
customers and
i
servers 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 case
L=∞
are presented below.
The probability to see
i
customers in the system
The mean number of customers in the system
The probability to see
n
busy servers
The mean number of busy servers
The mean number
Nidle
of idle servers
The vector
p^n
whose
(W¯(r−1)+ν+1)
-th entry is the joint probability to see
n
busy servers, the random environment in the state
r
and the process
νt
in the state
ν
The vector of conditional means of the number of busy servers under the fixed states of the random environment
The vector
p(a)(n)
whose
(W¯(r−1)+ν+1)
-th entry is the joint probability that an arbitrary arriving call sees
n
busy servers and the random environment in the state
r
and the state of the process
νt
becomes
ν
after the arrival epoch
The probability
p(a)(n)
that an arbitrary arriving call sees
n
busy servers
The vector
pb(a)(n)
whose
(W¯(r−1)+ν+1)
-th entry is the joint probability that an arbitrary arriving batch of size
k
sees
n
busy servers and the random environment in the state
r
and the state of the process
νt
becomes
ν
after the arrival epoch
where
The probability
pb(a)(kn)
that an arbitrary arriving batch of size
k
sees
n
busy servers
The vector
pb(a)(n)
whose
(W¯(r−1)+ν+1)
-th entry is the joint probability that an arbitrary arriving batch sees
n
busy servers and the random environment in the state
r
and the state of the process
νt
becomes
ν
after the arrival epoch
The probability
pb(a)(n)
that an arbitrary arriving batch sees
n
busy servers
The probability
Pimm
that an arbitrary customer will enter the service immediately upon arrival (without visiting a buffer)
5. Actual Sojourn Time
Let
νa(s)Res0
be the Laplace-Stieltjes transform (
LST
) of the sojourn time distribution and
ν¯a
be the mean sojourn time of the arbitrary customer in the system.
Theorem 6. The Laplace-Stieltjes transform
νa(s)
is calculated as follows
where
Proof. We derive the expression for the
LST
νa(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 variable
s
as 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
ℋ(s)
is the matrix
LST
of an arbitrary customer service time distribution. It is the
R
-size square matrix whose
(rr′)
entry is a probability that during the service time of a customer a catastrophe does not arrive and the process
rtt≥0
transits from the state
r
to the state
r′rr′=1R¯
It is defined by the formula:
Analogously, the entries of the matrix
LST
ℱ(s)
are the probabilities of no catastrophe arrival and corresponding transitions of the process
{rtmt(1)…mt(N)}t≥0
during the time interval from an arbitrary moment when all
N
servers 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
ν¯a
of an arbitrary customer in the system is calculated by
where
Proof. To get expression (18) for
ν¯a
we differentiate (17) at the point
s=0
and use the formula
ν¯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 the
BMAP
s (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 a
RE
and 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 a
RE
and their simple engineering approximations in cases of slowly and quickly varying
RE
(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 the
RE
which has two states (
R=2
). The generator of the random environment is
Q=(−5515−15)
The stationary distribution of the
RE
states is defined by the vector
q=(0.75,0.25)
The number of servers is
N=3
In the presented examples, we will use several different
MAP
s and
BMAP
s for description of the arrival process and two
PH
type distributions for description of the service processes under the fixed value of the
RE
For the use in the sequel, let us define these processes.
We consider four arrival processes
MAPrr=1,4¯
MAPr
is defined by the matrices
D0(r)
where
All these
MAP
s have fundamental rate
λ(r)=1.25
The
MAP1
has the squared variation coefficient
(cvar(1))2=4
and the coefficient of correlation of the lengths of successive inter-arrival times
ccor(1)=0.2
For the rest of the
MAP
s, the corresponding parameters are:
(cvar(2))2=4
ccor(2)=0.3
(cvar(3))2=1.097
ccor(3)=0.0052
(cvar(4))2=1.037
Based on these
MAP
s, we construct batch flows
BMAP
s as follows. If the
MAP
is defined by the matrices
D0(r)
and
D1(r)r=1,4¯
then the
BMAP
having the maximal size of a batch equal to
K
is defined by the matrices
D0(r)
Dk(r)=D1(r)qk−1(1−q)/(1−qK)k=1K¯r=1R¯
where
Following this way, we construct the
BMAP1
BMAP2
BMAP3
BMAP4
flows based on the
MAP1
MAP2
MAP3
MAP4
correspondingly, with
K=5
Note that the coefficients of variation and correlation of all
BMAP
s are the same as these coefficients for the corresponding
MAP
s. Fundamental rate
λ(r)
and the mean batch size
k¯(r)
of the
BMAP
s are the following:
λ(1)=λ(2)=λ(3)=λ(4)=3.488
The
PHrr=1,2¯
service processes are defined by the vectors
β(1)=(10)
β(2)=(0.2,0.8)
and the matrices
The mean rates of service are
μ(1)=2,μ(2)=14
The coefficients of variation of the service time distribution are defined by
(cvar(1))2=0.5
In the first experiment, we compare the dependence of
ν¯a
and
Pimm
on the system load
ρ
for the
BMAP
s with different correlations.
In the experiment we use service processes defined by
PH1
and
PH2
and four different input flows which are described by
BMAP1
BMAP2
BMAP3
and
BMAP4
having the same mean fundamental rate equal to 3.488 but different correlation coefficients.
We consider three queueing systems which have different combinations of the
BMAP
s under the first and second states of the
RE
The input flow in the first system is defined by
BMAP1
and
BMAP2
These
BMAP
s have large coefficients of correlation
ccor(1)=0.2
and
The input flow in the second system is defined by
BMAP3
and
BMAP4
These
BMAP
s have small coefficients of correlation
ccor(3)=0.0052
and
In the third system the input is defined by
BMAP1
and
BMAP4
The correlation coefficients of these
BMAP
s differ significantly.
Figures
1
and 2 show the dependence of the mean sojourn time
ν¯a
and the probability
Pimm
on the system load
ρ
for all these systems. Variation of the value of
ρ
in all experiments is performed by means of multiplying the entries of the matrices, which define the corresponding
BMAP
by some varying factor
γ
This implies the increase of the fundamental rate of all the
BMAP
by a factor
γ
Service time distributions are not modified. It is clear from Figure 1 that correlation in
BMAP
has a great impact on the sojourn time in the system. An increase of correlation at least in one of the
BMAP
s 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
BMAP
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
BMAP
s
In the second experiment we compare the values
ν¯a
and
Pimm
in the
BMAP/PH/N
system operating in the
RE
and in more simple queueing systems which can be considered as its simplified analogs. The first type analog is the
MX/PH/N
system in the
RE
where, under the fixed value of the
RE
the input flow is a group stationary Poisson with the same batch size distribution and intensity equal to fundamental rate of the corresponding
BMAP
in the original system. The second type analog is the system
MX/M/N
with parameters of arrival and service processes which are obtained by means of averaging, according to stationary distribution of the
RE
parameters of the original system.
Input flow is described by
BMAP1
and
BMAP2
Service processes are
PH1
and
Figures
3
and 4 show the dependence of the the mean sojourn time
ν¯a
and the probability
Pimm
on the value of
ρ
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 the
RE
has 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 by
PH1
and
PH2
distributions. Arrivals during these periods are defined by the stationary Poisson flow with the rates
λ1
and
λ2
correspondingly and initially we assume that
λ1≫λ2
It 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 rate
λ
should be 12.5 and consider four different situations: a huge difference of arrival rates
λ1=50λ2
a very big difference
λ1=10λ2
a big difference
λ1=3λ2
and equal arrival rates
λ1=λ2
The generator of the random environment is
Q=(−15155−5)
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 the
RE
is 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 the
RE
and calculation of performance measures in
BMAP/PH/N
system 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 the
RE
s with different rate which are characterized by the generators of the form
Q(k)=(−5515−15)10k
We vary the parameter
k
from -7 to 4 what corresponds to the variation of the
RE
rate from "very slow" to "very fast". In this and further experiments, the input flow is described by the
BMAP1
and
BMAP2
and the service process is defined by the
PH1
and
PH2
The 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 the
RE
and 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 the
RE
mean 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 the
RE
arrival 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"
RE
and the second one ("mixed parameters") can be applied to case of "very fast"
RE
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 the
RE
may 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 for
RE
rate (interval
k∈[−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
RE
rate
Figure 8.
Probability of immediate access to the servers as a function of the
RE
rate
Figure 9.
The mean number of customers in the system
Lqueue
as a function of the
RE
rate
In the fifth experiment we compare the mean number of customers, probability of immediate access to the servers and loss probability in the
BMAP/PH/N
and
BMAP/PH/N/L
systems operating in the
RE
for different values
L
of 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 disciplines
PA
and
CR
to their limits defined by the system with an infinite buffer is not very high. When we further increase the value
L
, we discover that even for the buffer capacity
L
about 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:
where
ν¯a
is the mean sojourn time in the system,
N
is the number of servers,
c1
is the charge for an unit of customer sojourn time in the system,
c2
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 as
c1=5
and
c2=3
Service time distribution at both states of the
RE
is exponential with intensities
μ(1)=1
On Figure 13, dependence of the cost criterion
J(N)
on
N
is presented along with the dependences of the summands
c1ν¯a
and
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 by
N=4
) can be significantly smaller than the values of the cost criterion for other values on
N
7. Conclusion
The
BMAP/PH/N/L
system 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.
