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Phase-Type Models of Channel-Holding Times in Cellular Communication Systems
Christensen, Thomas Kaare; Nielsen, Bo Friis; Iversen, Villy Bæk
Published in:
I E E E Transactions on Vehicular Technology
Link to article, DOI:
10.1109/TVT.2004.825803
Publication date:
2004
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Christensen, T. K., Nielsen, B. F., & Iversen, V. B. (2004). Phase-Type Models of Channel-Holding Times in
Cellular Communication Systems. I E E E Transactions on Vehicular Technology, 53(3), 725-733.
https://doi.org/10.1109/TVT.2004.825803
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 3, MAY 2004 725
Phase-Type Models of Channel-Holding Times in
Cellular Communication Systems
Thomas Kaare Christensen, Student Member, IEEE, Bo Friis Nielsen, Member, IEEE, and
Villy Bæk Iversen, Member, IEEE
Abstract—In this paper, we derive the distribution of the
channel-holding time when both cell-residence and call-holding
times are phase-type distributed. Furthermore, the distribution of
the number of handovers, the conditional channel-holding time
distributions, and the channel-holding time when cell residence
times are correlated are derived. All distributions are of phase
type, making them very general and flexible. The channel-holding
times are of importance in performance evaluation and simulation
of cellular mobile communication systems.
Index Terms—Call-holding time, cell-residence time, mobility.
I. INTRODUCTION
P
ERFORMANCE analysis of cellular mobile communica-
tion systems requires a revision of the models used for clas-
sical telephone systems. The limiting resource is radio channels.
In cellular mobile systems, a call may, by handover, change
radio channel during the call-holding time. There are several
reasons for this. The mobility of users may require a handover
to a neighboring cell. Also, propagation conditions may change
due to other subscribers or the environment and require han-
dover to another channel (either in the same cell or a neighboring
cell). The mobile system has high processing capabilities and it
is possible to implement advanced strategies for resource allo-
cation, e.g., in micro/macro cell systems or by exploiting the
overlap area between cells, allowing a user to choose among
different cells with acceptable transmission quality.
The channels of a cell are thus loaded by a mix of calls: new
calls initiated inside the cell and handover calls from neigh-
boring cells. Therefore, we need to distinguish between
cell-res-
idence time, the time a user is connected to the system (con-
trol or traffic channel) independent of whether there is a call
or not; channel-holding time of a specific traffic channel; and
call-holding time, which is all channel-holding times of a ses-
sion added together.
Many of the classic teletraffic models are insensitive to the
holding-time distribution, only the mean value being relevant
for the blocking probability. This is the case for all state-de-
pendent Poisson arrival processes [9]. We may assume that new
Manuscript received January 22, 2002; revised August 21, 2002 and March
3, 2003.
T. K. Christensen, deceased, was with Informatics and Mathematical Mod-
eling, Technical University of Denmark, Lyngby 2800, Denmark.
B. F. Nielsen is with Informatics and Mathematical Modeling, Technical Uni-
versity of Denmark, Lyngby 2800, Denmark (e-mail: bfn@imm.dtu.dk).
V. B. Iversen is with the Center for Communications, Optics, and Materials,
Technical University of Denmark, Lyngby 2800, Denmark (e-mail: vbi@com.
dtu.dk).
Digital Object Identifier 10.1109/TVT.2004.825803
calls arrive according to a Poisson model, but the arrival process
of handover calls are in general non-Poisson [13]. Thus, the
channel-holding time distribution becomes important.
Also, for simulation studies, it is important to know the
channel-holding time distribution to eliminate border effects
when simulating systems with many cells.
Many papers have studied the channel-holding time distribu-
tion. Some consider the geometry of the cells and derive the
channel-holding time distribution by assuming uniformly dis-
tributed users, directions, and speed. However, in the real world,
cells are irregular in size and shape and the users are distributed
according to buildings, roads, etc. Furthermore, a handover does
not always happen because the mobile station moves out of
range of the base station. Handovers to a new cell may be be-
cause of poor reception quality, which depends on things such as
possible obstacles and other interfering signals. Traffic conges-
tion can also result in a handover. It may thus be more appro-
priate to use probabilistic models, where distributions are ob-
tained by field observations.
Early papers (e.g., [7]) concluded that, for most practical
cases, both the channel-holding and the cell-residence times
were well approximated by negative exponential distributions.
More recent measurements [2], [3], however, show that this is
not fulfilled today, where cells are smaller and the handover
rate is higher. For Danish GSM1800 operators, there is, on
average, one handover per call. The increase in data traffic with
the introduction of new technologies, such as GPRS, Edge,
and UMTS, may also result in more general cell-residence
time distributions, since data traffic often is more variable
than predicted by the exponential distribution. Additionally,
a mixture of ordinary voice and data calls may not be well
approximated by the negative exponential distribution either.
In this paper, we derive the channel-holding time for phase-
type distributed call-holding and cell-residence times. This
provides a model that is non-Poissonian but still analytically
tractable and which easily can handle mixtures of different
kinds of traffic. To model correlated cell-residence times, a
Markovian arrival process (MAP) is used. This could be used to
model different speeds of customers. The channel-holding time
distribution has also been considered by Orlik and Rappaport
[12] for sum of hyperexponential (SOHYP) distributions and
by Fang and Chlamtac [5] for hyper-Erlang distributions, all of
which are included in the phase-type distribution. We use the
terminology shown in Fig. 1.
Forced handover due to, e.g., interference from channels in
the same cell can be included by keeping the same cell-resi-
dence time, but decomposing the channel-holding time into a
0018-9545/04$20.00 © 2004 IEEE
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726 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 3, MAY 2004
Fig. 1. Terminology applied in this paper. The call-holding time is the unencumbered call (session) holding time. The cell-residence time is the time interval a
user spends within a cell (independent of whether there is a call or not). The channel-holding time is the time interval that a radio channel in a cell is o
ccupied by
a call.
stochastic number of subchannel-holding times. The simplest
model will assume forced handover according to a Poisson
process, but more complex models are feasible at the cost of
increased complexity.
The physical cell structure such as sectorized cells has no in-
fluence on the models, which are based on cell-residence times.
Forced handover to overlapping neighboring cells due to inter-
ference or high traffic load can be included by modifying the
cell-residence time distribution.
This paper assumes that call-holding times are independent,
which is only valid for a congestion-free system. It is not pos-
sible to include load dependent handover probabilities or to cal-
culate congestion probabilities. The aggregated arrival process
becomes very complex and can only be modeled by making fur-
ther assumptions [13].
Phase-type models are very tractable for performance
evaluation. Building on Markov chain theory systems can be
analyzed by standard techniques in teletraffic and queueing
theory. Recent examples in mobile communications are Litjens
and Boucherie [11] and Christensen [4].
This paper significantly generalizes and enhances previous
work [5], [6], [12], [15] on channel-holding times. It contains
a very short introduction to phase-type distributions in Sec-
tion II-A and then goes on to describe the derivation of the
transition matrix between handovers in Section II-B, which
is essential to the following derivations. Using this result, the
distribution of the number of cells visited and channel-holding
times are derived in Sections II-C and D, respectively. This
is extended with conditional channel-holding times in Sec-
tion II-E. Using the framework established in the first part
of the paper, a final model incorporating correlation between
cell-residence times is derived in Section II-F. Finally, a case
study is presented in Section III, giving an example of the
application of the model, and concluding remarks are given in
Section IV.
II. M
ODEL OF CHANNEL-HOLDING TIMES
The phase-type model of channel-holding times is con-
structed in a stepwise fashion. Considering the evolution of the
call-holding time from handover to handover leads to an explicit
expression for the distribution of the channel-holding time.
Based on this, channel-holding time distributions conditioned
on termination of channel occupancy due to call termination
and handover are found. Finally, the model is extended to
include correlated cell-residence times by assuming that
handovers occur according to a MAP.
A. Phase-Type Distribution
The phase-type distribution is defined as the time until ab-
sorption in a Markov chain with
transient states and one ab-
sorbing state. It is parameterized by
, where
is the prob-
ability distribution of the phases at time zero,
is the proba-
bility of immediate absorption, and
is the transient part of the
generator matrix. The generator matrix for the entire Markov
chain is
The vector contains the intensities of absorption from the
transient states. The probability density function (pdf) is
(1)
In this paper,
is a column vector of a suitable dimension
with all elements equal to one. An introduction to phase-type
distributions is given in [10].
B. Derivation of the Transition Matrix Between Handovers
Let the stochastic variables
, , and , respectively, de-
scribe the remaining cell-residence time in the first cell from
the time a call is initiated, the cell-residence time after the
th
handover, and the remaining call-holding time in the last cell.
Let the stochastic variable
denote the number of handovers.
Then, the call-holding time is
,
for
, and otherwise. For the sake of simplicity,
intracell handovers are not considered; however, they can easily
be included in the model.
The distribution
is assumed to be of continuous phase-type
with states. The pdfs for , , and are
denoted
, , and , respectively.
Let
be the time instant of the th cell-residence time and let
be the phase of the remaining call-holding time distribution
at time
. Define such that is the
probability that the Markov chain at time
is in phase or,
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CHRISTENSEN et al.: PHASE-TYPE MODELS OF CHANNEL-HOLDING TIMES IN CELLULAR COMMUNICATION SYSTEMS 727
in other words, the probability of being in phase at the th
handover. The conditional probabilities are
implying that the unconditional probabilities are given by the
compound distribution
(2)
(3)
The elements
of are the joint probabilities of having
experienced
handovers and being in phase at that handover.
Thus, the elements of the vectors
will not sum to one.
We use uniformization [10] to rewrite the expression for
.
Let
. is then the absolute value
of the smallest element in the diagonal of
, since all diagonal
elements are negative. Define the uniformization matrix
; then, . is now the transition matrix of
a discrete Markov chain and
is the intensity of the exponential
distribution of time spent in each phase. The following property
is used:
(4)
Using (4) in the expression for
, we get
(5)
In a similar way, we get
(6)
with
(7)
Using (5) and the recursive formula (6), we get
(8)
and can be interpreted as transition matrices describing
the transitions between phases from one handover to the next.
The element
is the probability of being in phase at a hand-
over, conditioned on being in phase
at the previous handover.
A transition to the absorbing state means that the call termi-
nates in the cell. With the additional assumption that
and
are phase-type distributed with representations with
states,and with states, and are given by
(9)
(10)
since the exponentially weighted expectations are given by
Alternatively, let
. Then, is the absolute value of the smallest element
in the diagonal of
and , since all diagonal elements are
negative. Define the uniformization matrices
and
. Then, and .
The following property is used:
Then
(11)
and
(12)
The advantage of calculating
and in this manner is that
matrix inversion is no longer needed.
C. Distribution of the Number of Cells Visited
Let the stochastic variable
denote the number of hand-
overs. The probability of no handover is equivalent to the
probability of absorption in the underlying Markov chain of
the call-holding time before time instant
or, in other words,
the probability of absorption before the first handover
(13)
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728 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 3, MAY 2004
The probability of at least handovers, which is the same as
the probability that the call survives the
th cell, is given by
(14)
Obviously,
. The distribution of the number of
handovers is then a discrete phase-type distribution
as the vector describes the probabilities of the call sur-
viving the first handover in the different phases and the vector
describes the probabilities of the call surviving
handovers in the different phases of the underlying Markov
chain for the call-holding time. The scalar product with
then
gives the sum of the probabilities for all the phases.
The probability of exactly
handovers or visiting cells
for
is given by
(15)
Here,
is the vector of absorption probabilities
for the distribution of the number of handovers. The average
number of handovers is given by
(16)
Then the average number of cells visited is the average number
of handovers plus one:
(17)
A more direct way to derive this result would be to use the
mean of the discrete phase-type distribution. Since the proba-
bility that the call terminates in any given cell is
,
the fraction of ongoing channel occupancies in the
th cell is
given by
(18)
The fraction of channel occupancies terminating in the
th cell
is
(19)
The fraction of channel occupancies in the first cell is
(20)
The fractions of channel occupancies after the first
cell by handover from the respective phases are given by
the vector
(21)
D. Channel-Holding Time in a Cell
Define
as the channel occupancy or channel-holding time
in a cell. A channel occupancy is caused by a new call with
probability
, by a handover with probability
, or from the different phases with prob-
abilities
and , respectively.
Channel occupancy terminates either because the call has ter-
minated or because it has been handed over to a new cell. This
implies that the channel-holding time in a cell is the minimum
of the cell-residence time and the remaining call time. The min-
imum of two phase-type distributions is phase-type distributed
[10]. Thus,
, where using (21)
(22)
(23)
E. Conditional Channel-Holding Times
Absorption due to a handover occurs with the intensities
in the first cell and in any other cell and
absorption due to a terminated call occurs with the intensities
. The channel-holding time is now interpreted as the
time to absorption in a Markov chain with two absorbing states:
handover and terminated call. Thus, the channel-holding time
conditioned on a handover and a terminated call, respectively,
can be derived. Define
to be the channel-holding time
conditioned on absorption due to a handover and
as the
channel-holding time conditioned on absorption due to a
terminated call.
Then,
and [1]. de-
scribes the transient states from which the absorption occurred,
given the absorption was due to a handover, and
, from which
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