Completed analysis of cellular networks with PH-renewal arrival call

doi:10.1016/j.comcom.2008.12.016

Computer Communications

Volume 32, Issue 6, 27 April 2009, Pages 1039-1049

https://doi.org/10.1016/j.comcom.2008.12.016 Get rights and content

Abstract

In this paper, we apply the PH-renewal process to model new call and handoff processes, and apply the matrix-analytic approach to explore the performance measures of the drop and block probabilities. We examine the bursty nature of handoff call drops by means of conditional statistics with respect to alternating block and non-block periods. Five related performance measures are derived from conditional statistics, including the long-term new call block and handoff call drop probabilities, and the three short-term measures of average length of a block period and a non-block period, as well as the conditional handoff call drop probability during a block period. These performance measures greatly assist the priority reservation handoff mechanism in determining a proper threshold guard channel in the cell. Furthermore, we derive the handoff call drop probability from the short-term performance measures of average length of a block period and a non-block period, as well as the conditional handoff call drop probability during a block period. The results presented in this paper can provide guidelines for designing adaptive algorithms to adjust the threshold in the guard channel reservation handoff scheme.

Introduction

Handoff basically involves change of radio resources from one cell to an adjacent cell. It is well known that if an new call is blocked, it is not as disastrous as a handoff call being dropped. Therefore, it is important to provide a higher priority to handoff calls so that ongoing calls can be maintained [5], [16]. One way of assigning priority to handoff requests is by assigning guard channels to be used exclusively for handoff calls from among the available channels in a cell. This guard channel reservation handoff scheme has a tunable threshold for guard channel configuration. With a selected threshold, a block period is defined the interval of time during which the channel occupancy in a cell is at or above the threshold value, and a non-block period is the complementary interval of time. The available channels at or below the threshold is shared by new calls and handoff calls. Arriving new calls are blocked by the control scheme during a block period. Handoff calls are dropped only when the channel is full.

Choosing an appropriate threshold for the guard channel for handoff calls is the most significant design issue for wireless mobile networks. If a relatively low threshold is chosen, new calls will be excessively blocked, causing a low utilization of the channel capacity in a cell. On the other hand, if a fairly high threshold is chosen, handoff calls will be dropped more than expected because of the occupancy of the channel by new calls. This phenomena prevents the system from being able to meet the required drop probability for handoff calls. Hence, the threshold setting is a trade-off between system utilization and guaranteed drop probability for handoff calls. Performance analysis of a threshold policy is therefore necessary and desirable in order to assist the system in choosing a proper threshold.

The guard channel reservation handoff scheme has increasingly been receiving attention in cellular network design due to its simplicity in the implementation. Several performance evaluations have been conducted by examining the new call block and handoff call drop behavior of a cell with a guard channel reservation handoff scheme. Performance analysis of wireless cellular systems with Poisson handoff arrivals that specifically adopt the guard channel scheme has been carried out by a number of authors [8], [9]. In [19], [21], they made a successful drop analysis of a preemptive and priority reservation handoff scheme by considering both real-time and non-real-time service originating calls, and real-time and non-real-time handoff service request calls as Poisson arrival processes. All of these papers considered only the Poisson handoff call arrival process case.

So far, much research has been focused on exploring handoff call arrival processes. Orlik and Rappaport [12] addressed the issue of the handoff arrival process by considering a neighborhood of cells where all but one, the cell under study, is assumed to generate handoff arrivals according to either a Poisson or a two-state Markov-Modulated Poisson Process. Rajaratnam and Takawira [14] empirically showed that handoff traffic is a smooth process under negative exponential channel holding times. They characterize handoff traffic as a general traffic process and represent it using the first two moments of its offered traffic. Zeng and Chlamtac [20] have shown that the cell residence time distribution influences the handoff traffic statistics. They used a Gamma distribution for the cell residence times and showed that, for a large cell residence time variance in a non-blocking environment, handoff traffic cannot be characterized by a Poisson process.

From the above we can conclude that there is significant evidence that the handoff traffic cannot always be modeled as a Poisson process. The distribution type of the handoff inter-arrivals is still an open issue. Consequently, there is a need to develop performance models that allow for general distributions in handoff inter-arrivals. So far, several performance models with general distributions for handoff inter-arrivals have been conducted. Alfa and Li [1] derived a performance analysis method based on the Markovian arrival process (MAP) for arriving calls, and under the conditions that both the cell’s residence time and the requested call holding time possess the general phase type (PH) distribution. Therefore, Li and Alfa [7] proposed the queueing model with MAP arrival calls to investigate the related performance measures of the priority reservation handoff scheme. Dharmaraja et al. [3] successfully analyzed new call block and handoff call drop probabilities of a queue with a priority channel allocation scheme by considering new calls as Poisson processes and handoff call as renewal processes. Rajaratnam and Takawira [13] derived a performance analysis method based on Poisson new call arrivals, generalized handoff call arrivals, and using channel holding-time distributions that are more suitable and flexible than the simple negative exponential distribution.

The strategy and mathematical model used here to examine the performance measures of a guard channel reservation handoff scheme are different from those in the literature in one or more respects. In this paper, we use a PH distribution to model handoff call arrival processes due to the following conditions: (1) it is simple but good enough to fit field data and (2) the resulting queueing system model is tractable. The PH distribution [2], [6], [10] used in queueing systems in the past is known to provide a good approximation for a general distribution. In particular, the exponential distribution, the Erlang distribution, the hyper-exponential distribution and the Cox distribution, are all special cases of the PH distribution models. Horvath and Telek [4] developed a phase type fitting tool to determine the parameters of a PH distribution. Based on the above investigation, we use a PH-renewal process instead of general distributions to model handoff call arrival process so that the performance measures can be solved exactly using matrix-analytic techniques.

In addition to the evaluation of the new call block and handoff call drop probabilities, we examine the conditional handoff call drop during the block period. The threshold used to determine the block period splits the state space in two, allowing the use of two hypothesized Markov chains to describe the alternating renewal process. The distributions of various absorbing times in the two hypothesized Markov chains are derived to compute the average durations of the block period and the conditional handoff call drop probability during a block period. These performance measures will significantly assist the guard channel reservation handoff mechanism for determining a proper threshold. The overall analysis in this paper is based on the matrix-analytic approach [10], [11]. It is simple and efficient to compute the numerical results by any efficient mathematical tool.

The contributions of this paper are as follows. (1) We derive five related performance measures, including the long-term new call block and handoff call drop probabilities, and the three short-term measures of average length of a block period and a non-block period, as well as the conditional handoff call drop probability during a block period. To characterize the true call drop behavior of the guard channel reservation handoff, it is not adequate to examine only the long-term drop probabilities. For example, a handoff call may experience the drop of a string of consecutive handoff calls followed by bursty arrivals, even though the long-term handoff call drop probability is small. This phenomenon makes the handoff call suffer from a significant QoS degradation during that time period. Therefore, in light of the high correlation among consecutive handoff calls in wireless mobile networks, the handoff call drop behavior during a short-term interval, i.e., the conditional handoff call drop behavior, and the long-term one are necessary for characterizing the true handoff call drop behavior of a guard channel reservation handoff system. (2) We derive the handoff call drop probability from the short-term performance measures, average length of a block period and a non-block period, as well as the conditional handoff call drop probability during a block period. The results presented in this paper can be used to design the adaptive algorithm to adjust the threshold value of the guard channel in the guard channel reservation handoff system. Therefore, this system not only provides the drop probability guarantee for the handoff call, but also provides near-optimum utilization of the channel capacity for the new call.

This paper is organized as follows. In Section 2, the PH-renewal process as the new and handoff call model is briefly introduced. In Section 3, the new call block probability and handoff call drop probability are analyzed. Numerical results are computed and discussed in Section 4 to reveal the computational tractability of our analysis and to gain insight into the design of a guard channel reservation handoff scheme in wireless mobile networks. Some concluding remarks are given in Section 5.

Section snippets

PH-renewal process

Any distribution on $(0, \infty)$ , which can be obtained as the distribution of time until absorption in a continuous-time Markov chain which has a single absorbing state into which absorption is certain, is called Phase-type (PH-type).

Consider a Markov chain with $m + 1$ states, initial probability vector $(γ, 0)$ and transition rate matrix $Q_{PH} = [\begin{matrix} T & T^{0} \\ 0 & 0 \end{matrix}],$ where T is a nonsingular $m \times m$ matrix with $T_{ii} < 0$ , $T_{ij} > 0$ , for all $i \neq j$ , and $T^{0}$ is an $m \times 1$ vector such that $Te + T^{0} = 0$ .

The probability distribution of time until

Performance analysis

New arrival call will be modeled using a PH-renewal process with representation $(γ^{(n)}, T^{(n)})$ and handoff inter-arrival call will be modeled using a PH-renewal process with representation $(γ^{(h)}, T^{(h)})$ as described in Section 2. We assume that ongoing call (new or handoff) holding times are exponentially distributed with parameter $μ_{d}$ . The time spent in a given cell, before handing off, is called the cell dwell time. We assume this time is also exponentially distributed with parameter $μ_{h}$ . Note that

Numerical results and discussion

In this section, we will investigate the numerical results under PH-renewal new call and handoff call inter-arrival times. In our experiments, the numerical values of the PH-renewal parameters of handoff call used:

•
Generalized stage distribution: $\begin{matrix} γ^{(h)} = (0.4, 0, 0, 0.6, 0, 0), \\ T^{(h)} = η^{(h)} (\begin{matrix} - 7.5 & 7.5 & 0 & 0 & 0 & 0 \\ 0 & - 7.5 & 7.5 & 0 & 0 & 0 \\ 0 & 0 & - 7.5 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 2.1426 & 2.1426 & 0 \\ 0 & 0 & 0 & 0 & - 2.1426 & 2.1426 \\ 0 & 0 & 0 & 0 & 0 & - 2.1426 \end{matrix}), \\ T_{0}^{(h)} = η^{(h)} (\begin{matrix} 0 \\ 0 \\ 7.5 \\ 0 \\ 0 \\ 2.1426 \end{matrix}) \end{matrix}$

and the numerical values of the PH-renewal parameters of new call used:

•
Erlang distribution: $γ^{(n)} = (0.4, 0.6), T^{(n)} = η^{(n)} (\begin{matrix} - 2.5 & 0 \\ 0 & - \end{matrix})$

Conclusion

In this paper, we have presented an matrix-analytic performance model to study new call block and handoff call drop for PH-distribution handoff inter-arrival times. The proposed model may be of great interest in the design of 3G cellular mobile networks. In our exact performance model, the new call arrival process is Poisson but the handoff call arrival process is a PH-renewal process. We have investigated the handoff call drop probability during block period by means of conditional statistics

References (21)

A. Bobbio et al.
Acyclic discrete phase type distributions: properties and a parameter estimation algorithm
Performance Evaluation
(2003)
S. Dharmaraja et al.
Performance modeling of wireless networks with generally distributed handoff interarrival times
Computer Communications
(2003)
A.S. Alfa et al.
A homogeneous PCS network with Markov call arrival process and phase type cell residence time
Wireless Networks
(2003)
A. Horvath, M. Telek, PhFit: a general purpose phase type fitting tool, Tools 2002, Springer, LNCS 2324, 2002, pp....
B. Jabbari
Teletraffic aspects of evolving and next-generation wireless communication networks
IEEE Personal Communications Magazine
(1996)
M.A. Johnson et al.
An investigation of phased distribution moment-matching algorithms for use in queueing models
Queueing Systems – Theory and Applications
(1991)
W. Li et al.
A PCS network with correlated arrival process and splitted-rating channels
IEEE Journal on Selected Areas on Communications
(1999)
Y.B. Lin et al.
Queueing priority channel assignment strategies for pcs hand-off and initial access
IEEE Transactions on Vehicular Technology
(1994)
Y. Ma, J.J. Han, K.S. Trivedi, Call admission control for reducing dropped calls in code division multiple access...
M.F. Neuts
Matrix-Geometric Solutions in Stochastic Models – An Algorithmic Approach
(1981)

There are more references available in the full text version of this article.

Cited by (0)

^☆: This work was supported by the National Science Council, Taiwan, under Contract NSC92-2213-E-027-047.

View full text

Computer Communications

Completed analysis of cellular networks with PH-renewal arrival call☆

Abstract

Introduction

Section snippets

PH-renewal process

Performance analysis

Numerical results and discussion

Conclusion

Acyclic discrete phase type distributions: properties and a parameter estimation algorithm

Performance Evaluation

Performance modeling of wireless networks with generally distributed handoff interarrival times

Computer Communications

A homogeneous PCS network with Markov call arrival process and phase type cell residence time

Wireless Networks

Teletraffic aspects of evolving and next-generation wireless communication networks

IEEE Personal Communications Magazine

An investigation of phased distribution moment-matching algorithms for use in queueing models

Queueing Systems – Theory and Applications

A PCS network with correlated arrival process and splitted-rating channels

IEEE Journal on Selected Areas on Communications

Queueing priority channel assignment strategies for pcs hand-off and initial access

IEEE Transactions on Vehicular Technology

Matrix-Geometric Solutions in Stochastic Models – An Algorithmic Approach