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Congestion Probabilities in CDMA-Based Networks Supporting Batched Poisson Input Traffic

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Abstract

We propose a new multirate teletraffic loss model for the calculation of time and call congestion probabilities in CDMA-based networks that accommodate calls of different service-classes. The call arrival process follows a batched Poisson process, which is more “peaked” and “bursty” than the ordinary Poisson process. The call-admission-control policy is based on the partial batch blocking discipline. This policy accepts a part of the batch (one or more calls) and discards the rest, if the available resources are not enough to accept the whole batch. The proposed model takes into account multiple access interference, both the notion of local (soft) and hard blocking, the user’s activity, as well as interference cancellation. Although the analysis of the model does not lead to a product form solution of the steady state probabilities, we show that the call-level performance metrics, time and call congestion probabilities can be efficiently calculated based on approximate but recursive formulas. The accuracy of the proposed formulas are verified through simulation and found to be quite satisfactory. Comparison of the proposed model with that of Poisson input shows the necessity of the new model. We also show the consistency of the new model over changes of its parameters.

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Notes

  1. The peakedness factor \(z\) is the ratio of the variance over the mean of the number of arrivals; if \(z=1\), the arrival process is Poisson; if \(z<1\), the arrival process is quasi-random; if \(z>1\), the process is more peaked and bursty than Poisson (e.g. overflow traffic).

  2. Hard blocking occurs when the bandwidth requirement of a new call is higher than the available resources of the system. This type of blocking appears in wired networks.

  3. If we assume the existence of perfect power control and the same \(\left( {\frac{{{E_b}}}{{{N_0}}}} \right) \), data rate \(R\), activity factor \(v\) and consequently processing gain \(G\) and total received power \(p\) for all service-classes then Eq. (6) takes the form \({P_{own}}=\frac{{N\left( {{P_{other}} + {P_{noise}}} \right) }}{{(1-\beta )- N(1 - \beta ) + \frac{G}{{\left( {\frac{{{E_b}}}{{{N_0}}}}\right) }}}}\) where \(N\) is the total number of users in the reference cell.

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Author information

Authors and Affiliations

  1. Department of Informatics and Telecommunications, University of Peloponnese, 221 00 , Tripolis, Greece

    I. D. Moscholios

  2. BT Technology, Service and Operations, Ipswich, IP5 3RE, UK

    G. A. Kallos

  3. Department of Electronic Engineering, University of Surrey, Guildford, GU2 7XH, UK

    V. G. Vassilakis

  4. WCL, Department of Electrical and Computer Engineering, University of Patras, 265 04 , Patras, Greece

    M. D. Logothetis

Authors
  1. I. D. Moscholios
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  2. G. A. Kallos
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  3. V. G. Vassilakis
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  4. M. D. Logothetis
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Corresponding author

Correspondence to I. D. Moscholios.

Appendix

Appendix

Proof of Eq. (10).

Substituting Eq. (6) in Eq. (9) we have:

$$\begin{aligned}&\frac{{\frac{{{N_k}\left( {{P_{other}} + {P_{noise}}} \right) }}{{(1 - \beta ) - {N_k}(1 - \beta )+\frac{{{G_k}}}{{{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}}}}}+ {P_{\textit{other}}} + {P_{noise}}}}{{{P_{noise}}}} = \frac{1}{{1 - {\eta _{UL}}}} \Rightarrow \\&\frac{{\frac{{{N_k}{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}\left( {{P_{other}} + {P_{noise}}} \right) }}{{{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}\left[ {(1 - \beta ) - {N_k}(1 - \beta )} \right] + {G_k}}} + {P_{other}} + {P_{noise}}}}{{{P_{noise}}}} = \frac{1}{{1 - {\eta _{UL}}}} \Rightarrow \\&\frac{{\left( {{P_{other}} + {P_{noise}}} \right) \left[ {{N_k}{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k} + {{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}\left[ {(1 - \beta ) - {N_k}(1 - \beta )} \right] + {G_k}} \right] }}{{{P_{noise}}\left[ {{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}\left[ {(1 - \beta ) - {N_k}(1 - \beta )} \right] + {G_k}} \right] }} = \frac{1}{{1 - {\eta _{UL}}}}\mathop \Rightarrow \limits ^{\delta = \frac{{{P_{other}}}}{{{P_{noise}}}}} \\&\left( {\delta + 1} \right) \left( {1 - {\eta _{UL}}} \right) \left[ {{N_k}{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k} + {{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}\left[ {(1 - \beta ) - {N_k}(1 - \beta )} \right] + {G_k}} \right] \\&\quad = \left[ {{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}\left[ {(1 - \beta ) - {N_k}(1 -\beta )} \right] + {G_k}} \right] \Rightarrow \\&{N_k} = \frac{{{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}(1 - \beta ) + {G_k} - \left( {\delta + 1} \right) \left( {1 - {\eta _{ UL}}} \right) \left[ {{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}(1 - \beta ) + {G_k}} \right] }}{{{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}\left( {\delta + 1} \right) \left( {1 - {\eta _{UL}}} \right) \left[ {1 - (1 - \beta )} \right] + {{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}(1 - \beta )}}\Rightarrow \end{aligned}$$
$$\begin{aligned}&{N_k} = \frac{{\left[ {{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}(1 - \beta ) + {G_k}} \right] }}{{{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}}}\frac{{\left[ {1 - \left( {\delta + 1} \right) \left( {1 - {\eta _{UL}}} \right) } \right] }}{{\left[ {\beta \left( {\delta + 1} \right) \left( {1 - {\eta _{UL}}} \right) + (1 - \beta )} \right] }} \Rightarrow \\&{N_k} = \frac{{\left[ {{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}(1 - \beta ) + {G_k}} \right] }}{{{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}}}\frac{{\left[ {{\eta _{UL}}\left( {\delta + 1} \right) - \delta } \right] }}{{\left[ {1 + \beta \left( {\delta - {\eta _{UL}}\delta - {\eta _{UL}}} \right) } \right] }} \end{aligned}$$

or

$$\begin{aligned} {N_k} = \left[ {(1 - \beta ) + \frac{{{G_k}}}{{{{\left( {\frac{{{E_b}}}{{{N_0}}}} \right) }_k}}}} \right] \frac{{\left[ {{\eta _{UL}}\left( {\delta + 1} \right) - \delta } \right] }}{{\left[ {1 - \beta \left( {{\eta _{UL}}(\delta + 1} \right) - \delta )} \right] }}\hbox { which is eq. (10).} \end{aligned}$$

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Moscholios, I.D., Kallos, G.A., Vassilakis, V.G. et al. Congestion Probabilities in CDMA-Based Networks Supporting Batched Poisson Input Traffic. Wireless Pers Commun 79, 1163–1186 (2014). https://doi.org/10.1007/s11277-014-1923-8

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