Computer Communications 35 (2012) 1165–1171
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Computer Communications journal homepage: www.elsevier.com/locate/comcom
A new queueing model for spectrum renting in mobile cellular networks Tien Van Do a,⇑, Nam H. Do a, Ram Chakka b a b
Department of Telecommunications, Budapest University of Technology and Economics, H-1117, Magyar tudósok körútja 2, Budapest, Hungary Meerut Institute of Engineering and Technology (MIET), Meerut 250005, India
a r t i c l e
i n f o
Article history: Received 20 December 2010 Received in revised form 17 May 2011 Accepted 6 December 2011 Available online 5 April 2012 Keywords: Cellular networks Queueing model Fractional Guard Channel Spectrum renting
a b s t r a c t Spectrum renting is an operation practice that can be applied to relieve the temporary capacity shortages of a specific service area in wireless cellular networks. However, works in the literature do not take into account the specific feature of the present wireless technology. That is, the separate blocks of user channels are defined in each frequency band in the current standards for public mobile cellular networks, and each block should be controlled by a single network operator. This paper is the first attempt to model the spectrum renting policies and the call admission control in a realistic way. The comparison between a queueing model and a simulation model confirms that the proposed queueing model incorporating exponentially distributed call durations can be used to evaluate the performance of mobile cellular networks with call holding times following the lognormal distribution as well. Numerical results show that the variants of the Fractional Guard Channel Policy provide an efficient tool to guarantee the grade of service of handover calls at the expense of increased blocking probability of fresh calls. Furthermore, only spectrum renting can be used to decrease the blocking probability of fresh calls without compromising the grade of service of handover calls. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction At present, the exclusive access right to certain radio frequency bands is licensed to mobile network operators by governments. The license of frequency bands is guaranteed based on the result of spectrum auctions. However, it is recognized that the exclusive access may lead to the inefficient use of spectrum [1,2]. Network operators can utilize spectrum renting to increase the efficiency of the spectrum usage [1–6] and to relieve the temporary capacity shortage of a particular cell in a mobile cellular network. For example when the number of calls increases in a specific area, a network operator could decide to rent a frequency band from another operator to keep or enhance the grade of service of calls. In the literature, there are several works [4,5] which attempted to model the aspects of spectrum renting. However, the assumption [4,5] that user channels (one user channel is used to serve one subscriber) can be rented in one unit is not realistic because of the current specifications for mobile cellular networks. Since the separate blocks of user channels are defined in each frequency band, and each block should be controlled by a single network operator, it is impossible to rent one channel due to the control and security reason.
In this paper, we simultaneously integrate the aspects of renting policies involving a block of user channels and call admission control in wireless cellular networks in one analytical model. The main contributions of this paper are as follows. We take into account that (i) a rented frequency band accommodates the number of user channels. We propose (ii) the operation rule based on a hysteresis control with two thresholds for the network operator to rent or give back frequency bands based on the offered traffic. Furthermore, we incorporate (iii) several variants of the Fractional Guard Channel Policy [7,8] in our model. We show that (iv) the proposed queueing model can accurately evaluate mobile networks with call durations following the lognormal distribution and only spectrum renting can be used to decrease the blocking probability of fresh calls without compromising the grade of service of handover calls. The rest of this paper is organized as follows. We explain the technical background behind our modeling approach in Section 2. Modeling assumptions are presented in Section 3. A queueing model is described in Section 4. A numerical study is given in Section 5. Finally, Section 6 concludes the paper. 2. Technical background 2.1. Blocks of user channels
⇑ Corresponding author. Tel.: +36 1 4632070. E-mail addresses:
[email protected] (T.V. Do),
[email protected] (N.H. Do),
[email protected] (R. Chakka). 0140-3664/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comcom.2011.12.012
To satisfy the demand of voice calls from a large number of subscribers the whole coverage area of a specific mobile network operator is divided into a number of regular shaped cells, each
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assigned a number of frequency bands and controlled by a base station [9,10]. Note that radio frequency bands are taken from the spectrum licensed to the mobile network operator. At present there are two main alternatives for multiple access technology in the radio interface between mobile phones and base stations to support the simultaneous voice communication of subscribers. The first alternative is the Frequency and Time Division Multiple Access (FDMA/TDMA) technique that is specified in the most popular standard for Global System for Mobile Communications (GSM) networks [9]. The whole spectrum owned by the operators is divided into a number of single carrier channels of 200 kHz. Then, each carrier channel is subdivided into eight time slots, which accommodate eight full-rate or sixteen halfrate speech channels per radio frequency. Furthermore, a control channel such as Slow Associated Control Channel (SACC) is also defined in each carrier channel. The second alternative for the radio interface is the code division multiple access (CDMA) based on spread-spectrum technology and a code division scheme [11]. Normally, a number of spectrum bands of a certain size (e.g., 1.25 MHz in CDMA2000, 5 MHz in W-CDMA) are licensed to a specific mobile network operator. A spread spectrum multiple access technique assigns a different code, that is orthogonal to a code used by another user equipment, to each user to modulate their signal. The different codes will help to separate information of users in the same spectrum band. As a consequence, the allocation of codes must be carefully planned and controlled by an operator whom mobile phones request a channel from. At present the blocks of channels are defined in each frequency band, and each block should be controlled by a single network operator. Therefore, each rented frequency band comes with a number of channels for a mobile network operator.
2.2. An environment for spectrum renting Peha [1] outlined several examples of spectrum-sharing model with a spectrum policy reform. We assume that network operators cooperate with each other in a dynamic market to perform spectrum renting based on some economic principles and bid mechanism [2,3,12–15].
3. Modeling assumptions We consider a particular cell in a cellular mobile system with infinite user population (see Fig. 1). Without renting frequencies there are n channels to serve incoming calls. Let I(t) denote the number of occupied channels in a specific cell and JðtÞ (0 6 JðtÞ 6 L) be the number of rented frequencies, where L is the maximum number of rented frequency bands. The number of available channels is N j ¼ n þ jnR for JðtÞ ¼ j if nR is the number of channels in one rented frequency. The interarrival times of new calls and handover calls are exponentially distributed with rate kF and kH , respectively. Let k ¼ kF þ kH . We assume that call durations (of new calls and handover calls) in the cell are exponentially distributed with mean 1=l to obtain a mathematically tractable model. This assumption is normally applied in various queueing models for mobile networks [4,5,17– 20]. We will show in Section 5 that our model with the exponential distribution of channel holding times can be used to evaluate the performance of cellular networks where the holding times of calls follow the lognormal distribution. Note that Jedrzycki and Leung [21] first reported that the holding times of calls in cellular networks have the lognormal distribution. In what follows we explain the interaction between the arrival of calls, the admission policy and the spectrum renting policy illustrated in Fig. 1. 3.1. Call admission policy The Fractional Guard Channel (FGC) Policy has a call admission rule that accepts a new call with probability bi;j and handover calls with probability 1 for IðtÞ ¼ i and JðtÞ ¼ j. Following [22], several variants of the FGC policy can be established with the appropriate setting of parameters: The Limited average Fractional Guard Channel Policy (LFGC) is defined with the following values of the parameters: bi;j ¼ 1 (0 6 i 6 N j bgc 2), bNj bgc1;j ¼ 1 g þ bgc and bi;j ¼ 0 (N j bgc 6 i < N j ), where g denotes the real number of reserved channels.
Request to increase J(t) when Nj-I(t)) < ki;j Aj ði; kÞ ¼ il > : 0
Blocking probability
0.1
0.01
if k ¼ i þ 1; i ¼ 0; . . . ; Nj 1; if k ¼ i 1; 0 < i 6 Nj otherwise;
where ki;j ¼ ðkF bi;j þ kH Þ. Furthermore, Aj ðNj1 t 2 þ 1; Nj1 t 2 Þ ¼ 0 for j > 0. The ðN j þ 1Þ ðN jþ1 þ 1Þ matrix Bj (j ¼ 0; . . . ; L 1) includes the rate of transitions from state ði; jÞ to state ðk; j þ 1Þ. We obtain
0.001
(
PF, L=0 PF, L=3 PH, L=0 PH, L=3
0.0001
Bj ði; kÞ ¼
1e-005 0
5
10
λ/λH
15
20
a#ð1 #Þ1=#1 maxði Nj þ t1 ; 1Þ if k ¼ i; i P Nj t1 0
otherwise;
The elements of the ðN j þ 1Þ ðN j1 þ 1Þ matrix C j (j ¼ 1; . . . ; L) are the rates of transitions from state ði; jÞ to state ðk; j 1Þ. The nonzero elements of matrix C j (0 < j 6 L) are C j ðN j1 t 2 þ 1; N j1 t2 Þ ¼ ðN j1 t 2 þ 1Þl and C j ði; minði; N j1 ÞÞ ¼ g.
25
Fig. 6. Blocking probabilities vs k=kH for n ¼ 24; q ¼ 0:85; g ¼ 1:5; nR ¼ 8; t1 ¼ 3; t 2 ¼ 8; a ¼ 0:15 1/s, # ¼ 0:9; 1=g ¼ 3 h.
As a consequence, the generator matrix of the CTMC is written as
2 (3) The request process for a spectrum (the request is immediately started when the number of free channels reaches the threshold value t1 and the number of rented frequencies is below L. If the first request was unsuccessful, the retrial request for a frequency happens with rate a as long as the number of free channels remain below t 1 ); and (4) The expire of the lease time of rented frequencies. Therefore, the following types of transitions are possible between the states of the CTMC: ði; jÞ ) ði þ 1; jÞ for 0 6 i < N j and 0 6 j 6 L: these transitions are due to the acceptance of calls based on the call admission policy, ði; jÞ ) ði; j þ 1Þ for i P N j t1 : these transitions are due to the successful request for a new frequency, ði; jÞ ) ði 1; jÞ for i > 0 (if j ¼ 0) or 0 < i – N j t2 þ 1 (if j > 0): these transitions happen when a call departs from the system, ði; jÞ ) ði 1; j 1Þ for i ¼ N j1 t2 þ 1 and 0 < j 6 L: these transitions are initiated by the departure of a call and the decision to release the rented frequency, ði; jÞ ) ðminði; N j1 Þ; j 1Þ for 0 6 i 6 N j and 0 < j 6 L: these transitions happen when the lease time is expired and the rented frequency is taken back. The transitions between the states are ordered as follows: The ðN j þ 1Þ ðN j þ 1Þ matrix Aj (j ¼ 0; . . . ; L) contains the transition rate Aj ði; kÞ from state ði; jÞ to state ðk; jÞ. We can write
B0
C1
0
0
...
A1
B1
0
...
0 .. . 0
C2 .. . 0
A2 .. . ...
B2 .. .
...
0
0
...
ð1Þ
ð1Þ
C L1
... ð1Þ AL1 CL
...
0.001 0.0001 1e-005
3
7 ... 7 7 7 ... 7 7 7; 7 ... 7 7 BL1 7 5 ð1Þ AL
ð1Þ
Aj
8 A0 B0 > < A0 D D Aj Bj ¼ Aj D D D C j > : AL DAL DC L
if j ¼ 0; if 0 < j < L; if j ¼ L:
Z
Note that D ðZ ¼ Aj ; Bj ; C j Þ is a diagonal matrix whose diagonal element is the sum of all elements in the corresponding row of Z. The steady state probabilities are denoted by pi;j ¼ limt!1 Pr ðIðtÞ ¼ i; JðtÞ ¼ jÞ and pj ¼ ðp0;j ; . . . ; pNj ;j Þ. The balance equations can be expressed as follows:
p0 Að1Þ 0 þ p1 C 1 ¼ 0; pk1 Bk1 þ pk A1ðkÞ þ pkþ1 C kþ1 ¼ 0; 1 6 k < L; pL1 BL1 þ pL Að1Þ L ¼ 0:
ð1Þ ð2Þ ð3Þ
To compute the steady state probabilities, we shall proceed as follows. Let us define matrices Rk ’s such as pk ¼ pk1 Rk for k ¼ 1; 2; . . . ; L. Then, based on Eqs. (2) and (3), matrices Rk ’s can be recursively computed using
0.1
PF PH
0.01
1e-006
ð1Þ
A0
where
Blocking probability
Blocking probability
0.1
6 6 6 6 6 6 QX ¼ 6 6 6 6 6 4
PF PH
0.01 0.001 0.0001 1e-005
2
3
4
5
t1
6
7
8
1e-006
8
10
12
14
16
18
t2
Fig. 7. Blocking probabilities vs t 1 and t2 for n ¼ 24; q ¼ 0:85; p ¼ 0:1; g ¼ 1:5; L ¼ 3; nR ¼ 8; a ¼ 0:15 1/s, # ¼ 0:9; 1=g ¼ 3 h.
20
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T.V. Do et al. / Computer Communications 35 (2012) 1165–1171 ð1Þ
RL ¼ BL1 ðAL Þ1 ; ð1Þ
Rk ¼ Bk1 ½Ak þ Rkþ1 C kþ1 1 ;
ð4Þ ðk ¼ L 1; . . . ; 1Þ:
This means, pk (k ¼ 1; 2; . . . ; L) can be expressed in already computed Rk ’s as
pk ¼ p0
k Y
Rk ;
ðk ¼ 0; . . . ; LÞ;
ð5Þ
p0 and the
ð6Þ
i¼0
where R0 is the identity matrix by definition. Eq. (1) can be rewritten as
p0 ðA0ð1Þ þ R1 C 1 Þ ¼ 0:
ð7Þ
To compute p0 (and then the stationary probabilities) we utilize the P normalization equation Lk¼0 pk ek ¼ 1 and Eq. (7). The performance measures are as follows: The blocking probability of handover calls
PH ¼
L X pNj ;j ; j¼0
The blocking probability of fresh calls
PF ¼
Nj L X X
pi;j ð1 bi;j Þ:
j¼0 i¼0
5. Numerical results 5.1. Comparison with simulation In this section, numerical results obtained by our queueing model and a simulation model are compared. The channel holding times follow the log-normal distribution with the mean of 3.29 s and the standard deviation of 1.17 s in the simulation model (note that parameter values are taken from [21], which were obtained from fit2 ting with real traffic data). Therefore, 1=l ¼ e3:29þ1:17 =2 ¼ 53:22 s is chosen for the computation with our queueing model. Simulation runs were performed with the confidence level of 99.9%. The confidence interval is 0:6% of the collected data. The curves of the blocking probabilities in Figs. 2–4 for the LGFC Policy show the excellent agreement between the analytical and simulation results. Similar observations (the maximum discrepancy is less than 5% in the investigated scenarios) are obtained with wide ranges of parameters and other variants of the FGC Policy as well. The comparison shows that our queueing model can be used to evaluate the performance of networks accurately, where call holding times follow the lognormal distribution. 5.2. Impacts of spectrum renting policy As expected the variants of the Fractional Guard Channel Policy provide an efficient tool to guarantee and lower the blocking probability of handover calls at the expense of increased blocking probability of fresh calls (see Figs. 2–5). As observed from Fig. 5, the LFGC Policy is the most convenient way among the variants of the FGC Policy to balance the blocking probabilities. However, for high traffic loads we can not decrease the blocking probability of fresh calls below a certain threshold (e.g., 1% guaranteed by network operators) without increasing the probability of handover calls if no spectrum renting is performed. The curves in Figs. 2–7 clearly show that spectrum renting can decrease the blocking probability of fresh calls and handover calls as well. This means, renting spectrum really relieves the network from the temporary shortage of capacity. We observe from Fig. 7 that thresholds
t1 and t 2 have a strong impact on the reduction of the blocking probabilities as well. That is, the decrease of the blocking probabilities can be or more than one order. 6. Conclusions We have proposed an integrated analytical model for mobile cellular networks with spectrum renting and Fractional Guard Channel Policies. We have taken into account the aspects of renting policies involving a block of user channels and call admission control in wireless cellular networks in one analytical model. The comparison between the proposed queueing model and the simulation model incorporating the lognormal distribution of holding times has been performed, which confirms that our model with the assumption of the exponential distribution of channel holding times can be used to evaluate networks having the lognormal distribution of call holding times results with an acceptable accuracy. We show that the variants of the Fractional Guard Channel Policy provide an efficient tool to guarantee and enhance the grade of service of handover calls at the expense of increased blocking probability of fresh calls and the LFGC Policy is the most convenient way among the variants of the FGC Policy to balance the blocking probabilities. However, only spectrum renting can be used to improve the grade of service of fresh calls without compromising the blocking probability of handover calls. There are some research directions arising from our computational model. As an example, we can mention the consideration of the retrial phenomenon [17–20] of mobile subscribers. Furthermore, it would be an interesting issue to investigate the impact of some economic incentives (e.g., the fee structure of renting spectrum) to the spectrum renting policies. Acknowledgements This work is partially supported by the Hungarian Government through the TÁMOP-4.2.1/B-09/1/KMR-2010–0002 project at the Budapest University of Technology and Economics. References [1] J.M. Peha, Sharing spectrum through spectrum policy reform and cognitive radio, in: Proceedings of the IEEE. 97 (2009) pp. 708–719. [2] B. Jabbari, R. Pickholtz, M. Norton, Dynamic spectrum access and management dynamic spectrum management, IEEE Wireless Communications 17 (4) (2010) 6–15. [3] S. Gandhi, C. Buragohain, L. Cao, H. Zheng, S. Suri, Towards real-time dynamic spectrum auctions, Computer Networks 52 (4) (2008) 879–897. [4] S.-S. Tzeng, C.-W. Huang, Threshold based call admission control for QoS provisioning in cellular wireless networks with spectrum renting, in: T.M. Sobh, K.M. Elleithy, A. Mahmood (Eds.), Novel Algorithms and Techniques in Telecommunications and Networking, Springer, Berlin, Heidelberg, 2010, pp. 17–22. [5] S.-S. Tzeng, Call admission control policies in cellular wireless networks with spectrum renting, Computer Communications 32 (18) (2009) 1905–1913. [6] E. Noam, Taking the next step beyond spectrum auctions: open spectrum access, IEEE Communications Magazine 33 (1995) 66–73. [7] R. Ramjee, R. Nagarajan, D.F. Towsley, On optimal call admission control in cellular networks, INFOCOM (1996) 43–50. [8] R. Ramjee, D. Towsley, R. Nagarajan, On optimal call admission control in cellular networks, Wireless Networks 3 (1) (1997) 29–41. [9] Y.-B. Lin, I. Chlamtac, Wireless and Mobile Network Architectures, Wiley Computer Publishing, John Wiley & Sons, Inc., 2001. [10] E. Gelenbe, P. Kammerman, T. Lam, Performance considerations in totally mobile wireless, Performance Evaluation 36–37 (1–4) (1999) 387–399. [11] T.S. Rappaport, Wireless Communications Principles and Practice, PrenticeHall, 2002. [12] G. Kim, E. Gelenbe, Analysis of an automated auction with concurrent multiple unit acceptance capacity, in: Analytical and Stochastic Modeling Techniques and Applications, Springer Lecture Notes in Computer Science, 6148, Springer, 2010, pp. 382–396. [13] E. Gelenbe, Analysis of single and networked auctions, ACM Transactions on Internet Technology 9 (2) (2009).
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