Modeling UMTS power saving with bursty packet data traffic

Modeling UMTS Power Saving

with Bursty Packet Data Traffic

Shun-Ren Yang, Sheng-Ying Yan, and Hui-Nien Hung

Abstract—The universal mobile telecommunications system (UMTS) utilizes the discontinuous reception (DRX) mechanism to reduce the power consumption of mobile stations (MSs). DRX permits an idle MS to power off the radio receiver for a predefined sleep period and then wake up to receive the next paging message. The sleep/wake-up scheduling of each MS is determined by two DRX parameters: the inactivity timer threshold and the DRX cycle. In the literature, analytic and simulation models have been developed to study the DRX performance mainly for Poisson traffic. In this paper, we propose a novel semi-Markov process to model the UMTS DRX with bursty packet data traffic. The analytic results are validated against simulation experiments. We investigate the effects of the two DRX parameters on output measures including the power saving factor and the mean packet waiting time. Our study provides inactivity timer and DRX cycle value selection guidelines for various packet traffic patterns.

Index Terms—Bursty packet data traffic, discontinuous reception, power saving, universal mobile telecommunications system (UMTS).

Ç

1

I

NTRODUCTION

T

HE third-generation mobile cellular system universal

mobile telecommunications system (UMTS) offers high data transmission rates to support a variety of mobile applications including voice, data, and multimedia. In order to fulfill the high-bandwidth requirement of these different services, the mobile station (MS) power saving is a crucial issue for the UMTS network operation. Since the data bandwidth is significantly restricted by the battery capacity, most existing wireless mobile networks (includ-ing UMTS) employ discontinuous reception (DRX) to con-serve the power of MSs. DRX allows an idle MS to power off the radio receiver for a predefined period (called the DRX cycle) instead of continuously listening to the radio channel. Some typical DRX mechanisms are briefly described as follows:

. In Mobitex [15], all sleeping MSs are required to

synchronize with a specific hSVP6i frame and wake up immediately before the hSVP6i transmission starts. When some MSs experience high traffic loads, the network may decide to shorten the hSVP6i announcement interval to reduce the frame delay. As a result, the low-traffic MSs will consume extra unnecessary power budget.

. In Cellular Digital Packet Data (CDPD) [5], [12] and

IEEE 802.11 [8], a sleeping MS is not forced to wake up at every announcement instant. Instead, the MS may choose to skip some announcements for further reducing its power consumption. A wake-up MS has to send a receiver-ready (RR) frame to notify the network of its capability to receive the pending frames. However, such RR transmissions may collide with each other if the MSs tend to wake up at the same time. Thus, RR retransmissions may occur and extra power is unnecessarily consumed.

. UMTS DRX [2], [4] improves the aforementioned

mechanisms by allowing an MS to negotiate its DRX cycle length with the network. Therefore, the net-work is aware of the sleep/wake-up scheduling of each MS and only delivers the paging message when the MS wakes up.

In the literature, DRX mechanisms have been studied. Lin and Chuang [12] proposed simulation models to investigate the CDPD DRX mechanism. In [10], an analytic model was developed to investigate the CDPD DRX mechanism. This model does not provide a closed-form solution. Furthermore, the model is not validated against simulation experiments. In our previous work [21], we proposed a variant of the M=G=1 vacation model to explore the performance of the UMTS DRX. We derived the closed-form equations for the output measures based on the Poisson assumption. However, the Poisson distribution has been proven to be impractical when modeling bursty packet data traffic [20]. In [11] and [23], simulation and analysis were utilized to examine the UMTS DRX mechanism. The authors studied the impact of an inactivity timer on energy consumption for both real-time and non-real-time traffic. However, they do not consider the mean packet waiting time under DRX. This paper proposes a novel semi-Markov process to model the UMTS DRX for bursty packet data applications. The analytic results are validated against . S.-R. Yang is with the Department of Computer Science and Institute of

Communications Engineering, National Tsing Hua University, No. 101, Sec. 2, Kuang Fu Rd., Hsinchu, Taiwan 300.

E-mail: sryang@cs.nthu.edu.tw.

. S.-Y. Yan is with the Telecommunication Laboratories, Chunghwa Telecom Company, Ltd., 12, Lane 551, Min-Tsu Road Sec. 5 Yang-Mei, Taoyuan, Taiwan 32617. E-mail: siyan@cht.com.tw.

. H.-N. Hung is with the Institute of Statistics, National Chiao Tung University, 1001 Ta Hsueh Rd., Hsinchu, Taiwan 30050.

E-mail: hhung@stat.nctu.edu.tw.

Manuscript received 6 Oct. 2006; revised 9 Mar. 2007; accepted 3 Apr. 2007; published online 25 Apr. 2007.

For information on obtaining reprints of this article, please send e-mail to: tmc@computer.org, and reference IEEECS Log Number TMC-0267-1006. Digital Object Identifier no. 10.1109/TMC.2007.1072.

simulation experiments. Based on the proposed analytic and simulation models, the DRX performance is investi-gated by numerical examples. Specifically, we consider two performance measures:

. Power saving factor. This is the probability that the MS

receiver is turned off when exercising the UMTS DRX mechanism. This factor indicates the percen-tage of power saving in the DRX (compared with the case where DRX is not exercised).

. Mean packet waiting time. This is the expected waiting

time of a packet in the UMTS network buffer before it is transmitted to the MS.

2

UMTS DRX M

ECHANISM

As illustrated in Fig. 1, a simplified UMTS architecture consists of the core network and the UMTS terrestrial radio access network (UTRAN). The core network is responsible for switching/routing calls and data connections to the external networks, whereas the UTRAN handles all radio-related functionalities. The UTRAN consists of radio network controllers (RNCs) and Node Bs (that is, base stations) that are connected by an asynchronous transfer mode (ATM) network. An MS communicates with Node Bs through the radio interface based on the wideband CDMA (WCDMA) technology [7].

The UMTS DRX mechanism is realized through the radio resource control (RRC) finite-state machine exercised be-tween the RNC and the MS [1]. There are two modes in this finite-state machine (see Fig. 2). In the RRC Idle mode, the MS is tracked by the core network without the help of the UTRAN. When an RRC connection is established between the MS and its serving RNC, the MS enters the RRC

Connectedmode. In this mode, the MS could stay in one of

the four states:

. In the Cell_DCH state, the MS occupies a dedicated

traffic channel.

. In the Cell_FACH state, the MS is allocated a

common or shared traffic channel.

. In the Cell_PCH state, no uplink access is possible

and the MS monitors paging messages from the RNC.

. In the URA_PCH state, the MS eliminates the

location registration overhead by performing URA updates instead of cell updates.

In the Cell_DCH and Cell_FACH states, the MS receiver is always turned on to receive packets. These states correspond to the power active mode. In the RRC Idle mode and the Cell_PCH and URA_PCH states, the DRX is exercised to conserve the MS power budget. These states/ mode correspond to the power saving mode. Under DRX, the MS receiver activities could be described in terms of three periods (see Fig. 3):

. In the busy period (see Fig. 3a), the MS is in the

power active mode and the UMTS core network delivers packets to the MS through the RNC and Node B in the first-in, first-out (FIFO) order. Compared with WCDMA radio transmission, ATM is much faster and more reliable. Therefore, the ATM transmission delay is ignored in this paper and the RNC and Node B are regarded as a FIFO server. Furthermore, due to the high error rate and low bit rate nature of radio transmission, the Stop-and-Wait Hybrid Automatic Repeat request (SAW-Hybrid ARQ) flow-control algorithm [3] is executed to guarantee successful radio packet delivery: When Node B sends a packet to the MS, it waits for a positive acknowledgment (ack) from the MS before it can transmit the next packet. Hybrid ARQ was originally proposed for the High-Speed Downlink Packet Access (HSDPA) system and has also been adopted by next-generation wireless networks in-cluding the IEEE 802.16 WiMAX system. SAW-Hybrid ARQ is one of the simplest forms of ARQ, requiring very little overhead. Hybrid ARQ using this stop-and-wait mechanism offers significant improvements by reducing the overall bandwidth demanded for signaling and the MS memory. Due to its simplicity, SAW-Hybrid ARQ could also be implemented in the earlier UMTS releases without HSDPA support.

Fig. 1. A simplified UMTS network architecture.

Fig. 2. The RRC state diagram.

. In the inactivity period (see Fig. 3b), the RNC buffer is empty and the RNC inactivity timer is activated. If any packet arrives at the RNC before the RNC inactivity timer expires, then the timer is stopped. The RNC processor starts another busy period to transmit packets. In the inactivity period, the MS receiver is turned on and the MS is still in the power active mode.

. If no packet arrives within the threshold tI of the

RNC inactivity timer (see Fig. 3c), then the MS turns off its radio receiver and enters the sleep period to save power (see Fig. 3d). The MS sleep period

contains at least one DRX cycle tD. At the end of a

DRX cycle, the MS wakes up to listen to the paging channel. If the paging message indicates that some packets have arrived at the RNC during the last DRX cycle, then the MS starts to receive packets and the sleep period terminates. Otherwise, the MS returns to sleep until the end of the next DRX cycle. In the power saving mode, the RNC processor will not transmit any packets to the MS.

3

E

UROPEAN

T

ELECOMMUNICATIONS

S

TANDARDS

I

NSTITUTE

(ETSI) P

ACKET

T

RAFFIC

M

ODEL

The validity of traditional queuing analyses depends on the Poisson nature of the data traffic. However, in many real-world cases, it has been found that the predicted results from these queuing analyses differ substantially from the actual observed performance. In recent years, a number of studies have demonstrated that, for some environments, the data traffic pattern is self-similar [20] rather than Poisson. Compared with traditional Poisson traffic models, which typically focus on a very limited range of time scales and are thus short-range dependent in nature, self-similar traffic exhibits burstiness and correlations across an extremely wide range of time scales (that is, possesses long-range dependence). It has also been shown that heavy-tailed distributions such as Pareto and Weibull distributions are more appropriate when modeling data network traffic [14]. In this paper, we adopt the ETSI packet traffic model [6], where the packet size and the packet transmission time are assumed to follow the truncated Pareto distribution.

As shown in Fig. 4, we assume that the packet data traffic consists of packet service sessions. Each packet service session contains one or more packet calls depend-ing on the applications. For example, the streamdepend-ing video may comprise one single packet call for a packet session, whereas a Web surfing packet session includes a sequence of packet calls. An MS/mobile user initiates a packet call when requesting an information element (for example, the

downloading of a WWW page). If the request is permitted, then a burst of packets (for example, as a whole constituting a video clip in the WWW page) will be transmitted to the MS through the RNC and Node B. When the RNC receives the positive ack for the last packet from the MS, the current packet call transmission has completed. The time interval between the end of this packet call transmission and the beginning of the next packet call transmission is referred to as the interpacket call idle time tipc.

Having received all packets of the ongoing packet service session, the MS will then experience an even longer

intersession idle time tis. The tis period represents the time

interval between the end of the packet session and the beginning of the next packet session.

The statistical distributions of the parameters in our traffic model follow the recommendation in [6] and are summarized as follows. Note that, since we consider continuous time scale in this paper, the exponential distribution is used to replace the geometric distribution for continuous random variables:

. The intersession idle time tis is modeled as an

exponentially distributed random variable with mean 1=is.

. The number of packet calls Npc within a packet

service session is assumed to be a geometrically

distributed random variable with mean pc.

. The interpacket call idle time tipcis an exponential

random variable with mean 1=ipc.

. The number of packets Np within a packet call

follows a geometric distribution with mean p.

. The interpacket arrival time tip within a packet call

is drawn from an exponential distribution with mean 1=ip.

. The truncated (or cutoff) Pareto distribution is used

to model the packet size. Pareto distribution [9] has been found to match very well with the actual data traffic measurements [20]. A Pareto distribution has two parameters: the shape parameter  and the scale parameter l, where  describes the “heaviness” of the tail of the distribution. The probability density function is fxðxÞ ¼  l   _l x  þ1

and the expected value is

E½x ¼ 

 1

 

l:

If  is between 1 and 2, the variance for the distribution becomes infinity. We follow the

sugges-tion in [6] and define the packet size Sd with the

following formula:

Packet Size Sd¼ minðP ; mÞ;

where P is a normal Pareto distributed random variable with  ¼ 1:1 and l ¼ 81:5 bytes and m ¼

66;666 bytes is the maximum allowed packet size.

According to the above parameter values, the average packet size is calculated as 480 bytes. The Fig. 4. ETSI packet traffic model.

above , l, and m parameter settings for the packet size distribution have been validated by the ETSI technical bodies [6]. Many telecommunications vendors and operators adopted these settings to conduct the UMTS field trials. These configurations were also followed by a number of analytic and simulation studies in the literature [11], [18] to investigate the performance of UMTS networks.

. Let the packet service time tx denote the time

interval between when the packet is transmitted by the RNC processor and when the corresponding

positive ack is received by the RNC processor. The tx

distribution has mean value 1=x. In our model, we

suppose that txis proportional to the packet size Sd

and is defined as tx¼

Packet Size Sd

Transmission Bit Rate:

Six types of transmission bit rates are proposed in [6] for the WWW surfing service: 8, 32, 64, 144, 384, and 2,048 kilobits per second (Kbps).

The ETSI model for bursty packet data traffic with long-range dependence can be justified in [14]. Paxson and Floyd [14, Appendix D] have shown that the M=G=1 model with infinite-variance Pareto distribution can be used to generate self-similar traffic. In the ETSI packet traffic model, the packet arrival process is governed by the exponential and geometric distributions with memoryless property, whereas

the packet service time tx is a truncated Pareto random

variable.

4

A

N

A

NALYTIC

M

ODEL FOR

UMTS P

OWER

S

AVING

Based on the ETSI packet traffic model defined in the previous section, this section proposes an analytic model to study the UMTS power saving mechanism. The notation used in the analytic model is listed in Appendix A. Let the

two UMTS DRX parameters tI and tD be of fixed values

1=Iand 1=D, respectively. We first describe a semi-Markov

process [13], [17]. We then show how this semi-Markov process can be used to investigate the performance of the UMTS power saving mechanism (including the power saving factor and the mean packet waiting time). As illustrated in Fig. 5, this semi-Markov process consists of four states:

. State S1 includes a busy period tB and then an

interpacket call inactivity period t

I1.

. State S2 includes a busy period tB and then an

intersession inactivity period tI2.

. State S3includes a sleep period tS1, which is entered

from state S1.

. State S4includes a sleep period tS2, which is entered

from state S2.

If we view this semi-Markov process only at the times of state transitions, then we could obtain an embedded

Markov chain with state-transition probabilities pi;j(where

i, j 2 f1; 2; 3; 4g). These state-transition probabilities are derived as follows:

. p1;1and p1;2. In state S1, the RNC inactivity timer is

activated at the end of the busy period t

B, and then

the MS enters the interpacket call inactivity period t

I1.

Note that, based on the burstiness nature, our analytic model assumes that a busy period corresponds to the transmission duration of a packet call. That is, the

interpacket arrival time tip within a packet call is

significantly shorter than the packet service time tx,

and a busy period will not terminate until the end of the corresponding packet call delivery. When the first packet of the next packet call arrives at the RNC before the inactivity timer expires (with probability q1¼ Pr½tipc< tI ¼ 1  eipc=I), the timer is stopped,

and another busy period begins. In this case, if the new arriving packet call is the last one of the ongoing

session (with probability q2¼ 1=pc, that is, the

memoryless property of geometric distributions),

then the MS enters state S2. Otherwise (with

prob-ability 1  q2), the ongoing session continues, and the

MS enters state S1again. From the above discussion,

we have p1;1¼ q1ð1  q2Þ ¼ 1 e ipc I   1 1 pc   and p1;2¼ q1q2¼ 1 e ipc I   ₁ pc :

. p2;1 and p2;2. The derivations of p2;1 and p2;2 are

exactly the same as that of p1;1 and p1;2 except that

the interpacket call idle period tipcis replaced by the

intersession idle period tis and q1 is replaced by

q3¼ Pr½tis< tI ¼ 1  eis=I. Therefore, we have p2;1¼ q3ð1  q2Þ ¼ 1  e is I   1 1 pc   and p2;2¼ q3q2¼ 1  e is I  _{ 1} pc :

. p1;3 and p2;4. In state S1, if no packet arrives before

the inactivity timer expires (with probability 1  q1),

then the MS enters the sleep period tS1(state S3, that

is, the power saving mode). Therefore, p1;3¼ 1  q1¼ e

ipc

I_{: ð1Þ}

Similarly, p2;4can be derived by substituting q3for q1

in (1), and we have

p2;4¼ 1  q3¼ e

is I_:

. p3;1 and p3;2. In state S3, if the next packet call

terminates the ongoing session (with probability q2),

then the MS will move to S2 at the next state

transition. Otherwise (with probability 1  q2), the

MS will switch to state S1. Thus,

p3;1¼ 1  q2¼ 1  1 pc and p3;2¼ q2¼ 1 pc :

. p4;1 and p4;2. Similar to p3;1 and p3;2, the transition

from state S4to state S1or S2also depends only on

whether the coming packet call is the end of the packet session. Therefore, we conclude that

p4;1¼ p3;1¼ 1  q2¼ 1  1 pc and p4;2¼ p3;2¼ q2¼ 1 pc :

The transition probability matrix P ¼ ðpi;jÞ of the embedded

Markov chain can thus be given as

P ¼ q1ð1  q2Þ q1q2 1 q1 0 q3ð1  q2Þ q3q2 0 1 q3 1 q2 q2 0 0 1 q2 q2 0 0 0 B B @ 1 C C A: Let e

i ði 2 f1; 2; 3; 4gÞ denote the probability that the

embedded Markov chain will stay at Si in the steady

state. By using P4_i¼1ei ¼ 1 and the balance equations

e i ¼

P4

j¼1ejpj;i, we can solve the stationary distribution

e_{¼ ð}e iÞ and obtain e¼ e 1¼ 1q2 1þð1q1Þð1q2Þþq2ð1q3Þ e₂¼ q2 1þð1q1Þð1q2Þþq2ð1q3Þ e₃¼ ð1q1Þð1q2Þ 1þð1q1Þð1q2Þþq2ð1q3Þ e 4¼ q2ð1q3Þ 1þð1q1Þð1q2Þþq2ð1q3Þ: 8 > > > > < > > > > : ð2Þ

Let Hiði 2 f1; 2; 3; 4gÞ be the holding time of the

semi-Markov process at state Si. We proceed to derive E½Hi:

. E½H1. In state S1, the MS experiences a busy

period tB and then an interpacket call inactivity

period t

I1. Hence,

E½H1 ¼ E½tB þ E½t 

I1: ð3Þ

Since a busy period is identical to the duration of a

packet call delivery, a t

B consists of Np packet

service times tx. From Wald’s theorem [13,

Theo-rem 5.18], we have E½t B ¼ E½NpE½tx ¼ p x : ð4Þ As shown in Fig. 3, t

I1¼ minðtipc; tIÞ. If the next

packet arrives before the inactivity timer expires (that is, tipc< tI), then tI1¼ tipc, and the next busy

period follows (see Fig. 3b). Otherwise (the next packet arrives after the inactivity timer has expired; that is, tipc tI), tI1¼ tI, and the next sleep period

follows (see Fig. 3c). Therefore, E½t I1 ¼ E½minðtipc; tIÞ ¼ Z 1 x¼0 Pr½minðtipc; tIÞ > xdx ¼ Z 1=I x¼0 Pr½tipc> xdx ¼ Z 1=I x¼0 eipcx_dx ¼ 1 ipc   1 eipcI   : ð5Þ

Substitute (4) and (5) into (3) to yield E½H1 ¼ p x þ 1 ipc   1 eipcI   : ð6Þ

. E½H2. State S2 contains a busy period tB and an

intersession inactivity period t

I2. Therefore,

E½H2 ¼ E½tB þ E½tI2: ð7Þ

Similar to the derivation of E½t

I1, E½tI2 is E½t_I2 ¼ E½minðtis; tIÞ ¼ 1 is   1 eisI h i : ð8Þ

Substituting (4) and (8) into (7), E½H2 is expressed as

E½H2 ¼ p x þ 1 is   1 eisI h i : ð9Þ

. E½H3. State S3 contains a sleep period tS1 from

state S1. Suppose that there are Nd1 DRX cycles in a

t

S1 period. Due to the memoryless property of the

exponential tipc distribution, Nd1 has geometric

distribution with mean 1=pd1. pd1 is the probability

that packets arrive during a DRX cycle and is derived as follows:

pd1¼ Pr½tipc tD ¼ 1  e

ipc

D_{: ð10Þ}

Since Nd1 is a stopping time, from (10) and Wald’s

theorem, we have E½H3 ¼ E XNd1 i¼1 tD " # ¼ E½Nd1tD¼ 1 1 eipcD ! 1 D   : ð11Þ

. E½H4. State S4 comprises a sleep period tS2 from

state S2. Assume that tS2 consists of Nd2DRX cycles.

Likewise, Nd2 is a geometric random variable with

mean 1=pd2, where pd2¼ Pr½tis tD ¼ 1  e is D_: Thus, we obtain E½H4 ¼ E XNd2 i¼1 tD " # ¼ E½Nd2tD¼ 1 1 eisD ! 1 D   : ð12Þ Based on the semi-Markov process, we derive the power

saving factor Psand the mean packet waiting time E½tw in

the next two sections.

4.1 Power Saving Factor Ps

The power saving factor Psis equal to the probability that

the semi-Markov process is at S3 or S4 (that is, the sleep

period or power saving mode) in the steady state. We note that, at the end of every DRX cycle, the MS must wake up for a short period  so that it can listen to the paging information from the network. Therefore, the “power

saving” period in a DRX cycle is tD . Let E½H30 and

E½H0

4 be the mean “effective” sleep periods in states S3and

S4, respectively. Then, E½H30 and E½H40 can be obtained by

replacing the tD in (11) and (12) with tD  and we have

E½H0 3 ¼ 1 1 eipcD ! 1 D     ð13Þ and E½H₄0 ¼ 1 1 eDis ! 1 D     : ð14Þ From [17, Theorem 4.8.3], Ps¼ lim

t!1Pr½the MS receiver is turned off at time t

¼ e 3E½H30 þ e4E½H40 P4 i¼1 e iE½Hi : ð15Þ

Substituting (2), (6), (9), and (11)-(14) into (15), we derive the

closed-form equation for the power saving factor Ps.

4.2 Mean Packet Waiting Time E½tw

In order to derive the mean packet waiting time E½tw, we

first need to compute the expected total number of packets

E½Nt that are processed in states S1 and S2 and the

expected total waiting time E½Wt of all these packet arrivals

(note that no packets are processed in states S3 and S4).

Then, E½tw can be expressed as

E½tw ¼

E½Wt

E½Nt

: ð16Þ

Let E½Ni;j (i 2 f1; 2g and j 2 f1; 2; 3; 4g) be the mean

number of packets delivered to the MS in state Si, given that

the previous state transition is from state Sj. Denote E½Wi;j

as the mean total waiting time of the associated Ni;j packet

arrivals. Using expectation by conditioning technique [16], we have E½Nt ¼ X2 i¼1 X4 j¼1

e_jpj;iE½Ni;j and E½Wt ¼

X2

i¼1

X4

j¼1

e_jpj;iE½Wi;j:

ð17Þ

We proceed to derive E½Ni;j and E½Wi;j:

. E½N1;1 and E½W1;1. Given the previous state S1, the

transmitted N1;1packets in state S1 correspond to a

packet call and have geometric Np distribution with

mean pand variance pðp 1Þ. Therefore,

E½N1;1 ¼ p: ð18Þ

Since these N1;1 packet transmissions constitute the

busy period tB in state S1, we have the mean total

waiting time: E½W1;1 ¼ E X Np1 i¼1 ðitx itipÞ " # ¼ E NpðNp 1Þ 2   E½tx tip ¼  2 pþ pðp 1Þ  p 2 " # E½tx tip ¼ pðp 1Þ 1 x  1 ip   : ð19Þ

. E½N1;2 and E½W1;2. If the previous state is S2, then

the number of transmitted packets N1;2 in state S1

also has geometric Np distribution. Thus,

E½N1;2 ¼ E½N1;1 ¼ p ð20Þ and E½W1;2 ¼ E½W1;1 ¼ pðp 1Þ 1 x  1 ip   : ð21Þ

. E½N1;3 and E½W1;3. To derive E½N1;3 and E½W1;3,

besides those N1;300 packets that arrive during

state S1, we also need to consider the N1;30 packets

that are accumulated during the sleep period of

the previous state S3. Suppose that the first of the

N_1;30 packets arrives at time t within the DRX

cycle tD. Due to the memoryless property, t has

the truncated exponential tipc distribution with the

following density function: fðtÞ ¼ 1 1 eipcD ! ipceipct; ð22Þ where 0  t  1

D. Since the interpacket call idle time

tipc (several hundred seconds; see the suggested

value in [6]) is significantly longer than the suitable

tDperiods (which will be elaborated on in Section 5),

we also assume that at most one packet call could

appear in a tD period. Under these assumptions,

three cases for the N1;3 packet arrivals are possible

one packet, in case 2, there are N1;30 and N1;300 packets

in state S3and state S1, respectively, and in case 3, all

packets of the packet call arrive in state S3. For given

t, let ið1  i  3Þ be the probability that case i

occurs. It is clear that 1¼

1 p

: ð23Þ

Denote tH as the interval between the first packet

arrival and the last packet arrival of the packet call.

Then, tH has the Erlang-Npdistribution with rate ip

and 2can be derived as follows:

2¼ 1  1 p   Pr½tH > tD t ¼ 1  1 p  _{ X}1 n¼1 Pr½Np¼ n Pr½tHjNp¼n> tD t ¼ 1  1 p  _{ X}1 n¼1 1 p 1 1 p  n1 eipð_D1tÞX n1 k¼0 ½ipð_1_D tÞk k! ( ) ¼ 1  1 p   1 p eipð_D1tÞ X1 k¼0 ½ipð1D tÞ k k! X1 n¼k 1 1 p  n ¼ 1  1 p   eipð_D1tÞ eipð_D1tÞð1_p1Þ ¼ 1  1 p   eippð 1 DtÞ_: ð24Þ From (23) and (24), we have

3¼ 1  1 2¼ 1 1 p   1 eippð 1 DtÞ   : ð25Þ

Next, we compute E½N_1;3ji0  and E½N00

1;3ji and the

associated mean total waiting time E½W0

1;3ji and

E½W00

1;3ji for each of the three cases in Fig. 6, where

1 i  3. In case 1, the only packet of the packet call

arrives at time t of the tD period, and its mean

waiting time is 1=D tin state S3. Therefore,

E½N_1;3j10  ¼ 1; E½N_1;3j100  ¼ 0; E½W_1;3j10  ¼ 1

D

 t; and E½W1;3j100  ¼ 0:

ð26Þ

In case 2, according to the decomposition property of

Poisson processes [17, Proposition 2.3.2], N0

1;3j2 is a

shifted Poisson random variable (that is, including

the packet at t) with mean and the second moment

E½N0 1;3j2 ¼ 1 þ ip 1 1 p   1 D  t   ; E½N02 1;3j2 ¼ V ar½N1;3j20  þ ðE½N1;3j20 Þ 2 ¼ ip 1 1 p   1 D  t   þ ðE½N_1;3j20 Þ2: ð27Þ

The mean total waiting time of these N0

1;3j2packets is E½W_1;3j20  ¼ 1 D  t   þ ðE½N_1;3j20   1Þ1 2 1 D  t   þ E X N0 1;3j2 k¼1 ðk  1Þ 1 x 2 4 3 5: ð28Þ In (28), the first term represents the waiting time in state S3of the first packet arriving at t. The second

term reflects the mean total waiting time in state S3

of the other N0

1;3j2 1 packets (note that these packet

arrivals are uniformly distributed on the interval

ðt; tDÞ [17, Theorem 2.3.1] and the expected waiting

time in state S3 is thus 1₂ð_1_D tÞ for each of these

packets). The third term corresponds to the mean

total waiting time in state S1 of the N_1;3j20 packets.

Substituting (27) into (28), we derive the conditional

expectation E½W0

1;3j2 for given t (see Appendix B

for the details). Now, consider the mean total

waiting time E½W00

1;3j2 of the N1;3j200 packets that

arrive in state S1. Clearly, N1;3j200 is a geometric

random variable with mean and the second moment E½N_1;3j200  ¼ p; E½N1;3j2002  ¼ 22p p: ð29Þ Then, E½W00 1;3j2 ¼ E X N00 1;3j2 k¼1 N_1;3j20 x þk 1 x  k ip ( ) 2 4 3 5: ð30Þ

In (30), the first term represents the mean total

service time of the N0

1;3j2packet arrivals in state S3.

The second term corresponds to the mean total service time of the first k  1 packet arrivals in

state S1. Substituting (27) and (29) into (30), we

derive E½W00

1;3j2 for given t(see Appendix C for the

details). Fig. 6. Three cases for the N1;3packet arrivals.

In case 3, all packets of the packet call arrive in

state S3and, thus,

E½N00

1;3j3 ¼ 0; E½W1;3j300  ¼ 0: ð31Þ

Assume that the last packet of the packet call arrives at twithin the tDperiod (see Fig. 6). Given t, from

the decomposition property of Poisson processes,

tH ¼ t tis an exponential random variable with

rate ip

p (see also the derivation in (24)). Therefore,

t has the following conditional probability density

function: fjðrjtÞ ¼ 1 1 eippð 1 DtÞ " # ip p   eippðrtÞ_{; ð32Þ}

where t  r  1=D. The conditional mean E½tjt

and the conditional second moment E½t2

jt of t

can then be derived from (32) and are provided in

Appendix D. We proceed to derive E½N0

1;3j3 and

E½W0

1;3j3. Given t, similar to N1;3j20 , N1;3j30 is a

shifted Poisson random variable (that is, including

the packets at tand t) with mean and the second

moment E½N_1;3j30  ¼ 2 þ ip 1 1 p   E½tjt  t ð Þ;

E½N_1;3j302  ¼ V ar½N_1;3j30  þ ðE½N_1;3j30 Þ2

¼ ip 1 1 p   E½tjt  t ð Þ þ ðE½N_1;3j30 Þ2: ð33Þ Similar to the derivation in (28), the mean total

waiting time of these N_1;3j30 packets is expressed as

E½W_1;3j30  ¼ E½N_1;3j30  1 D 1 2ðE½tjt þ tÞ   þ E X N0 1;3j3 k¼1 ðk  1Þ 1 x 2 4 3 5: ð34Þ

Combining (22)-(31), (33), and (34), E½N1;3 and

E½W1;3 are therefore

E½N1;3 ¼ Z 1 D t¼0 X3 i¼1

iðE½N1;3ji0  þ E½N1;3ji00 Þ

( ) fðtÞdt; E½W1;3 ¼ Z 1 D t¼0 X3 i¼1

iðE½W1;3ji0  þ E½W1;3ji00 Þ

( )

fðtÞdt:

ð35Þ

Note that, since the N1;3packets constitute a packet

call, E½N1;3 in (35) is further simplified to have

E½N1;3 ¼ p:

The mean total waiting time E½W1;3 in (35) has no

closed-form solution and can be easily computed by using numerical computing software, for example, Matlab [19].

. E½N1;4 and E½W1;4. The derivations of E½N1;4 and

E½W1;4 are identical to that of E½N1;3 and E½W1;3,

except that ipc should be replaced by is. This is

because, in state S4, the MS would experience an

intersession idle time tiswith mean 1=israther than

an interpacket call idle time tipcwith mean 1=ipc.

. E½N2;j and E½W2;j (j 2 f1; 2; 3; 4g). Note that the

only difference between state S1and state S2is that

the busy period in state S1 is followed by an

interpacket call inactivity period, whereas the busy

period in state S2 is followed by an intersession

inactivity period. Therefore, the packets processed in

state S2have the same statistical properties as those

processed in state S1and we have

E½N2;j ¼ E½N1;j and E½W2;j ¼ E½W1;j;

where j 2 f1; 2; 3; 4g.

Substitute the above E½Ni;j and E½Wi;j values (i 2 f1; 2g

and j 2 f1; 2; 3; 4g) into (17) to yield E½Nt and E½Wt.

Finally, substituting the obtained E½Nt and E½Wt into (16),

we derive the mean packet waiting time E½tw.

5

N

UMERICAL

R

ESULTS

The analytic model has been validated against simulation experiments. These simulation experiments are based on a discrete-event simulation model (including Packet arrival, Packet departure, Sleep, Reading, and Wakeup events), which simulates the MS power saving behaviors according to the UMTS DRX mechanism. The interested reader is referred to [22] for the details of the simulation model. Table 1 compares the analytic and simulation results, where ip¼ 5x, ipc¼ x=800, is¼ x=16;000, tD¼ 20E½tx,

¼ E½tx, pc¼ 5, and p¼ 25. The table indicates that,

for the power saving factor Ps, the discrepancies between

the analytic analysis and the simulation are less than 0.01 percent in most cases. For the mean packet waiting

time E½tw, the discrepancies are less than 0.1 percent in

most cases. It is clear that the analytic analysis is consistent with the simulation results. Based on the analytic model, we investigate the DRX performance. Specifically, we consider the bursty packet data traffic. Figs. 7, 8, 9, and

10 plot the Psand E½tw curves. In these figures, tx has the

cut-off Pareto distribution with shape parameter  ¼ 1:1

and mean E½tx ¼ 0:5 seconds and tD and tI are of fixed

values. The parameter settings are described in the captions of the figures.

Effects of tipc. Fig. 7a indicates that the power saving

factor Ps curves decrease and then increase as the

interpacket call idle time tipc increases. This phenomenon

is explained as follows: For tipc< 4;000E½tx, when tipc

approaches zero, the packet arrivals in a packet service

TABLE 1

The Comparison between the Analytic and Simulation Results (ip¼ 5x, ipc¼ x=800, is¼ x=16;000,

session degenerate into a single packet train. After a burst of these packet transmissions, the MS will immediately

experience a long intersession idle time tis and will

eventually be switched into the sleep mode to reduce the power consumption. Therefore, we have high power saving factor Psin this case. As tipcincreases, Psis affected by the

operation of the RNC inactivity timer. Specifically, the MS is more likely to be found in the inactivity period when the

next packet call arrives. Consequently, Psdecreases as tipc

increases. On the other hand, if tipc> 4;000E½tx, then more

of the interpacket call idle times will be longer than the

RNC inactivity timer threshold tI as tipc increases. As a

result, the MS is more likely to be in the power saving

mode when subsequent packet calls arrive. Therefore, Ps

increases as tipc increases. Fig. 7b illustrates that the mean

packet waiting time E½tw is an increasing function of tipc.

When tipc is sufficiently small (for example, tipc< 300E½tx

in this experiment), the value of E½tw is mainly dominated

by the intersession idle time tis. Therefore, decreasing tipc

will insignificantly affect the E½tw performance. We also

note that E½tw is more sensitive to tipcfor a large tDthan a

small tD.

Effects of tis. Fig. 8a shows the intuitive result that Psis

an increasing function of the intersession idle time tis. We

observe that increasing tiswill not significantly improve the

E½tw performance in Fig. 8b. In our traffic model, if the

current packet call is not served completely, then the next packet call will not be generated. Therefore, not more than one packet call will wait in the MS sleep period and

increasing tis will not change the E½tw value.

Effects of tI. Fig. 9 indicates that, by increasing the RNC

inactivity timer threshold tI, Psand E½tw decrease. When tI

is small (for example, tI < 200E½tx in this experiment), it is

likely that the MS is found in the power saving mode as the next packet call arrives. Consequently, we observe high

power saving factor Ps. However, the mean packet waiting

time E½tw is unacceptably high in this case. As tI ! 1, it is

more likely that the MS will never enter the sleep mode and

the E½tw decreases. Due to the characteristic of the packet

burstiness within the packet call, the server state is likely to

be busy when the next packets arrive. As a result, E½tw is

bounded at 20E½tx in this example and increasing tI will

not enhance E½tw. We also note that E½tw is more sensitive

to tI for a large tD than a small tD.

Effects of tD. Fig. 10 shows that Ps and E½tw are

increasing functions of the DRX cycle tD. We observe that,

when tD is large (for example, tD> 100E½tx in Fig. 10a),

increasing tD will not improve the Psperformance. On the

Fig. 7. Effects of tipc (ip¼ 5x, is¼ x=80;000, tI¼ 2;000E½tx, ¼ E½tx, pc¼ 5, and p¼ 25).

Fig. 8. Effects of tis (ip¼ 5x, ipc¼ x=4;000, tI¼ 2;000E½tx, ¼ E½tx, pc¼ 5, and p¼ 25).

Fig. 9. Effects of tI (ip¼ 5x, ipc¼ x=4;000, is¼ x=80;000, ¼ E½tx, pc¼ 5, and p¼ 25).

Fig. 10. Effects of tD (ip¼ 5x, ipc¼ x=4;000, is¼ x=80;000, tI¼ 2; 000E½tx, pc¼ 5, and p¼ 25).

other hand, when tDis small (for example, tD< 10E½tx in

Fig. 10b), decreasing tD insignificantly improves the E½tw

performance. Therefore, for tI ¼ 2;000E½tx, tD should be

selected in the range 10E½t½ x; 100E½tx.

Effects of . Fig. 10a illustrates the impacts of the

wakeup cost  on Ps. When tD is large (for example,

tD> 100E½tx),  is a small portion of a DRX cycle and

thus only has insignificant impact on Ps. When tD is small

(for example, tD< 20E½tx), Ps increases as  decreases.

Fig. 10b demonstrates an intuitive result that E½tw is not

affected by .

6

C

ONCLUSION

The UMTS utilizes the DRX mechanism to reduce the power consumption of MSs. DRX permits an idle MS to power off the radio receiver for a predefined sleep period and then wake up to receive the next paging message. The sleep/wake-up scheduling of each MS is determined by

two DRX parameters: the inactivity timer threshold tI and

the DRX cycle tD. Analytic and simulation models have

been developed in the literature to study the DRX performance mainly for Poisson traffic. In this paper, we proposed a novel semi-Markov process to model the UMTS DRX with bursty packet data traffic. The analytic results were validated against simulation experiments. We inves-tigated the effects of the two DRX parameters on output

measures including the power saving factor Ps and the

mean packet waiting time E½tw. Our study indicated the

following:

. The power saving factor Ps curves decrease and

then increase as the interpacket call idle time tipc

increases.

. The mean packet waiting time E½tw is an increasing

function of tipc.

. When tipc is sufficiently small, decreasing tipc will

insignificantly affect the E½tw performance.

. E½tw is more sensitive to tipc for a large tD than a

small tD.

. Ps is an increasing function of the intersession idle

time tis.

. By increasing the RNC inactivity timer threshold tI,

Psand E½tw decrease.

. E½tw is more sensitive to tI for a large tD than a

small tD.

. For the parameter settings considered in this paper,

tD should be selected in the range 10E½t½ x; 100E½tx

for better Psand E½tw performance.

. When tDis large,  is a small portion of a DRX cycle

and only has insignificant impact on Ps.

A

PPENDIX

A

N

OTATION

L

IST

. E½H0

3: the mean “effective” sleep period in state S3.

. E½H0

4: the mean “effective” sleep period in state S4.

. E½Ni;j: the mean number of packets delivered to

the MS in state Si, given that the previous state

transition is from state Sj.

. E½Nt: the expected total number of packets that are

processed in states S1and S2.

. E½Wi;j: the mean total waiting time of the associated

Ni;j packet arrivals.

. E½Wt: the expected total waiting time of the Nt

packet arrivals in states S1and S2.

. 1=D: the length of each DRX cycle tD in a sleep

period.

. 1=I: the length of the RNC inactivity timer

thresh-old tI.

. 1=ip: the expected value for the tip distribution.

. 1=ipc: the expected value for the tipcdistribution.

. 1=is: the expected value for the tisdistribution.

. 1=x: the expected value for the txdistribution.

. p: the expected value for the Np distribution.

. pc: the expected value for the Npc distribution.

. Np: the number of packets within a packet call.

. Npc: the number of packet calls within a packet

service session.

. Ps: the power saving factor.

. tD: the DRX cycles in a sleep period.

. tip: the interpacket arrival time within a packet call.

. tipc: the time interval between the end of a packet call

transmission and the beginning of the next packet call transmission (that is, the interpacket call idle time).

. tis: the time interval between the end of a packet

session transmission and the beginning of the next packet session transmission (that is, the intersession idle time).

. tI: the threshold of the RNC inactivity timer.

. tw: the packet waiting time in the RNC buffer.

. tx: the time interval between when the packet is

transmitted by the RNC processor and when the corresponding positive ack is received by the RNC processor.

A

PPENDIX

B

E½W

0 1;3j2



FOR

G

IVEN

t



E½W

_1;3j20

 ¼

1



_D

 t







þ ðE½N

_1;3j20

  1Þ

1

2

1



_D

 t







þ E

X

N0 1;3j2 k¼1

ðk  1Þ

1



_x

2

4

3

5

¼

1



D

 t







þ ðE½N

_1;3j20

  1Þ

1

2

1



D

 t







þ

1



x

 

E

N

0 1;3j2

ðN

1;3j20

 1Þ

2

"

#

¼

1



D

 t







þ ðE½N

_1;3j20

  1Þ

1

2

1



D

 t







þ

1

2

x





ðE½N

_1;3j202

  E½N

_1;3j20

Þ;

where E½N0

A

PPENDIX

C

E½W

00 1;3j2



FOR

G

IVEN

t



E½W

_1;3j200

 ¼ E

X

N00 1;3j2 k¼1

N

_1;3j20



x

þ

k

 1



x



k



ip

(

)

2

4

3

5

¼

1



_x

 

E½N

_1;3j200

E½N

_1;3j20

 

1



_ip





E½N

_1;3j200



þ

1



x



1



ip





E

N

00 1;3j2

ðN

1;3j200

 1Þ

2

"

#

¼

1



x

 

E½N

_1;3j200

E½N

_1;3j20

 

1



ip





E½N

_1;3j200



þ

1

2

1



x



1



ip





ðE½N

_1;3j2002

  E½N

_1;3j200

Þ;

where E½N0

1;3j2, E½N1;3j200 , and E½N1;3j2002  are given in (27)

and (29).

A

PPENDIX

D

E½t



jt





AND

E½t

2

jt





E½t



jt



 ¼ t



þ

1

1

 e

 ip pð 1 DtÞ

"

#



p



_ip



1



_D

 t



þ



p



_ip





e

 ip pð 1 DtÞ





;

E½t

2_

jt



 ¼ t

2

þ

h

₁

1

 e

ippðD1tÞ

in

2t





p



ip





þ 2



p



ip





2



h

2t



1



D

 t



þ



p



ip





þ 2



p



ip





2

þ2



p



ip





1



D

 t







þ ð

1



_D

 t



Þ

2

i

e

 ip pð 1 DtÞ

o

_:

A

CKNOWLEDGMENTS

The authors would like to thank the anonymous re-viewers. Their valuable comments have significantly enhanced the quality of this paper. Yang’s work was supported in part by the National Science Council of Taiwan (NSC) under Contracts NSC-94-2752-E-007-003-PAE, NSC-94-2213-E-007-072, and NSC-94-2219-E-009-024. Hung’s work was supported in part by the NSC Grant NSC-94-2118-M-009-003.

R

EFERENCES

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[2] 3GPP, “3rd Generation Partnership Project; Technical Specifica-tion Group Services and Systems Aspects; General Packet Radio Service (GPRS); Service Description; Stage 2,” Technical Specifica-tion 3G TS 23.060, version 3.6.0 (2001-01), 2000.

[3] 3GPP, “3rd Generation Partnership Project; Technical Specifica-tion Group Radio Access Network; UTRA High Speed Downlink Packet Access,” Technical Specification 3G TR 25.950, version 4.0.0 (2001-03), 2001.

[4] 3GPP, “3rd Generation Partnership Project; Technical Specifica-tion Group Radio Access Network; UE Procedures in Idle Mode and Procedures for Cell Reselection in Connected Mode,” Technical Specification 3G TS 25.304, version 5.1.0 (2002-06), 2002. [5] CDPD Forum, “Cellular Digital Packet Data System Specification:

Release 1.1,” technical report, CDPD Forum, Inc., Jan. 1995. [6] ETSI, “Universal Mobile Telecommunications System (UMTS);

Selection Procedures for the Choice of Radio Transmission Technologies of the UMTS,” Technical Report UMTS 30.03, version 3.2.0, Apr. 1998.

[7] H. Holma and A. Toskala, WCDMA for UMTS. John Wiley & Sons, 2000.

[8] Wireless Medium Access Control (MAC) and Physical Layer (PHY) Specifications, Draft Standard 802.11 D3.1, IEEE, Apr. 1996. [9] N.L. Johnson, Continuous Univariate Distributions 1. John Wiley &

Sons, 1970.

[10] S.J. Kwon, Y.W. Chung, and D.K. Sung, “Queueing Model of Sleep-Mode Operation in Cellular Digital Packet Data,” IEEE Trans. Vehicular Technology, vol. 52, no. 4, pp. 1158-1162, July 2003. [11] C.-C. Lee, J.-H. Yeh, and J.-C. Chen, “Impact of Inactivity Timer on Energy Consumption in WCDMA and cdma2000,” Proc. Third Ann. Wireless Telecomm. Symp. (WTC ’04), May 2004.

[12] Y.-B. Lin and Y.-M. Chuang, “Modeling the Sleep Mode for Cellular Digital Packet Data,” IEEE Comm. Letters, vol. 3, no. 3, pp. 63-65, Mar. 1999.

[13] R. Nelson, Probability, Stochastic Processes, and Queueing Theory. Springer-Verlag, 1995.

[14] V. Paxson and S. Floyd, “Wide Area Traffic: The Failure of Poisson Modeling,” IEEE/ACM Trans. Networking, vol. 3, no. 3, pp. 226-244, June 1995.

[15] RAM Mobile Data, “Mobitex Interface Specification,” technical report, RAM Mobile Data, 1994.

[16] S.M. Ross, Introduction to Probability Models, fifth ed. Academic Press, 1993.

[17] S.M. Ross, Stochastic Processes, second ed. John Wiley & Sons, 1996. [18] M. Sarvagya and R.V. Raja Kumar, “Performance Analysis of the UMTS System for Web Traffic over Dedicated Channels,” Proc. Third Int’l Conf. Information Technology: Research and Education (ITRE ’05), June 2005.

[19] The MathWorks, MATLAB User’s Guide. The MathWorks, Inc., 1993.

[20] W. Willinger, M.S. Taqqu, R. Sherman, and D.V. Wilson, “Self-Similarity through High-Variability: Statistical Analysis of Ether-net LAN Traffic at the Source Level,” IEEE/ACM Trans. Network-ing, vol. 5, no. 1, pp. 71-86, Feb. 1997.

[21] S.-R. Yang and Y.-B. Lin, “Modeling UMTS Discontinuous Reception Mechanism,” IEEE Trans. Wireless Comm., vol. 4, no. 1, pp. 312-319, Jan. 2005.

[22] S.-R. Yang and S.-Y. Yan, “The Supplement to ‘Modeling UMTS Power Saving with Bursty Packet Data Traffic’,” technical report, Nat’l Tsing Hua Univ., http://www.cs.nthu.edu.tw/~sryang/ submission/MUPSwBPDT.pdf, 2006.

[23] J.-H. Yeh, C.-C. Lee, and J.-C. Chen, “Performance Analysis of Energy Consumption in 3GPP Networks,” Proc. Wireless Telecomm. Symp., May 2004.

Shun-Ren Yang received the BS, MS, and PhD degrees in computer science and information engineering from the National Chiao Tung University, Hsinchu, Taiwan, in 1998, 1999, and 2004, respectively. From April 2004 to July 2004, he was appointed as a research assistant in the Department of Information Engineering, the Chinese University of Hong Kong. Since August 2004, he has been with the Department of Computer Science and the Institute of Communications Engineering, National Tsing Hua University, Taiwan, where he is an assistant professor. His current research interests include the design and analysis of personal communications services networks, computer telephony integration, mobile computing, and performance modeling.

Sheng-Ying Yan received the BS degree in computer science and information engineering from the National Chiao Tung University, Hsinchu, Taiwan, in 2004 and the MS degree in computer science from the National Tsing Hua University, Hsinchu, in 2006. In 2007, he joined the Telecommunication Laboratories, Chunghwa Telecom, Taiwan. His current re-search interests include the design and analysis of personal communications services networks and performance modeling.

Hui-Nien Hung received the BS degree in mathematics from the National Taiwan Univer-sity, Taipei, in 1989, the MS degree in mathe-matics from the National Tsing-Hua University, Hsinchu, Taiwan, in 1991, and the PhD degree in statistics from the University of Chicago, Chicago, in 1996. He is currently a professor in the Institute of Statistics, National Chiao Tung University, Hsinchu. His research interests include applied probability, biostatistics, statis-tical inference, statisstatis-tical computing, and industrial statistics.

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