Performance analysis of queueing strategies for multiple priority calls in multiservice personal communications services

Computer Communications 23 (2000) 1069–1083 www.elsevier.com/locate/comcom

Performance analysis of queueing strategies for multiple priority calls in multiservice personal communications services D.C. Lee a,*, S.J. Park b, J.S. Song b a

Department of Computer Science, Howon University, 727 Wolha-Ri Impi, KunSan, ChonBuk, South Korea b Department of Computer Science, Yonsei University, Seoul, South Korea Received 8 April 1999; received in revised form 15 December 1999; accepted 15 December 1999

Abstract Supporting multiple priority calls, we propose the queueing strategies which efficiently manage the queue to reduce the blocking probability with multiclass calls in multiservice personal communications services (MPCS). The two queueing schemes are proposed and are shown the analytic model with …n ⫹ 1† class calls. Numerical results demonstrate that the two proposed schemes show the improved performance compared to the previous scheme with two types of traffic in terms of the blocking probability. The proposed schemes are flexibly adaptable to MPCS. 䉷 2000 Elsevier Science B.V. All rights reserved. Keywords: Queueing strategies; Multiple priority calls; Multiservice personal communications service

1. Introduction Technological advances and rapid development of handheld wireless terminals have facilitated the rapid growth of wireless communications and mobile computing. Taking ergonomic and economic factors into account and considering the new trend in the telecommunications industry to provide ubiquitous information access, the population of many users will continue to grow at a tremendous rate. Another important developing phenomenon is the shift of many applications to multimedia platforms in order to present information more effectively. The tremendous growth of the wireless/mobile user population, coupled with the bandwidth requirements of multimedia applications, requires an efficient use of the scarce radio spectrum allocated to wireless/mobile communications. It has been anticipated that demands for multiple types of service from low-bandwidth applications such as voice to high-bandwidth applications such as data, image, and video will grow for future mobile personal communications. For delivering the desired levels of quality of service (QoS) in multiservice personal communications services (MPCS) to multiple type of mobile users, an improved channel allocation mechanism is required. It is to obtain a high admitted traffic and to reduce blocking probability while guaranteeing the protection of calls in restricted channels. The * Tel.: ⫹ 82-654-450-7523; fax: ⫹ 82-654-450-7525. E-mail address: [email protected] (D.C. Lee).

channel allocation scheme has been introduced such as fixed channel allocation (FCA), dynamic channel allocation (DCA), hybrid channel allocation (HCA), distribute channel allocation (DCA), borrowing channel allocation (BCA), and adaptive channel allocation (ACA) [16,17]. For reducing blocking probability of call, those schemes make use of queueing in personal communications services (PCS). We classify calls in PCS as voice call and data call: new voice call, new data call, hand-off voice call, and hand-off data call. Intuitively, when channels are busy, the blocking probability of the hand-off data call can be reduced by waiting only the hand-off data call in the queue. The blocking probability of voice call can be also reduced by giving priority to voice call in the queue environment [1,3,4,7,10–13]. Another priority control method is the measured based priority scheme (MBPS). It is the scheme in which the handoff call requests which can not be served are queued and served according to a signal power level based strategy [14]. MBPS always shows the better performance than FIFO scheme does. However, the previous schemes consider only two class calls and have limits in reducing the blocking probability of multiple priority calls by allocating channels efficiently in MPCS. An efficient queueing management with priority methods for multiclass calls can reduce the blocking probability of calls in restricted channels in a cell. In this paper, we consider the queueing strategies for multiple priority calls in MPCS and propose the two

0140-3664/00/$ - see front matter 䉷 2000 Elsevier Science B.V. All rights reserved. PII: S0140-366 4(00)00171-7

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D.C. Lee / Computer Communications 23 (2000) 1069–1083

Fig. 1. The previous queueing scheme with two types of traffic.

queueing schemes. As main performance measures, the blocking probability and the mean waiting time of multiclass calls are considered. The organization of the paper is as follows. Section 2 describes the previous queueing scheme with two types of traffic. Section 3 describes the two proposed queueing schemes of …n ⫹ 1† class calls, which show the blocking probability and the mean waiting time of priority calls as main performance measures. In Section 4, the analytic results of two schemes are verified by performing the simulation and compared to the previous scheme with two types of traffic (voice calls and data calls). Additionally, we compare the two proposed schemes of each other with four class calls. Finally, we conclude this paper in Section 5.

scheme is depicted in Fig. 1 and the state diagrams are given in two-dimensional case. Since voice call has a preemptive priority over data packet in the service facility and HOL priority control in the queue, the probabilities Pi,0 for i ˆ 0; 1; 2; …; K are given by the M/M/C/K formulae modified for the purpose of covering the mobility in queue, i.e. Pi;0 ˆ P0;0

Pi;0

ri i!

iⱕC 3

2

7 6 7 rC 6 li⫺C 7 6 2 ˆ P0;0 7 6 i⫺C 7 Y C! 6 4 …Cm ⫹ nmQ † 5

…1† C⬍iⱕK

nˆ1

2. The previous queueing scheme The previous scheme considers only two types of traffic (voice calls and data packets), which is supported by a set of C channels plus a buffer of size K-C [12,13]. The arrival rates are l 1 and l 2, respectively and the channel holding time in a cell (the time in which a call or a packet occupies a channel while its terminal is cell) follows exponential distributions for both types of traffic with means 1=m1 and 1=m2 ; respectively. Any type of arrival has access to any facility but voice call can preempt the service of data packet which return to the queue next to the last voice call arrival. Thus, this scheme has a system with preemptive priority in the C channels and Head-of-the-Line (HOL) priority in the queue, where voice has a priority over data packet. A call in such a system is blocked only if there are already K calls in the system while a data packet is blocked if the system is full. Moreover, any type of traffic must leave the queue after a finite time because the vehicle has to leave the cell. In this scheme, it is assumed that the time TQ per voice call or data packet is allowed (the dwelling time in queue), which follows an exponential distribution with mean 1=mQ dependent mainly on the system structure, i.e. the cell length and the vehicle speed. The previous queueing

where r ˆ l1 =m1 Moreover, it can use the normalizing condition below. K X K X i

Pi;j ˆ 1

i ˆ 0; 1; …; K j ˆ 0; 1; …; k

…2†

j

The systems (1) and (2) are now sufficient for the evaluation of the state probabilities Pi,j. Evaluating those probabilities, the main performance parameters are defined to be: 1. The blocking probabilities (Pbi, voice calls …i ˆ 1† and data packets …i ˆ 2†† are given by Pb1 ˆ

K X iˆC

Pi;j PQ1

and

Pb2 ˆ

K X

KX ⫺i

Pi;j PQ2

iˆ0 jˆC ⫺ i ⫹ 1

…3† where PQi is the probability of leaving the cell. 2. The average time Wi in the queue is given by     1 1 and E…N2 † ˆ l2 E…N1 † ˆ l1 W1 W2

…4†

where E(Ni) is the mean number of call in the queue.

D.C. Lee / Computer Communications 23 (2000) 1069–1083

3. The two proposed queueing schemes Priority traffic control mechanisms for multiclass calls are usually classified into two types: the space priority traffic control method and the time priority traffic control one. The space priority traffic control method includes the separate routing (SR) control method, the push-out (PO) control method, the partial buffer sharing (PBS) method, etc. The time priority traffic control method includes the head of line (HOL) control method, etc. In the space priority traffic method, the PO control method allows the queue access for a given class, if the queue is not full or if the arriving call can replace a call with lower priority in the queue. The selection of the call, which may be discarded, is controlled by the replacement strategy. The SR control method uses separate routes within mobile network for different traffic classes. This is the simplest way to establish space priority capabilities in mobile network. No special management actions are taken at the call level, since the priority processing is executed at the connection level by the routing function. This method is only applicable if the priorities will be assigned on a connection basis. PBS control method allows the queue access for any class if the queue space less than any threshold value is occupied. Obviously, the class with the highest priority call has access to the whole queue. This method is simple for algorithmic implementation and guarantees traffic order [5,15]. In time priority traffic control method, since high priority calls prior to low priority calls holds the idle channel, the higher the priority call is, the lower is the mean waiting time. This control method is called HOL [2]. To model the queueing scheme with multiple priority calls which is based on priority control methods, we propose the two queueing schemes which efficiently manage …n ⫹ 1† class calls in the queue for reducing the block of channel allocation in MPCS.

T2, it is also terminated by force. If the queue size M is full, all of …n ⫹ 1† class calls are terminated by force irrespective of their priorities. To analyze the queueing model, we assume the following: (a) The each arrival call is modeled as Poisson distribution with arrival rates l i of i class traffic. (b) The waiting time and the channel holding time of multiclass calls have the exponential distributions with service rate mq and service rate m, respectively. (c) Multiple Priorities of arrival multiclass calls have li ⬎ li⫹1 ; 1 ⬍ i ⬍ n: (d) During a call holds the channel, arriving calls have non-preemptive priorities. (e) Queue size M is finite (K–C) and FIFO discipline is served in each threshold value. (f) A cell is equipped with C permanently assigned channels. (g) The model for multiclass call is M/M/ C/M/K. Fig. 3 shows the state transition diagram for scheme I. Let Pi …t† ˆ Pr ‰I…t† ˆ iŠ denote the probability that the process is in state i at time t, and Pi ˆ limt!∞ Pi …t† denote the steady state probability that the process is in state i. Using the assumed parameters, the state probabilities of state i are given by nX ⫹1

PT1 ˆ

!c

li

1

iˆ1

c!mc

T1 Y

…l1 †T1 ⫺c P0

… jmq ⫹ cm†

jˆ1 nX ⫹1

PT2 ˆ

!c

li

1

iˆ1

c!mc

T2 Y

…l1 †T1 ⫺c …l2 †T2 ⫺T1 P0

…jmq ⫹ cm†

jˆ1

.. . nX ⫹1

3.1. Scheme I Fig. 2 shows the proposed queueing model of …n ⫹ 1† class calls which is based on the HOL control method. The queue is logically divided by threshold values Ti, 1 ⬍ i ⬍ n ⫹ 1: When multiclass calls l i, 1 ⬍ i ⬍ n ⫹ 1 arrive in the queue, each call waits in intervals of each threshold value Ti, which high priority call prior to low priority call waits in the queue. In the queue with threshold values, when each call arrives in the queue of which state is over its threshold values, it is terminated by force and blocked. These calls must leave the queue or the cell because the channels can not be allocated for them. If the highest priority call l 1 arrives in the queue of which state is over the first threshold value T1, it is terminated by force and blocked because the channels in a cell can not be allocated for it. If the second higher priority call l 2 arrives in the queue of which state is over the second threshold value

1071

PTi ˆ

!c

li

1

iˆ1

c!m

Ti Y

c

…jmq ⫹ cm†

jˆ1

 …l1 †T1 ⫺c …l2 †T2 ⫺T1 ……li †Ti ⫺Ti⫺1 P0 .. . nX ⫹1

PM ˆ

!c

li

1

iˆ1

c!mc

M Y

…jmq ⫹ cm†

jˆ1

 …l1 †T1 ⫺c …l2 †T2 ⫺T1 ……ln⫹1 †M⫺Tn P0

(5)

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D.C. Lee / Computer Communications 23 (2000) 1069–1083

Fig. 2. The basic queueing model for …n ⫹ 1† class calls (scheme I).

From Eq. (5), the steady state probabilites Pi depending on threshold values Ti are given by 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 Pi ˆ 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

nX ⫹1

!i

li

iˆ1

nX ⫹1

0⬍iⱕC

P0

i!mi

!c

li

1

iˆ1 i Y

c!mc nX ⫹1

!c

li

1 T1 Y

…l1 †T1 ⫺C

… jmq ⫹ cm†

jˆ1

.. . nX ⫹1

li

1

iˆ1

nX ⫹1

TY n⫺1

!c

… jm q ⫹ c m †

1

iˆ1 Tn Y jˆ1

…jmq ⫹ cm†

…l1 †T1 ⫺c

i Y jˆT1 ⫹ 1

…l2 †i⫺T1 P0 > > > > …jmq ⫹ cm† ;

8 > > > > < > > > > :

8 > > > > < > > > > :

9 > > > > =

1

> > > > :

…l1 †T1 ⫺c

jˆ1

li

c!mc

8 > > > > <

!c

c!mc

C ⫹ 1 ⬍ i ⱕ C ⫹ T1

… jmq ⫹ cm†

jˆ1

iˆ1

c!mc

…l1 †i⫺C P0

1 i Y

…ln †i⫺Tn⫺1

… jmq ⫹ cm†

jˆTn ⫺ 1 ⫹ 1

1 i Y jˆTn ⫹ 1

…jmq ⫹ cm†

…ln⫹1 †i⫺Tn

C ⫹ T1 ⫹ 1 ⬍ i ⱕ C ⫹ T2

9 > > >8 > ⫺1 = > > > ;

9 > > >8 > n =< Y : jˆ2 > > > > ;

.. . 9 =

…lj †Tj ⫺Tj⫺1 P0 ;

9 =

…lj †Tj ⫺Tj⫺1 P0 ;

C ⫹ Tn⫺1 ⬍ i ⱕ C ⫹ Tn

C ⫹ Tn ⫹ 1 ⬍ i ⱕ C ⫹ M

…6†

D.C. Lee / Computer Communications 23 (2000) 1069–1083

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…1† From Eq. (6), we obtain initial probability P…i† 0 which represents equations from initial probability P0 by the first threshold …n† value T1 to initial probability P0 by nth threshold value Tn. The results are given by

P0 ⫹

M X

P…i† 0 ˆ1

…7†

iˆ1

2 P…1† 0

6 6 6 ˆ 61 ⫹ 6 4

nX ⫹1

!c

li

iˆ1

c!m

nX ⫹1 T1 X

c iˆ1

1 i Y

…l1 †

i⫺c

⫹ …l1 †

T1 ⫺C

li

M X

iˆ1

c!m

… jmq ⫹ cm†

c

1

iˆT1 ⫹ 1

jˆ1

P…2† 0

T1 Y jˆ1

2 6 6 6 ˆ 6P…1†⫺1 ⫹ …l1 †T1 ⫺C …l2 †T2 ⫺T1 T 6 0 2 Y 4

nX ⫹1

1

> > > … jmq ⫹ cm† > :

li

c!mc

93⫺1 > > > 7 > =7 1 i⫺T1 7 …l2 † 7 i 7 > Y > 5 > > …jmq ⫹ cm† ; jˆ1

3⫺1

!c M X

iˆ1

…jmq ⫹ cm†

8 > > > > <

!c

1 i Y

iˆT2 ⫹ 1

…l3 †

7 7 7 7 5

i⫺T2 7

…jmq ⫹ cm†

jˆT2 ⫹ 1

jˆ1

.. . 2 P…n† 0

6 6 6 ˆ 6P0…n⫺1†⫺1 ⫹ …l1 †T1 ⫺C …l2 †T2 ⫺T1 ……ln †Tn ⫺Tn⫺1 T 6 n Y 4

nX ⫹1

1 …jmq ⫹ cm†

li

iˆ1

c!mc

jˆ1

3⫺1

!c M X iˆTn ⫹ 1

1 i Y

…ln⫹1 †

7 7 7 7 5

i⫺Tn 7

…jmq ⫹ cm†

jˆTn ⫹ 1

…8† Also, initial probability P0 is obtained by modifying results of Eq. (8) 2

nX ⫹1

!i

nX ⫹1

3⫺1

!c

7 6 li li T1 7 6 C X X 1 7 6 iˆ1 iˆ1 i⫺C 7 61 ⫹ … ⫹ l † 1 c i i 7 6 Y c! m i! m 7 6 iˆ1 iˆ1 7 6 … j m ⫹ c m † q 7 6 7 6 jˆ1 7 6 !c 7 6 n ⫹ 1 7 6 X 7 6 7 6 li T 2 7 6 X 1 1 7 6 ⫹ iˆ1 T1 ⫺C i⫺T1 … l1 † …l2 † 7 6 c T i 1 7 6 Y c! m Y iˆT1 ⫹ 1 7 6 … jmq ⫹ cm† … jmq ⫹ cm† 7 6 7 6 jˆT1 ⫹ 1 jˆ1 7 6 7 6 7 6 .. 7 P0 ˆ 6 7 6 . 7 6 8 9 ! 7 6 c nX ⫹1 7 6 > > 7 6 > > 8 9 > > li 7 6 > > Tn nY ⫺1 < = < = 7 6 X 1 1 6 ⫹ iˆ1 T1 ⫺C i⫺Tn⫺1 Tj⫺ Tj⫺1 7 … … l † l † … l † 6 1 n j c TY i : jˆ2 ;7 > > n⫺1 7 6 Y c!m > > iˆTn ⫺ 1 ⫹ 1 7 6 > > > > … j m ⫹ c m † … jmq ⫹ cm† 7 6 q : ; 7 6 jˆTn ⫺ 1 ⫹ 1 jˆ1 7 6 7 6 8 9 ! 7 6 c nX ⫹1 7 6 > > > > 7 6 8 9 > > li 6 > > M n = 7 < = > Y 7 6 c!m Y > > ⫹ 1 iˆT n > > 5 4 > > … jmq ⫹ cm† … jmq ⫹ cm† : ; jˆ1

jˆTn ⫹ 1

…9†

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D.C. Lee / Computer Communications 23 (2000) 1069–1083

Fig. 3. State transition diagram of scheme I.

From Eqs. (6)–(9), the blocking probabilities…P…i† b ; 1 ⱕ i ⱕ n ⫹ 1† depending on i priorities are given by P…1† b ˆ Prob‰T1 ⬍ iŠ ˆ

T2 X

Pi ⫹

iˆT1 ⫹ 1

T3 X iˆT2 ⫹ 1 nX ⫹1

P…2† b

Pi

iˆTn ⫹ 1

!c

li

T2 X

iˆ1

ˆ Prob‰T1 ⬍ i ⬍ T2 Š ˆ 1 ⫺

M X

Pi ⫹ … ⫹

c!mc

1

iˆT1 ⫹ 1

T1 Y

…l1 †T1 ⫺C

… jmq ⫹ cm†

jˆ1

8 > > > > < > > > > :

9 > > > > =

1

i Y jˆT1 ⫹ 1

…l2 †i⫺T1 P0 > > > > … jmq ⫹ cm† ;

.. . nX ⫹1

P…n† b

!c

li

iˆ1

ˆ Prob‰Tn⫺1 ⬍ i ⬍ Tn Š ˆ 1 ⫺

c!mc

Tn X

1

iˆTn ⫺ 1 ⫹ 1

TY n⫺1

…l1 †T1 ⫺C

… jmq ⫹ cm†

jˆ1



8 ⫺1
> > > > :

1 i Y

…l2 †i⫺Tn⫺1

… jmq ⫹ cm†

jˆTn ⫺ 1 ⫹ 1

9 > > > > = > > > > ;

9 = …lj †Tj ⫺Tj⫺1 P0 ;

nX ⫹1

Pb…n⫹1† ˆ Prob‰Tn ⬍ i ⬍ MŠ ˆ 1 ⫺

8 > > > > <

!c

li

iˆ1

c!mc

M X

Tn Y

iˆTn ⫹ 1 jˆ1

1 …l †T1 … jmq ⫹ cm† 1

1 i Y

… jmq ⫹ cm†

8 9 n
jˆTn ⫹ 1

…10† P…1† b is the blocking probability with which the highest priority call l1 occurs when it arrives in the queue of the state which is over the first threshold value T1, and Pb…n⫹1† is the blocking probability with which the lowest priority call ln⫹1 occurs when it arrives in the queue of the state which is full from n th threshold value Tn to queue size M. The total blocking probability P…total† is b given by ˆ P…total† b

1 nX ⫹1

…2† … ⫹ ln P…n† ⫹ ln⫹1 P…n⫹1† Š ‰l1 P…1† b ⫹ l2 Pb ⫹ b b

…11†

li

iˆ1

From Eqs. (6) and (7), the mean waiting times …Wq…i† ; 1 ⱕ i ⱕ n ⫹ 1† depending on i priorities can be written as Eq. (12) using

D.C. Lee / Computer Communications 23 (2000) 1069–1083

1075

Little’s formula [6]. nX ⫹1

wq…1†

1 ˆ l1

!c

li

iˆ1

c!mc

T1 X

i

iˆ0

1 i Y

…l1 †i⫺C P0

… jmq ⫹ cm†

jˆ1

2 wq…2†

6 1 6 6 ˆ 6l1 Wq…1† ⫹ l2 6 4

nX ⫹1

8 > > > > <

!c

li

T2 X

iˆ1

c!mc

i…l1 †

T1 ⫺C

iˆT1 ⫹ 1

1 T1 Y jˆ1

> > > … jm q ⫹ c m † > :

1 i Y

…l2 †

i⫺T1

… jm q ⫹ c m †

jˆT1 ⫹ 1

3

9 > > > > =

7 7 7 P0 7 7 > > 5 > > ;

.. . 2 w…n† q

6 1 6 6 ˆ 6ln⫺1 Wq…n⫺1† ⫹ ln 6 4

nX ⫹1

!c

li

Tn X

iˆ1

c!mc

i

iˆTn ⫺ 1 ⫹ 1

1 TY n⫺1

T1 ⫺C

…l1 †

… jm q ⫹ c m †

jˆ1

2 wq…n⫹1†

6 1 6 6 ˆ 6ln Wq…n† ⫹ ln⫹1 6 4

nX ⫹1

!c

li

iˆ1

c!mc

M X iˆTn ⫹ 1

i

1 Tn Y jˆ1

… jm q ⫹ c m †

T1 ⫺C

…l1 †

8 > > > > < > > > > :

1 i Y jˆTn ⫺ 1 ⫹ 1

8 > > > > < > > > > :

…ln †

… jmq ⫹ cm†

1 i Y

…ln⫹1 †

… jmq ⫹ cm†

jˆTn ⫹ 1

i⫺Tn

i⫺Tn⫺1

9 > > 8 > > ⫺1 = : jˆ2 > > > ;

9 > > 8 > > n = > > > ;

3 …lj †

Tj ⫺Tj⫺1

9 =

7 7 7 P0 7 ; 7 5

3 Tj ⫺Tj⫺1

…lj †

9 =

7 7 7 P0 7 ; 7 5

…12† Here, Wq…1† is the mean waiting time that the highest priority call l1 waits in the queue by the first threshold value T1, and Wq…n⫹1† is the mean waiting time that the lowest priority call ln⫹1 waits in the queue by queue size M. 3.2. Scheme II Fig. 4 shows the proposed queueing model for …n ⫹ 1† class calls which is based on the PBS control method [8,9]. When multiclass calls lj arrive in the queue, each call waits in the queue that is divided by threshold values Ti and shares the queue partially irrespective of priority. But when each call arrives in the queue of the state which is over its threshold values, it

Fig. 4. The basic queueing model for …n ⫹ 1† class calls (scheme II).

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D.C. Lee / Computer Communications 23 (2000) 1069–1083

Fig. 5. State transition diagram of scheme II.

is terminated by force. If the lowest priority call ln⫹1 arrives in the queue of the state which is over the first threshold value, it is terminated by force and blocked. The channel can not be allocated for this call ln⫹1 because it shares the queue by the first threshold value T1. If multiclass calls arrives in the queue of the state which is over the second threshold value T2, the lowest priority call ln⫹1 and the second lower priority call ln are also terminated by force. The channels can not be allocated for those calls ln ; ln⫹1 because they only share the queue by the first threshold value and the second threshold value. If queue size M is full, all of …n ⫹ 1†class calls are terminated by force irrespective of their priorities. According to the previous description, the state transition diagram of our scheme II is given in Fig. 5. The state probabilities of state i are given by 0

nX ⫹1

@

1c

lj A

0 1

jˆ1

PT1 ˆ

c!mc

T1 Y

nX ⫹1

@

1T1 ⫺C

lj A

P0

jˆ1

… jmq ⫹ cm†

jˆ1

0

nX ⫹1

@

1c

lj A

0 1

jˆ1

PT2 ˆ

c!mc

T2 Y

nX ⫹1

@

1T1 ⫺C 0 1T2 ⫺T1 n X @ lj A lj A P0

jˆ1

… jmq ⫹ cm†

jˆ1

jˆ1

.. . 0 @

nX ⫹1

…13†

1c

lj A

0 1

jˆ1

PTi ˆ

c!mc

Ti Y

@

nX ⫹1

1T1 ⫺C 0

lj A

jˆ1

… jmq ⫹ cm†

@

n X

1T2 ⫺T1 0

lj A

…@

n ⫺X j⫹2

1Ti ⫺Ti⫺1

lj A

P0

jˆ1

jˆ1

jˆ1

.. . 0 @ PM ˆ

nX ⫹1

1c

lj A

0 1

jˆ1

c!mc

M Y jˆ1

… jmq ⫹ cm†

@

nX ⫹1 jˆ1

1T1 ⫺C 0

lj A

@

n X jˆ1

1T2 ⫺T1

lj A

……l2 ⫹ l1 †Tn ⫺Tn⫺1 …l1 †M⫺Tn P0

D.C. Lee / Computer Communications 23 (2000) 1069–1083

1077

From Eq. (13), the steady state probabilities Pi depending on threshold values Ti are given by 20

Pi ˆ

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

@

nX ⫹1

1c

lj A

jˆ1

0 @

c!mc nX ⫹1

0⬍iⱕC

P0 1c

lj A

0 1

jˆ1 i Y

c!mc 0 @

nX ⫹1

1c

lj A

@

nX ⫹1

1i⫺C

lj A

jˆ1

8 > 1T1 ⫺C > > > nX ⫹1 < 1 @ A l j T1 > Y > jˆ0 > > … jmq ⫹ cm† :

c!mc

jˆ1

@

nX ⫹1

1c

lj A

c!mc

@

nX ⫹1

C ⫹ T1 ⫹ 1 ⬍ i ⱕ C ⫹ T2

jˆT1 ⫹ 1

.. .

8 > > 1T1 ⫺C 0 1T2 ⫺T1 > > nX ⫹1 n < X 1 T ⫺T n⫺1 n⫺2 @ A @ A l l … l ⫹ l ⫹ l † j j 3 2 1 TY > n⫺1 > jˆ0 jˆ0 > > … jmq ⫹ cm† : 0

jˆ1

0

9 > > 1i⫺T1 >0 > ⫹1 = nX 1 @ lj A P0 i > Y > jˆ0 > … jmq ⫹ cm† > ;

0

jˆ1

0

C ⫹ 1 ⬍ i ⱕ C ⫹ T1

P0

jˆ0

… jmq ⫹ cm†

1c

jˆ1

lj A

0 1

jˆ1

c!mc

Tn Y

@

nX ⫹1

1T1 ⫺C 0

lj A

@

jˆ0

… jmq ⫹ cm†

n X

1T2 ⫺T1

lj A

…l2 ⫹ l1 †Tn ⫺Tn⫺1

jˆ0

jˆ1

8 > > > > < > > > > :

i Y jˆTn ⫺ 1 ⫹ 1

9 > > > > = 1 i⫺Tn⫺1 …l2 ⫹ l1 † P0 > > > > … jmq ⫹ cm† ; 9 > > > > =

1

i Y jˆTn ⫹ 1

…l1 †i⫺Tn P0 > > > > … jmq ⫹ cm† ;

.. .

C ⫹ Tn⫺1 ⫹ 1 ⬍ i ⱕ C ⫹ Tn

C ⫹ Tn ⫹ 1 ⬍ i ⱕ C ⫹ M

…14† P…i† 0

which represents equations from initial probability From Eq. (14), we obtain the initial probability by nth threshold value. The results are given by value to initial probability P…n† 0 0

2

P0…1†

1c

nX ⫹1

0

nX ⫹1

P…1† 0

by the first threshold

1c

@ 6 lj A 1i⫺C @ lj A 0 6 T1 nX ⫹1 6 X jˆ1 jˆ1 1 6 @ ˆ 61 ⫹ lj A ⫹ c i 6 Y c! m c!mc jˆ1 iˆ1 6 … jmq ⫹ cm† 4 jˆ1

2



M X iˆT1 ⫹ 1

0 @

nX ⫹1 jˆ1

1T1 ⫺C

lj A

1 T1 Y jˆ1

2 P0…2†

… jmq ⫹ cm†

6 6 6 6 6 4

3

3⫺1

1i⫺T1 77 7 77 77 @ lj A 77 i 77 Y jˆ1 57 … jmq ⫹ cm† 5 0

1

n X

jˆT1 ⫹ 1

0

1c

nX ⫹1

6 lj A 1T1 ⫺C 0 1T2 ⫺T1 @ 0 6 nX ⫹1 n 6 ⫺1 X jˆ1 6 @ lj A ˆ 6P…1† ⫹@ lj A 6 0 c!mc jˆ1 jˆ1 6 4

3

2 M X iˆT2 ⫹ 1

1 T2 Y

… jm q ⫹ c m †

jˆ1

6 6 6 6 6 4

0 1 i Y

… jmq ⫹ cm†

jˆTn ⫺ 1 ⫹ 1

@

nX ⫺1 jˆ1

3⫺1

1i⫺T2 77 7 77 7 7 lj A 77 77 57 5

.. . 2 P0…n†

0

nX ⫹1

1c

@ 6 lj A 1T1 ⫺C 0 6 nX ⫹1 6 jˆ1 ⫺1 6 ˆ 6P…n⫺1† ⫹@ lj A ……l2 ⫹ l1 †Tn ⫺Tn⫺1 6 0 c!mc jˆ1 6 4

2 M X iˆTn ⫹ 1

1 Tn Y jˆ1

… jm q ⫹ c m †

6 6 6 6 6 4

3

3⫺1

7 77 7 7 1 77 …l1 †i⫺Tn 77 i 7 7 Y 57 … jmq ⫹ cm† 5

jˆTn ⫹ 1

(15)

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D.C. Lee / Computer Communications 23 (2000) 1069–1083

Also, initial probability P0 is obtained by modifying the results of Eq. (15). 2

0

nX ⫹1

1

0

1c

nX ⫹1

3⫺1

@ @ 7 6 0 1i⫺C lj A lj A 7 6 T1 C nX ⫹1 X X 7 6 jˆ1 jˆ1 1 i 7 61 ⫹ @ A ⫹ lj 7 6 c i i Y 7 c! m 6 m i! iˆ1 iˆ1 jˆ0 7 6 … jmq ⫹ cm† 7 6 7 6 jˆ1 7 6 7 6 0 1 c 7 6 3 2 nX ⫹1 7 6 7 6 @ A 0 1 0 1 lj 7 6 T1 ⫺C 6 i⫺T1 7 7 6 T n ⫹ 1 n 7 6 2 X X X jˆ1 1 1 7 6 7 6 @ A @ A 7 6⫹ l l 7 6 j j c 7 6 T i 7 6 1 Y c!m Y 7 6 iˆT1 ⫹ 1 jˆ1 jˆ1 5 4 7 6 … j m ⫹ c m † … j m ⫹ c m † q q 7 6 7 6 jˆT1 ⫹ 1 jˆ1 7 6 7 6 7 6 P0 ˆ 6 ... 7 7 6 7 6 0 1c 7 6 n ⫹ 1 7 X 6 7 6 @ A 0 1T1 ⫺C 0 1T2 ⫺T1 lj 7 6 Tn nX ⫹1 n 7 6 X X jˆ1 1 1 7 6 i⫺T n⫺1 @ A @ lj A … 7 6⫹ … l l ⫹ l † j 2 1 c TY i 7 6 n⫺1 Y c! m 7 6 iˆTn ⫺ 1 ⫹ 1 jˆ1 jˆ1 7 … jmq ⫹ cm† 6 … j m ⫹ c m † q 7 6 jˆTn ⫺ 1 ⫹ 1 7 6 jˆ1 7 6 0 1 7 6 c 7 6 nX ⫹1 7 6 @ A 7 6 0 1 0 1 l j T1 ⫺C T2 ⫺T1 7 6 M nX ⫹1 n 7 6 X X jˆ1 1 1 T ⫺T i⫺T 6⫹ n n⫺1 n 7 @ A @ A … … … l l l ⫹ l † l † 7 6 j j 2 1 1 c Tn i Y 7 6 c!m Y iˆTn ⫹ 1 jˆ1 jˆ1 5 4 … j m ⫹ c m † … jm ⫹ cm† q

q

jˆTn ⫹ 1

jˆ1

…16† From Eqs. (14)–(16), the blocking probabilities nX ⫹1

P…1† b ˆ Prob‰i ˆ MŠ ˆ

…P…j† b ;

!c

0

li

1

iˆ1 M Y

c!mc

@

… jmq ⫹ cm†

1 ⱕ j ⱕ n ⫹ 1† depending on j priorities are given by

nX ⫹1

1T1 0 1T2 ⫺T1 n X ……l1 †M⫺Tn P0 lj A @ lj A

jˆ0

jˆ0

jˆ1 nX ⫹1

P…2† b

!c

li

iˆ1

ˆ Prob‰Tn ⬍ iŠ ˆ

c!mc

8 > 1T2 ⫺T1 > > > nX ⫹1 n < X 1 @ A @ A … l l j j Tn > Y > jˆ0 jˆ0 > > … jmq ⫹ cm† : 0

M X iˆTn ⫹ 1

1T1 0

jˆ1

9 > > > > = 1 i⫺Tn … l † P0 1 i > Y > > > … jmq ⫹ cm† ; …17† jˆT ⫹ 1 n

.. . P…n† b ˆ Prob‰T2 ⬍ iŠ ˆ

T3 X iˆT2 ⫹ 1

Pb…n⫹1† ˆ Prob‰T1 ⬍ iŠ ˆ

M X

Pi ⫹ … ⫹

T2 X iˆT1 ⫹ 1

Pi

iˆTn ⫹ 1

Pi ⫹

T3 X iˆT2 ⫹ 1

Pi ⫹ … ⫹

M X

Pi

iˆTn ⫹ 1

…n⫹1† is the P…1† b is the blocking probability with which the highest priority call l1 occurs when the queue size M is full. Pb blocking probability with which the lowest priority call ln⫹1 occurs when arrives in the queue of the state which is over the first

D.C. Lee / Computer Communications 23 (2000) 1069–1083

1079

threshold value. The total blocking probability P…total† is given by b  nX ⫹1 … ⫹ l1 P…1† † ˆ …ln⫹1 Pb…n⫹1† ⫹ ln P…n† lj P…total† b b ⫹ b

…18†

jˆ1

From Eqs. (14)–(16), the mean waiting times …Wq…j† ; 1 ⱕ j ⱕ n ⫹ 1† depending on j priorities can be written as Eq. (19) using Little’s formula. 0

2 Wq…1†

1c

nX ⫹1

3

@ 1 60 lj A 6 X n jˆ1 1 6 @ lj A…Wq…2† † ⫹ ˆ n⫹1 6 X 6 c!mc 6 jˆ0 lj 4

0

M X

i

iˆTn ⫹ 1

jˆ0

Tn Y

@

nX ⫹1

lj A

@

jˆ0

… jmq ⫹ cm†

n X

1T2 ⫺T1

lj A

60n ⫺ 1 1 1 6 6@ X A …3† ˆ X lj …Wq † ⫹ 6 n 6 lj 4 jˆ0

nX ⫹1

…

jˆTn ⫹ 1

3

!c

li

0

Tn X

iˆ1

c!mc

7 7 7 i⫺Tn 7 l † P … 1 0 7 i Y 7 5 … jmq ⫹ cm† 1

jˆ0

jˆ1

2 Wq…2†

1

1T1 ⫺C 0

i

iˆTn ⫺ 1 ⫹ 1

jˆ0

1 TY n⫺1

@

… jmq ⫹ cm†

nX ⫹1

1T1 ⫺C

lj A

1

…

i Y

jˆ0

…l2 ⫹ l1 †

… jm q ⫹ c m †

j⫺Tn⫺1

7 7 7 P0 7 7 5

jˆTn ⫺ 1 ⫹ 1

jˆ1

.. . 2 Wq…n†

6 6 1 6 ˆ 6l1 …Wq…n⫹1† † ⫹ …l2 ⫹ l1 † 6 4

nX ⫹1

3

!c

li

c!mc

0

T2 X

iˆ1

i

iˆT1 ⫹ 1

1 T1 Y

… jmq ⫹ cm†

jˆ1 nX ⫹1

Wq…n⫹1†

1 ˆ l1

!c

li

iˆ1

c!mc

T1 X iˆ0

0 i

1 i Y

… jm q ⫹ c m †

@

nX ⫹1

@

nX ⫹1 jˆ0

1 T1

lj A

0 1 i Y

… jm q ⫹ c m †

@

n X jˆ1

1j⫺T1

lj A

7 7 7 P0 7 7 5

jˆT1 ⫹ 1

1i

lj A P0

(19)

jˆ0

jˆ1

Here, Wq…1† is the mean waiting time that the highest priority call l1 waits in the queue by queue size M and Wq…n⫹1† is the mean waiting time that the lowest priority call ln⫹1 waits in the queue by the first threshold value T1.

4. Numerical results In this section, we compute the blocking probabilities and the mean waiting time for the two proposed queueing schemes under priority policy. We compare the numerical results of the two proposed schemes with those of the previous scheme and verify the numerical results by simulation. The simulation model can be characterized as follows: (a) Simulation is performed by SIMSCRIPT II. 5 using discrete time event-scheduling. (b) Modules of simulation have PREAMBLE, MAIN, INITIAL, ARRIVAL, DEPARTURE, STOP. SIM. (c) Arrival distribution is Poisson distribution and service time distribution is exponentially distributed. (d) Arriving calls have106~107 calls in a cell. In Fig. 6, the two proposed schemes are compared to the

previous scheme with two types of traffic (voice call and data call) in aspect of the blocking probability. Numerical results are given for the following parameters: high priority call l 1 ˆ voice call, low priority call l 2 ˆ data call, queue size M ˆ 40; threshold value T ˆ 20; traffic intensity r ˆ li =cm ˆ li =M mq ; i ˆ 1; 2 and number of channel percell C ˆ 40: Compared to the previous scheme, the proposed scheme I shows the same performance in terms of the blocking probability because the scheme I is based on HOL priority with two types of traffic. However, the proposed scheme II shows a better performance. In Fig. 7, the two proposed schemes are also compared to each other in aspect of the blocking probability with four class calls vs. traffic intensity r: Consider four class calls as image call l1 ;voice call l2 ; delay sensitive data call l3 ; and data call l4 : Image call l1 is the highest priority call and data call l4 is the lowest priority call. We also assume the following: queue size M is 40 and threshold values T1,T2,T3 are 10, 20, 30, respectively, which have equivalent intervals and mixing ratios of priority traffic l1 ; l2 ; l3 ; l4 are 0.1, 0.1, 0.1, 0.1, respectively and

1080

D.C. Lee / Computer Communications 23 (2000) 1069–1083

Fig. 6. Comparison of blocking probabilities with two class calls vs. traffic intensity …r†.

channel number C is 40. The scheme II has lower blocking probability than scheme I under same priority. It is because each call shares the queue partially in scheme II where it waits in the queue by only each threshold value in scheme I. In scheme I, blocking probabilities with four class calls have

nearly same results irrespective of priorities but the image call l1 has also the lowest blocking probability. It is because each call has the same intervals of threshold value and mixing ratio of priority traffic in the queue. In scheme II, the image call l1 has the lowest blocking probability, where

Fig. 7. Blocking probabilities with four class calls vs. traffic intensity (r ).

D.C. Lee / Computer Communications 23 (2000) 1069–1083

1081

Fig. 8. Blocking probabilities with four class calls vs. traffic intensity (r ).

the other calls have almost same results of blocking probability. It implies that the image call l1 shares the queue of whose size is M. Fig. 8 shows the blocking probability vs. traffic intensity under variable intervals of threshold value T1 : T2 : T3 ˆ

20 : 30 : 35 and variable mixing ratio of traffic l1 : l2 : l3 : l4 ˆ 0:1 : 0:5 : 0:5 : 0:5. In scheme I, the blocking probabilities of voice call l2 ; D.S data call l3 ; and data call l4 have almost same results but the image call has the lowest blocking probability. It is because the

Fig. 9. Mean waiting time with four class calls vs. call arrival rate (l ).

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D.C. Lee / Computer Communications 23 (2000) 1069–1083

Fig. 10. Blocking probabilities with four class calls vs. the first threshold value (T1).

image call has larger interval of threshold value than the other calls and has the lowest traffic load 0.1. In scheme II, blocking probabilities of call are similar to that in Fig. 7 but three class calls l2 ; l3 ; l4 have the higher blocking probabilities than image call l1 . Fig. 9 shows the mean waiting time Wq vs. arrival rate l ˆ l1 ˆ l2 ˆ l3 ˆ l4 under same performance parameters in Fig. 7. The scheme I has shorter mean waiting time than scheme II under same priority condition. It is because image call waits in the queue by only first threshold value T1 in scheme I, where the call waits in the queue by queue size M in scheme II. In scheme I, the higher the priority is, the shorter is the mean waiting time. In scheme II, the higher the priority call is, the longer is the mean waiting time. Fig. 10 shows the blocking probability vs. the first threshold value T1, where queue size M is 40 and traffic intensity r is 0.5 and the mixing ratios of traffic l1 ; l2 ; l3 ; l4 are 0.1, 0.1, 0.1, 0.1, respectively, and the second threshold value T2 and the third threshold value T3 are 20, 30, respectively. scheme II has also lower blocking probability than scheme I under same priority condition. In scheme I, whenever the first threshold value T1 is increased, blocking probability of data call l4 is decreased. When the first threshold value T1 is close to the second threshold value T2, it shows that blocking probabilies of two class calls have same result. Because the first threshold value of data call l4 has same bandwidth of

the threshold value T2 of D.S data call l3 : In scheme II, whenever the first threshold value T1 is also increased, blocking probability of data call l4 is decreased and has the same effect as that of scheme I. Finally, the proposed schemes will achieve a better performance by changing variable performance indexes: threshold values, queue size, mixing ratio of traffic, and priorities.

5. Conclusions We propose the queueing strategies with multiple priority calls to reduce the block of channel allocation in multiservice personal communications services. In the two proposed schemes, we present the efficient queueing schemes and show the analytic model of …n ⫹ 1† class calls. Compared to the previous scheme with two class calls, the proposed scheme I have the same performance to that of previous scheme in terms of blocking probability. However, the proposed scheme II shows an improved performance. In the two proposed schemes, scheme II has a better performance than scheme I with respect to the blocking probabilities with four class calls, where scheme I has better results than scheme II in terms of the mean waiting time. The proposed schemes can be flexibly applied to multiservice personal communications services.

D.C. Lee / Computer Communications 23 (2000) 1069–1083

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