Modelling Service Time Distribution in Cellular Networks Using Phase ...

Modelling Service Time Distribution in Cellular Networks Using Phase-Type Service Distributions 1

Aruna Jayasuriya1, David Green2, John Asenstorfer1 Institute for Telecommunication Research, Cooperative Research Centre for Satellite Systems, University of South Australia, Mawson Lakes, SA 5095, Australia. 2 Teletraffic Research Centre, University of Adelaide, Adelaide SA 5000, Australia. Tel: +61 8 8302 3879 Fax: +61 8 8302 3873 [email protected]

Abstract --– Third generation mobile communication networks are designed to provide a variety of high data rate services with higher Quality of Service (QoS) than second-generation systems. Handover becomes one of the major problems in such a mobile environment as it is of utmost importance to provide a higher guarantee that the users are able to continue their service during the entire length of the transmission, without it being blocked during the handover or loss of quality or data. Careful dimensioning of the network and the underlying teletraffic analysis plays a major role in determining the various Grade of Service (GoS) parameters that service providers can provide at various network loads. The channel holding time of a cell is one of the major parameters that needs to be accurately modelled in the teletraffic analysis. This paper focuses on using Phase-type distributions of Generalised Erlang form to model channel holding time in a mobile environment. We also present the Quasi-BirthDeath process, which characterises the queuing models with Generalised Erlang service and exponential interarrival distributions. Further we investigate the use of channels exclusively reserved for handover users to improve the handover performance.

I. INTRODUCTION One of the major problems that needs to be addressed in mobile communication networks is the continuity of a service during a handover without any data loss, as the user moves from cell to cell. This is called seamless handover [1]. Most of the new services, such as video-conferencing and e-commerce applications require a higher guarantee of being able to continue the service across cell boundaries, than is provided by current mobile services [1]. The blocking probability encountered at handover is an important Grade of Service (GoS) parameter for mobile users. Due to the fact that higher GoS is required from future networks, it is of utmost importance to carefully dimension the network to provide the guaranteed GoS levels. During this dimensioning phase, the number of channels needed to guarantee acceptable blocking probabilities for users will be estimated under various network loads. Furthermore the size of the cells will be selected to provide enough resources to service the predicted amount of traffic in a particular area. Section II briefly describes some of the techniques that were used to model mobile communication networks and some of their drawbacks. This is followed by a statement of a model for the channel occupancy time in mobile networks. Section III introduces the Phase-Type (PH) distributions, which are the distributions that we used to characterise channel occupancy

time in this study, followed by the phase-type approximation for the channel occupancy time. Section IV briefly describes the Quasi-Birth-Death (QBD) processes, which result from multi-server queue with phasetype interarrival and service distributions. We then describe how to derive the QBD process for the mobile cell server under investigation. Section VI introduces the use of channels reserved exclusively for handover users to improve the handover performance. This is followed by the results obtained in this study, which shows that the optimum terrestrial system capacity that can be achieved is 80% of the total capacity, with a single channel reserved exclusively for handover users. Section VIII concludes the paper with closing remarks and some future research directions. II. TELETRAFFIC ANALYSIS OF MOBILE NETWORKS Channel holding (occupancy) time is a very important parameter in analysing mobile communication networks. Previously, analyses used exponential distributions to model channel holding time for the sake of tractability [2]. But experimental data showed that actual channel occupancy distributions are significantly different from exponential distributions used in these analyses [3]. Various other methods such as the Sum of Hyperexponential (SOHYP) [4] and general distributions [5] were proposed but the complexity of the analysis has increased considerably with these techniques. A. Channel Holding Time in Mobile Communication Networks Users in mobile communication networks can be grouped into two major categories according to the time they occupy resources in a particular cell; namely new and handover users. Assuming call duration is exponentially distributed with parameter µ Fang and Chlamatac [5] showed that the Laplace transform of the distribution of service time for new users, f*nh , and the Laplace transform of the distribution of service time for handover users, f*hh, are given by s µ (1) + f * nh ( s ) = f * r (s + µ ) , s+µ s+µ where f * r ( s) is the Laplace transform of the distribution of time between the start of a call and the first handover; and f * hh ( s ) =

µ s + f * c (s + µ ) , s+µ s+µ

(2)

where f c * (s ) is the Laplace transform of the distribution of cell residue time. Assuming new and handover users have exponential interarrival time distributions with parameters λn and λh, then the channel holding time, t, in a cell can be expressed as [5] λn λh (3) f *t ( s) = f *nh ( s) + f *hh ( s) . λn + λh

λn + λh

where f t * ( s ) is the Laplace transform of the distribution of t. This result shows that the channel holding time in a cellular network can be expressed as a combination and convolution of several exponential distributions. In this research we approximate the channel holding time using Phase-type distributions. Phase-type (PH) distributions represent a set of distributions that are combinations and convolutions of different exponential distributions. III. PHASE-TYPE DISTRIBUTIONS Phase-type distributions can be used to approximate virtually any renewal process, with the dimensionality of the phase-type distribution increasing with the complexity of the particular process being modelled [6,7]. Any continuous distribution, X, on [ 0 , ∞ ) , which can be obtained as the distribution of time until absorption in a continuous time finite state Markov chain (which has a single absorbing state into which absorption is certain) is said to be of phase-type. The initial state may be chosen randomly and all states are transient except the absorbing state [6-7]. From the above definition we can define a Markov process on the states {0,1…,n} with initial probability vector (τ 0 , 2 ) and infinitesimal generator or rate transition matrix 0 0  Q= , t T 

(4)

where 2 is a row vector of size n, T is an n × n matrix, and t is a column vector of size n. As Q is a rate transition matrix of a Markov process [6], Te + t = 0 , where e is a column vector of 1’s. Further elements of T and t satisfy the following, Tii < 0 , t i ≥ 0 , Tij ≥ 0 for 1 ≤ i ≠ j ≥ n For an initial probability vector (τ 0 , 2 ) ; we also have that

τ 0 + 2H = 0 . The above phase-type distribution is denoted as PH(τ,T). A Modelling Channel Occupancy Time Using Phase-Type Distributions In this study it was intended to characterise the channel holding time of cellular users using phase-type distributions. A mobile network environment was set up in OPNET network modeller to obtain typical user channel holding times in mobile environments. This network consists of four cells with radii of 2 kms. A Total of 500 users were randomly distributed within

the network. These users were given different mobility patterns to simulate the mobility patterns followed by cellular users in urban areas. This was simulated for a 10hour period. Interarrival times for handover and new users in different cells, and the times new and handover users spend in each cell were recorded. The times new and handover users spent in each cells were used to create the probability distribution for channel holding time for the above network. Fang and Chlamatac show that an Erlang distribution is much better suited than an exponential distribution to characterise the channel holding time in cellular networks [5]. Therefore in this study a phase-type distribution of Generalised Erlang form was used to approximate the channel holding time distribution obtained from the simulation data. The Expectation Maximisation (EM) method was used to find the parameters of the Generalised Erlang distribution that best fits the experimental data [8]. Although the accuracy of the approximation increases with the dimensionality of the distributions, it also increases the complexity of the analysis. Therefore a distribution of two phases was selected in this study as it was observed that the error between the actual and the fitted distribution does not improve significantly after the number of phases were increased beyond 2. Parameters of the PH distribution selected are given below. - 1.68567 T = 0 

1.68567  − 0.0856

and 2 = [1 0]

IV QUEUING MODEL FOR CHANNELS OF CELLULAR NETWORKS A cell in a mobile network having n channels can be modelled as a PH/PH/n/n queue. The rate transition matrix for this queue has a characteristic block tri-diagonal form. As both new and handover arrivals are assumed to be exponential in cellular networks, the collective channels in mobile cells can be modelled as a M/PH/n/n queue with the rate transition matrix, Q, of the form given in (5). Assuming an exponential arrival distribution with parameter λ and phase-type service distribution with parameters PH(τ,T), matrices B 0 , B1 and B 2 are given in (6).

 .  .  .      .    A  

 B1 B0 0 0  B2 A11 A01 0 0 A 22 A12 A02   Q= A2i A1i  ⋅ ⋅ ⋅ ⋅  ⋅ ⋅ ⋅ 0 ⋅ ⋅ ⋅ ⋅ ⋅ 0 

0i

. . .

. . .

. .

. .



.

A2,n−1 A1,n−1 A0,n−1 0 A2,n A1,n

            

(5)

B1 = λ ,

B0 = λ ⊗ τ ,

B2 = t

(6) k-1,m-k+1

Where symbol ⊗ represents the Kronecker product between the two matrices. However, the construction of A0i , A1i and A2i for general phase-type distributions are very complex and are beyond the scope of this article.

A simplification can be made to the Q matrix by observing some properties of the Erlang distribution and the behaviour of servers in mobile cells. In servers with Erlang distributions, the users always start the service in the first phase and move on to the next phase with probability 1 once the sojourn in that phase is over. Users who finish the sojourn in the last phase depart the system. When there are m users in the system it is irrelevant which user of these m users is at which server and similarly who finishes the service first. This leads us to combine all the users in the same service phase to a single server with the service rate equivalent to the combined rate of all the servers in that phase. Using this simplification and a service distribution of two phases, the state of the system can be represented by (n1, n2), where n1 is the number of users in phase 1 and n2 is the number of users in phase 2. In order to obtain the characteristic tri-diagonal form of the Q matrix, it is necessary to perform a linear ordering of the states of this two-dimensional continuous time Markov chain. In this case it is the simple ordering {0, (1,0), (0,1), (2,0), (1,1), (0,2),...., (n,0), (n-1,1),..., (1,n-1), (0, n)}. We can define levels where level m is the combination of all states when the number of users in the system are m and there are m+1 possible states at level m. The events, which change the state of the system and rates of leaving the current state at those events are shown in table 1. Fig. 1 shows the transitions listed in table 1. The rate transition matrix for the M/PH/n/n QBD process can be constructed by observing the transitions between states given in fig. 1 and using table 1 to get the transition rates between different states. TABLE 1: TRANSITION RATES BETWEEN DIFFERENT STATES

From 0 (n1,n2) (n1,n2) (n1,n2)

To 1,0 (n1+1,n2) (n1,n2-1) (n1-1,n2+1)

Rate λ λ n2µ2 n1µ1

Event Arrival arrival departure phase change

n1 ≥ 1 n2 ≥1 n2,,n1 ≥ 1

(m-k) µ 2

k µ1

λ

λ k,m-k

k-1,m-k

V M/PH/N/N SERVER WITH A GENERALISED ERLANG SERVICE DISTRIBUTION A drawback in using the above model to represent the mobile network cells is the fact that size of the rate transition matrix increases exponentially with the number of channels available in a cell [9]. Future mobile networks intend to provide a potentially large number of channels per cell to support the high data rate services that will be available in the future. Therefore the methods available in [9] cannot be used to find the blocking probabilities experienced by new and handover users in the mobile network environment.

k,m-k+1

(m-k) µ 2

k+1,m-k

(k+1) µ 1

k+1,m-k-1

k,m-k-1 m-1 users in the system

m users in the system

m+1 users in the system

Fig. 1. State transitions for M/PH/n/n queue. The ordering given above allows us to generate the Q matrix of the form (5) and the matrices A2i , A1i , A0i are given below. For T = - µ1 µ1   0 

 − µ2 

A0i is a matrix of size (i + 1) × (i + 2)

              0

  A0i =    

(7)

A1i is a matrix of size (i + 1) × (i + 1)



. . . −ζ0 iµ1    i − − ζ µ ( 1 ) . 1 1     .   −ζk (i −k)µ1 A1i =  .   .    µ1   .  −ζi  

 



where for k = 0,1,..,i ζ k = λ + (i − k)µ1 + kµ2 . A2i is a matrix of size (i + 1) × i

      2       i 

0   2 A2i =     

(8)

   

(9)

(10)

2

2

An extra boundary condition is required at level n, where n is the total number of channels available in the system. New arrivals are not accepted to the system at this level thus making the arrival rate zero. Therefore the ζ k s for k = 0,1,..,n in the matrix A1n have the following form.

ζ k = (n − k)µ1 + kµ2

(11)

Stationary probabilities, x, of this QBD process can be found by solving (12), which states that effective probability flow out of a state is balanced by the probability flow into the state.

xQ = 0

(12)

n

Pblock _ ho = p (n) ,

By taking the transpose Q , Q and rearranging the equation, Q x=0,

(13)

the stationary probabilities of this system are given by the nullspace of Q t , which was found using Singular Value Decomposition (SVD) methods. Once the stationary probabilities, x , of the system have been obtained where xe = 1 , where e is a column vector of 1’s, we can find the probability of having i users in the system p (i ) as follows, p (0) = x (1) ,

p (i ) =

S i +1+ i

∑ x( j )

for 1 ≤ i ≤ n

(14)

j = S i +1 i

where

Si =

∑ j = i(i + 1) / 2 j =1

If the system does not distinguish between new and handover users no priority will be given to one class over the other. In such a system the blocking probability for any user, Pblock will be given by, (15) Pblock = p (n) and the average load of the system at this blocking probability is given by n

L=

∑ ip(i)

(16)

i =1

VI IMPROVING HANDOVER PERFORMANCE As continuing an existing call is more important than accepting a new user into the network, handover users should have priority over new users during resource allocation. In this study, priority for handover users was achieved by reserving a certain number of channels exclusively for handover users [10]. In this scenario only handover users are accepted to the system when the total load exceeds a certain threshold. This is modelled by varying the interarrival parameter with the number of users in the system and blocking probabilities are calculated by adding the probabilities of the states where particular types of users are not allowed to visit. When there are C HO channels exclusively reserved for handover users, in equations (7), (8) and (9), the value for λ has to be modified as follows. For i = 0,1,..,n ; If i ≥ n − C HO , λ = λ h Otherwise , λ = λ n + λ h Where n is the total number of channels available in the system and, λ n and λ h are the parameters of exponential interarrival rates for new and handover users respectively. Blocking probabilities for the handover and new users, Pblock _ ho and Pblock _ new can be calculated as follows.

∑ p( j )

(17)

j =n −C HO

t

t

Pblock _ new =

VII RESULTS The results obtained through the queuing models abovedescribed are presented in this section. In particular analysis has been performed to determine the blocking probabilities that handover and new users experience with channels exclusively allocated for handover users. In these analyses it was assumed that a typical terrestrial cell has 50 channels. It was also assumed that the GoS blocking probabilities for new and handover users must be less than 5% and 2% respectively [1]. The operating point of the system is taken to be the maximum capacity that leads to the blocking probabilities less than or equal to the above stated values. Further it was observed that with an average call length of 3 minutes users make 0.8 handovers per call. Therefore we selected,

λ h = 0.8λ n . A Blocking Probabilities With Channels Reserved for Handover Users Fig. 2 shows the performance of the system with various numbers of channels reserved exclusively for handover users. In the case where no channels are reserved for handover users, to meet the GoS for handover users (2% blocking) 79 % (or 39 channels) of the total system capacity is found to be the operating point of the system. In this case new users enjoy the same blocking probability of 2% which is 3% less than their guaranteed GoS blocking probability. Therefore mechanisms have to be used to distinguish between the two types of users so that the operating capacity can be maximised while guaranteeing the GoS. This can be achieved through reserving a certain portion of the channels exclusively for handover users. Fig. 2 shows the results obtained with this strategy. It shows that the operating capacity can be increased to 80% when 1 channel is reserved for handover users. Subsequent allocation of more handover only channels leads to a drop in the operating capacity. This is due to the fact that with higher number of handover only channels, new users have only a smaller number of channels to contend, resulting that the GoS blocking probabilities can only be met at lighter network loads. Fig.3 shows the behaviour of operating capacity with the number of channels exclusively reserved for handover users. This shows that the operating capacity drops by 7% when the number of handover-only channels are increased from 1 to 4. In terms of the system capacity, increase in handover-only channels from 1 to 4 corresponds to reserving 6% more of the total capacity exclusively to handover users. Therefore the optimum operating condition is to reserve a single channel for handover users, which results in the

maximum capacity while satisfying GoS for both new and handover users. Fig. 2 shows that the handover users experience very low blocking probabilities (0.0032 with 4 handover only channels) at the GoS operating points with higher number of handover only channels. In third generation networks these operating conditions may be of interest due to the need of very low handover blocking probabilities for services such as video conference and e-commerce applications. VIII CONCLUSIONS AND FUTURE DIRECTIONS In this study we modelled the channel holding time of a cellular network as a generalised Erlang distribution. We also generated the block tri-diagonal rate transition matrix for the M/PH/n/n queue, which we used to model the collective channels of a single cell in a cellular network. Also we investigated the performance of the system when handover user

Blocking Probabilities for new and handover users

0.08

0.07 4 3

0.06

2

1

0.05

In third generation networks some services may require very small blocking probabilities, lower than that guaranteed in current systems. These results show that this can be achieved by lowering the operating point considerably until the required GoS is met. Another possible strategy is to make the handover decision making strategy more intelligent in order to support GoS required by different types of users. One of the techniques that can be utilised for system improvement is the use of mobility prediction to estimate the probability of a user requiring a handover to a particular cell and dynamically reserving a certain portion of the resources from these cells. The authors are currently investigating the use of intelligent systems to improve the handover procedures through movement predictions and early reservations [10]. ACKNOWLEGMENT This work was supported by the Australian Government’s Cooperative Research Centre Program. REFERENCES

channel reserved for handover = 0

0.03

1

0.02

2 3

0.01

0

72

74

76 78 System capacity or utilisation (%)

[1] [2]

handover users new users 80

82

Fig. 2. Blocking probabilities all users with channels exclusively reserved for handover users. 81

80

79

78

Operating capacity

These results show that the system can operate at a maximum capacity of 80 % while guaranteeing that the new users will suffer blocking probability of less than 0.05 while handover users will experience blocking probabilities less than 0.02. This operating condition is achieved when only 1 terrestrial channel is reserved exclusively for handover users.

0.04

4

77

76

75

74

73

72

were given priority in the system through reserving channels exclusively for handover users.

0

1

2

3

Number of handover only channels ( total number of channels = 50)

Fig. 3. Operating capacity with number of channels allocated for handover users.

4

UMTS Task Force Report, tech. rep., Mar. 1996. D. Hong and S. S. Rappaport, “Traffic Model and Performance Analysis for Cellular Mobile Radio Telephone systems with Prioritized and Non-Prioritized Handoff Producers,” IEEE Trans. Veh. Tech., pp. 77-92, Aug. 1986. [3] C. Jedrzycki and V. Leung, “Probability Distribution of Channel Holding Time in Cellular Telephony Systems”, Proc. IEEE Veh. Technol. Conf., pp. 247-251, May 1996. [4] P. Orlik and S. Rappaport, “A Model for Teletraffic Performance and Channel Holding Time Characterization in Wireless Cellular Communication with General Session and Dwell Time Distributions”, in IEEE J. Sele. Areas of Commun., vol. 16, pp. 788-803, June 1998. [5] Y. Fang and I. Chlamtac, “Teletraffic Analysis and Mobility Modelling of PCS Networks,'' in IEEE Trans. Commun., vol. 47, pp. 1062-1072, July 1999. [6] M. Neuts, Matrix-Geometric Solutions in Stochastic Models, John Hopkins Publishers, pp. 41-129, 206-215 1981. [7] G. Latouche, Introduction to Matrix Analytic Methods in Stochastic Modelling, ASA-SIAM publishers, pp. 3-50, 1996. [8] S. Asmussen, O. Nerman, and M. Olsson, “Fitting Phase-type Distributions via the EM algorithm”, Scandinavian Journal of Statistics, vol 23, pp 419-441, 1996. [9] V. Ramaswami and D. Lucantoni, “Algorithms for multi-server queue with phase-type service”, Comm. Statistic Stochastic Models, vol. 1, no. 3, pp. 393-417, 1985 [10] A. Jayasuriya et. al., Protocol for Internetwork Handover between Terrestrial UMTS and Satellite UMTS networks, Using Intelligent Systems, Proc. IEEE Int. workshop on Intelligent signal Process. and Commun. Systems, pp. 417-421, Nov. 1998.