blocked and hand-off calls lost are presented for any general call arrival distribution in ... no product form solution for a multi-cell analysis exists, and the resulting ...
Approximation Models of Wireless Cellular Networks Using Moment Matching K. Mitchell, K. Sohraby, A. van de Liefvoort and J. Place University of Missouri – Kansas City Computer Science Telecommunications Program 5100 Rockhill Kansas City, MO 64116
Abstract—In this paper we present an analytical model for micro- and pico-cell wireless networks for any arbitrary topology in a high mobility environment. We introduce an approximation technique which uses a single cell decomposition analysis which incorporates moment matching of hand-off processes into the cell. The approximation technique is novel in that it can provide close approximations for non-Poisson arrival traffic and it is easily parallelized. Performance measures such as new calls blocked and hand-off calls lost are presented for any general call arrival distribution in a non-homogeneous traffic environment. We produce some numerical examples for some simple topologies with varying mobility for several call arrival distributions and compare our results to those from simulation studies. Keywords—Pico-cell, wireless networks, moment matching
I. I NTRODUCTION The design of wireless cellular networks requires a thorough understanding of traffic characteristics incorporating mobility throughout the network. Cellular networks must be designed with adequate capacity in order to provide an acceptable Quality of Service (QoS). At the call level, QoS in a cellular network is generally characterized by new call blocking ( PBi ), hand-off blocking (PHij ) [10], [11], forced-termination, and dropout. New call blocking occurs when no free channel is available in the cell where the mobile subscriber initiates the call. Hand-off blocking occurs when a subscriber enters a new cell while a call is in progress and there is no available channel to service the call. Hand-off blocking is considered to be more serious than new call blocking. In order to minimize blocking, an increase in capacity can be achieved by increasing the number of channels in a cell, reducing the cell size, or introducing efficient power control algorithms. New generations of personal communications networks supporting multimedia traffic are expected to have small cell sizes, resulting in higher hand-off rates between cells than those of traditional cellular networks. Analytic models for traditional cellular networks generally involve a single cell analysis based on Poisson hand-off processes [1], [3], [7]. However, for micro- or pico- cellular networks where high mobility rates are anticipated, these models do not provide accurate performance measurements [10], [11]. Therefore, models which provide a multiple cell analysis with an arbitrary mobility pattern must be developed. Unfortunately, no product form solution for a multi-cell analysis exists, and the resulting state space explosion prohibits an exact analysis [4], [18]. Clearly numerical approximation techniques must be employed. The fixed-point approximation (FPA) used by Kelly [4] for blocking provides a good approximation for large networks with low mobility. However, FPA is based on the assumption
that all arrivals are Poisson and independent. The Poisson assumption makes it impossible to distinguish between new arrival traffic and hand-off traffic [10], [11]. In this paper we present two novel approaches to modeling cellular networks. The first approach is an approximation technique which is based on a generalization of the FPA to include non-Poisson new call and hand-off arrival processes. Using techniques presented by Lipsky et al. [6], we derive the interdeparture distribution of the cell hand-off process. Using an algorithm by Van de Liefvoort [14], the hand-off arrival process into a cell is characterized as a matrix exponential marginal distribution created by matching an arbitrary number of moments of the hand-off departure processes of the adjacent cells. Our findings show that matching three moments provides sufficient accuracy, and this can be done in most cases with a secondorder matrix representation. The second approach addresses the problem of convergence. Both Sidi [10] and Sohraby [11] observed that the FPA does not converge to the exact solution in small networks with high mobility rates because one subscriber may repeatedly visit the same cell. These models assume random mobility. However, we find that in many underlying topologies, subscribers do not travel in a random fashion. Subscribers move along fixed paths which may be roadways, hallways, commuter rail lines, walking paths, etc., from a starting point to a destination point, and may pass through a single or many cells depending on the mobility rate. At any time along this path a user may begin or terminate a call. For small networks with a linear topology, we find that traffic moving in one direction is independent of traffic moving in the opposite direction. In order to model the two independent streams, we separate the traffic into two feedforward networks each represented by a separate class. Each class is a traffic flow which shares the same resource (channels). Our method is easily parallizable, as calculations for the departure process of each cell can be carried out in parallel with an exchange of moments occurring at each iteration of the FPA. This allows for scalability not possible with simulation models. The rest of this paper is organized as follows: In section II of this paper, we present the Linear Algebra Queueing Theory (LAQT) techniques used in our performance model. In section III, we present the derivation and solution for new call blocking probabilities and hand-off blocking probabilities in a linear network. We then generalize the model to include mobility in two dimensions. In section IV we produce numerical results for new calls blocked and hand-off calls lost for several call arrival
distributions and section V concludes this paper.
B=
II. LAQT T ECHNIQUES A. Matrix Exponential Distribution A matrix exponential (ME) distribution is defined as a probability distribution with representation (p; B ; e), i.e.,
F (t) = 1 p exp ( B t) e0 ; t 0;
E [X n ℄ = n!pV n e0 ;
(2)
where V is the inverse of B . The class of matrix exponential distributions is identical to the class of distributions that possess a rational LaplaceStieltjes transform. Given a rational Laplace-Stieltjes transform for a density function f (t) of order m in the form Pm
=0 Pim
1 si i
i i=0 i s
;
(3)
one mapping of the coefficients i and i to (p; B ; e) given in [15] is p = 00 ; 01 ; ; ( 1)m 1 m 0 1 2
; 0; .. .;
6 6
B = 66 4 (
1)
m 1 0 ;
; 0; .. .;
0
1
(
1)
..
m 2 1 ; 2
1
3
7 7 .. 7 : 4 . 5 6
e0 = 66
..
0
.. .
.
.;
;
1
(
1)
0 m 1
3 7 7 7 7 5
0
(4)
0
Such a representation is referred to as companion normal form. A representation is not unique. If F (t) has representation 1 BX ; X 1 e) is also a representa(p; B ; e), then (pX ; X tion for any non-singular matrix X . The only limitations on (p; B ; e) stem from the requirement that F (t) must form a distribution function. As there are no prescribed structural or domain restrictions on the components (p; B ; e), one can choose a physically based representation, or an algorithmic representation. For example, consider the Erlang- 2 distribution. A phasetype representation, where B = T [8], appears in LAQT as follows,
p=
;
1
0
; 0;
2
2 2
e0 =
1
1
:
(5)
To illustrate that a representation is not unique, applying equation (4) to the Laplace-Stieltjes transform produces the companion canonical form
p=
(1)
where p is the starting operator for the process, B is the process rate operator, and e0 is a summing operator [5]. The minus sign in exp ( B t) represents a natural generalization from a scalar exponential process to a vector process. The order of the representation is indicated by the dimension of the the matrix B , and the degree of the distribution F (t) is the minimal order of all its representations. The n-th moment of the matrix exponential distribution is given by
f (s) =
B=
; 2 4 ; 0
1
;
1
0
e0 =
4
1 0
:
(6)
Although the class of second degree matrix exponential distributions is equivalent to that of physically based (phasetype) distributions, third and higher degree distributions may not have physical representations. For example, the density f (t) = 8 sin2 (t)exp( 2t), with Laplace-Stieltjes transform f (s) = s3 +6s216 +16s+16 , is not a phase-type distribution, but it does have a representation in LAQT using equation (4). The above described distribution has a squared coefficient of variation ( 2 ) of 1/4, which is smaller than what can be represented by any phase-type distribution of the same degree. Thus, the use of matrix exponential distributions allows for a greater range in representing a sequence of moments for a given order of representation. When incorporating matrix exponential distributions in performance models, any representation can be chosen, as LAQT solution methods are not dependent on process representation. The true power of LAQT is the ability to choose a purely algorithmic representation. This creates a great deal of freedom in the algebraic manipulation of these processes as demonstrated in the next section describing the moment matching process. B. Constructing a Matrix Exponential Distribution Using Moment Matching Suppose we have a set of power moments from a continuous R distribution F (t) such that E [X n ℄ = tn dF (t), n = 0; 1; : : : and we want to determine F (t) from the set of moments. This is a variation of the famous Stieltjes Moment Problem [12], [13] in which a continuous distribution F (t) is re-created from its moments. The Stieltjes Moment Problem has not yet been solved for the general case. By exploiting the algebraic representation of the matrix exponential distribution and extending work by Gragg and Lindquist [2], Van de Liefvoort [14] has developed an algorithm which produces a matrix exponential distribution (p; B ; e) from a set of moments E [X n ℄. Define
rn =
E [X n℄ n!
(7)
as the set of normalized or reduced moments of the distribution. Applying Van de Liefvoort’s algorithm to the first three
where
reduced moments yields
p= 2
;
1
r1 ;
B 1=4
r1 r13 2r1 r2 +r3
r12 r2 ; r1
e0 =
0
1
r12
5
r2
(8)
We use the above explicit representation in our model of the hand-off arrival distribution into a cell. Van de Liefvoort’s algorithm returns the matrix exponential representation of a distribution which has a rational LaplaceStieltjes transform and has the prescribed moments. There are occasions when the returned function is not a probability distribution function (pdf). This can occur if only a finite sequence of moments is specified or if they are corrupted by noise. In general, the power moments (n = E [X n ℄) of a given sequence must satisfy [17]
p
p
p
1 2 3 3 4 4 : : : : For second degree distributions, the range of the second and third moments are more limited than the above equation suggests. The second moment is bound such that 2 1=2. For 2 1=2 1, the third moment is bounded on both sides 1 2
2
3
For 2
1+
2
1
2 (r1 )3 r3 2 (r1 )3 :
p
2
2
(9) , the third moment is bounded only from below
1
r3 1=4( 2 + 1)2 (r13 );
(10)
where rn is defined by equation (7). If the squared coefficient of variation or the third moment falls outside of these bounds, a higher-order representation is required. If 2 = 1, B will become singular. To avoid this situation, either the process is assumed to be Poisson or a higher order representation to capture the third and higher moments may be required. For a third-order representation, the first five moments are mapped into (p; B ; e)
p= 2 6
6 B 1 = 66 4
1
r1 ; r12 r2 ; r1
;
0
0
0
r2 r23 2r1 r2 r3 +r32 +r4 r12 r4 r2 ; 2 2 (r1 r2 ) r1 2
e0 = 4
1 0 0
=
r24 r1 + 3r22 r12 r3 2r1 r2 r32 2r2 r13 r4 + 2r22 r1 r4 r32 r13 + r33 + 2r3 r4 r12 2r3 r2 r4 + r5 r14 2 2 2r5 r1 r2 + r5 r2 r23 2r1 r2 r3 + r32 + r4 r12 r4 r2 r12 r2 :
The moment bounds for a third-order representation are as of yet unknown. In the instances where moments may fall out of bounds for a second-order representation and a third-order representation must be used, care must be taken to insure that the results of the moment matching process yields a pdf. C. Combining Different Processes Call arrival and hand-off processes into a cell are concurrently active and are statistically independent of one another, meaning that operators in one space should leave the space of the other process invariant. In order to preserve the independence of each space, the disjoint spaces are combined into the system space by using the Kronecker product whose operator is denoted as ” ”. Let us note that all operations on call arrivals are said to operate in arrival space and all operations on hand-offs into a cell operate in hand-off space. An operator that operates on arrival space only is embedded in the system space by taking the Kronecker product of it with the identity operator from the hand-off space. For example, B a is joined b a . Similarly, p with I h to form B a I h which is written as B h bh . Each prois embedded in the product space by I a ph = p cess embedded into system space is differentiated from the nonembedded process by the use of hats, thus the product space is sometimes referred to as hat space, see also [16]. We prefer to use the more compact hat notation rather than explicitly using Kronecker products in our equations. III. W IRELESS C ELLULAR M ODEL A. Linear Topology Fig. 1 depicts a linear network. We begin with mobility in one direction only, from left to right. Cell j has new calls arriving according to a matrix exponential arrival process denoted by (paj ; B aj ; eaj ). Cell j also has hand-off traffic from cell i
r1 ; r13 2r1 r2 +r3 ; r12
=
3
:
0
0
r1
3 7 7 7 7 5
i
j
3
5;
(11) Fig. 1. Linear Network
k
2
~ Q j=
6 6 6 6 6 6 6 6 6 4
T ; T Æ AÆ ; 0; W ; T W ; T Æ AÆ ; 0; 2W ; T 2W ; .. .;
..
; 0;
..
.;
.;
; ;
; 0;
0
0
;
; 0 .. .; 0; 0 .. .. .. .; .; . .. Æ Æ .; T A; 0 (N 1)W ; T (N 1)W ; T Æ AÆ 0; N W; T + T Æ AÆ N W
denoted as (phij ; B hij ; ehij ). The call holding time in cell j is assumed to be exponential with mean 1=j and the distribution of the time before a hand-off to cell k is also assumed to be exponential with mean 1= jk . Fig. 2 depicts the Markov state diagram for a single cell j . Each state n represents the number of active calls in the cell. The number of active calls increases by one each time a new call or a hand-off from cell i occurs. The number of active calls decreases by one each time a call terminates or is handed off to cell k . The aggregate arrival process is the superposition of two matrix exponential processes. The superposition of new call arrivals (paj ; B aj ; eaj ), and hand-off arrivals (phij ; B hij ; ehij ), is in Kronecker product space, (see section IIC), with representation 0 e0 : bh ;b e paj pbhij ; Bb aj + B ij hij aj The superposition of the exponential call holding and hand-off processes in system space is again exponential with rate
j + jk : Based on the Markov state diagram of Fig. 2, the single cell
0
0
0
= =
j
ij
ij
j
j
ij
ij
From the state diagram of Fig. 2 and the generator matrix, we can derive the following balance equations.
0
=
j (0)T + (j + jk )j (1) Bb a be0a pba + Bb h be0h pbh
Bb a be0a pba + Bb h be0h pbh
I + I )
(
n2
n1
n0
I + I )
2(
(13)
Bb a be0a pba + Bb h be0h pbh
...
Bb a be0a pba + Bb h be0h pbh nN
N ( I + I )
Fig. 2. Markov State Diagram for a Single Cell
j (N
1)
1
(14)
T Æ AÆ + j (N ) (T + T Æ AÆ N W ) :
1)
j Q~ j = 0:
(16)
Normalization is provided by the residual equation
a pa V a h pbh Vb h =
N X n=0
j (n);
(17)
where V a = B a 1 and V h = B h 1 . Equations (16) and (17) can be solved by any conventional method. Closed form solutions exist and can be found in Lipsky [5, pp. 246-54] or Neuts [8, pp. 92-95]. The expression for new call blocking to a single cell j is given by
PBj
=
j (Nj )Bb a be0a : ba b e0a n=0 j (n)B
PNj
j
j
j
(18)
j
The expression for hand-off blocking from cell i into cell j is given by 0 bh b j (Nj )B ij ehij PHij = PNj (19) 0 : bh b ij ehij n=0 j (n)B
bh ba +B B Bb a be0a pba + Bb h be0h pbh (j I + jk I ) : j
T Æ AÆ + j (n) (T nW ) +(n + 1)(j + jk ) j (n + 1); n = 1; : : : ; N
j (n
(12)
The steady state probability vector j is the solution to
above in equation (12), where =
=
7 7 7 7 7 7 7 7 7 5
(15)
j is modeled as a ME=M= = system. Using Neuts’ notation ~ ([8, p. 92]) where = N , the generator matrix Q j is given
T T Æ AÆ W
=
3
B. Departures From a Cell The hand-off process (phij ; B hij ; ehij ) from cell i into cell in the analysis presented above is a matrix exponential approximation of the actual departure process of cell i. It is constructed by matching moments of the semi-Markov departure process from cell i to cell j . Likewise, the moments of the departure process from cell j to cell k are used to construct the hand-off process (phjk ; B hjk ; ehjk ) used in the analysis of cell k . Fig. 3 depicts the departure process from cell j . The Markov chain is partitioned into two matrices, B djk and Ldjk . State transitions which cause a hand-off departure event from cell j to cell k are mapped into the Ldjk matrix. All other
j
2 ~ Q j=
6 6 6 6 6 6 4
T ; T Æ2 AÆ2 ; 0; T Æ1 AÆ1 ; 0; 0 Æ Æ W 2; T W 2; T 2 A2 ; 0; T Æ1 AÆ1 ; 0 0; 2W 2 ; T + T Æ AÆ 2W 2 ; 0; 0; 0 W 1; 0; 0; T W 1; T Æ2 AÆ2 ; T Æ1 AÆ1 Æ Æ 0; W 1; 0; W 2; T + T A W 1 W 2; 0 0; 0; 0; 2W 1 ; 0; T + T Æ AÆ 2W 1
b ab B e0a pba + Bb h be0h pbh
b ab B e0a pba + Bb h be0h pbh
b ab B e0a pba + Bb h be0h pbh
...
n=2
n=1
n=0
I
I
n=N
NI
2
jk I
b ab B e0a pba + Bb h be0h pbh
jk I
N jk I
2
Fig. 3. Markov State Diagram for the Cell Hand-off Departure Process
state transitions are considered internal and are mapped into the B djk matrix. This process is similar to the Markov Arrival Process (MAP) described by Neuts [9, pp. 393–398], where B djk = A0 and Ldjk = A1 and the block matrix components may have imaginary phases (i.e., matrix exponential ). The Ldjk matrix for the cell hand-off process from cell j to cell k is shown below. 2
Ld
jk =
The matrix pression
6 6 6 6 6 6 6 4
;
; jk I ; 0; 0; 2 jk I ; .. .. .; .; 0; 0; 0
Bd
jk
;
; 0; 0;
0
0
..
..
.;
..
.;
..
.;
.; N jk I ;
;
0
T T Æl AÆl T Æ AÆ W1 W2
(21)
Ld
jk
Bd
jk =
Qj :
0 (22) 0
The expression for the reduced moments of the cell departure process from cell j to k is
rn =
j Ld n 0 j Ld e0 V d e ; jk
jk
jk
(20)
= = = = =
b h2 b a2 + B b h1 + B b a1 + B B Bb al be0al pbal + Bb hl be0hl pbhl ; l 2 f1; 2g T Æ1 AÆ1 + T Æ2 AÆ2 1 I + 1 I 2 I + 2 I : j
j
j
j
jk
j
jk
ij
j
ij
j
ij
ij
ij
The balance equations are
0
can then be derived using the following ex~
7 7 7 7 7 7 5
case and incorporate directionality in the mobility pattern, we partition the cell traffic into two classes. In the linear case, (e.g., highway), we will have a stream of traffic with one class continuing from left to right, and another class from right to left. Fig. 4 shows the Markov state representation for the two class cell. For illustrative purposes we describe a small two channel cell. Each cell is now treated as a two class GI=M= = system or a (GI1 + GI2 )=(M1 + M2 )= = system. Each state is represented by the tuple (n1 ; n2 ) where nl ; l 2 f1; 2g represents the number of active calls in the cell for each class. Note that n1j +n2j Nj where Nj is the total number of channels available in cell j . The global balance equations and ~ matrix is similar to that described for the mapping into the Q the single class system in the previous section. See equations ~ for cell j is given (13)-(15) and (16). The generator matrix Q in equation (20) above, where
3
7 0 7 7 .. 7 . 7 7 7 5 0
3
0 0
= =
= =
(23)
where V djk is the inverse of B djk . The unknown j is the solution to equations (16) and (17) for cell j . Using equations (8) or (11), these moments are then mapped directly into the B hjk matrix of the hand-off arrival process into the next cell. C. Linear Topology in Two Directions We expand our analysis to a linear network with traffic flowing in two directions. In order to generalize on the feed-forward
0
=
0
=
j (0; 0)T + j (0; 1)W 2 + j (1; 0)W 1 (24) j (0; 0)T Æ2 AÆ2 + j (0; 1) (T W 2 ) + 2j (0; 2)W 2 + j (1; 1)W 1 (25) Æ Æ Æ Æ j (0; 1)T 2 A2 + j (0; 2) (T 2W 2 + T A ) (26) j (0; 0)T Æ1 AÆ1 + j (1; 0) (T W 1 ) (27) + j (1; 1)W 2 + 2 j (2; 0)W 1 Æ Æ Æ Æ j (0; 1)T 1 A1 + j (1; 0)T 2 A2 (28) + j (1; 1) (T W 1 W 2 + T Æ AÆ ) Æ Æ j (1; 0)T 1 A1 + j (2; 0) (T 2W 1 + T Æ AÆ ) : (29)
As in the single class system, the steady state probability vector j is the solution to
j Q~j = 0:
(30)
b a2 b B e0a2 pba2 + Bb h2 be0h2 pbh2
;
b a2 b B e0a2 pba2 + Bb h2 be0h2 pbh2
;
(0 0)
;
(0 1)
(0 2)
2 I + 2 I )
2 I + 2 I )
(
1 I + 1 I )
2(
( 1 I + 1 I ) 0 b a1 b B ea1 pba1 + Bb h1 be0h1 pbh1
(
b a1 b b a2 b e0a1 pba1 + Bb h1 be0h1 pbh1 B e0a2 pba2 + Bb h2 be0h2 pbh2 + B
b a1 b B e0a1 pba1 + Bb h1 be0h1 pbh1
b a2 b B e0a2 pba2 + Bb h2 be0h2 pbh2 b hb e0h pbh Bb a be0a pba + B
;
;
(1 0)
(1 1)
2 I + 2 I )
(
b a1 b b a2 b e0a1 pba1 + Bb h1 be0h1 pbh1 B e0a2 pba2 + Bb h2 be0h2 pbh2 + B
1 I + 1 I )
2(
b a1 b B e0a1 pba1 + Bb h1 be0h1 pbh1
;
(2 0)
b a1 b b a2 b e0a1 pba1 + Bb h1 be0h1 pbh1 B e0a2 pba2 + Bb h2 be0h2 pbh2 + B
Fig. 4. Markov State diagram of a two class cell with two channels
Normalization is provided by the residual equation
Nj X n=0
;
j (n)
=
h1 ph1 V h1 h2 ph2 V h2 ; ij
ij
ij
ij
ij
;
jk =
0; 2jk I ; 0; 0; 0; 0;
; ;
; 0; ; 0; 2 2jk I ; 0; 0; 0; 0; 0; 0; 0; 2jk I ; 0; 0; 0; 0
0
0
0
; ; 0; 0; 0; 0; 0
0
0
0 0 0 0
;
(1 0)
(1 1)
2jk I
where V alj = B al1j and V hlij = B hl1ij for l 2 f1; 2g. The hand-off departure transitions for class l = 2 are represented in Fig. 5. These are the transitions which are mapped to the Ld2 matrix. Note that all transitions are to states to the immediate left. For l = 1, all transitions will be to states immediately above. 6 6 6 6 6 6 4
(0 2)
2jk I
Bb a be0a pba + Bb h be0h pbh
(31)
2
;
(0 1)
2jk I
a1j pa1j V a2j a2j pa2j V a2j ij
Ld2
;
(0 0)
;
(2 0)
Fig. 5. Hand-off departure transitions of the two class cell
3
7 7 7 7 (32) 7 7 5
0
The departure moments (as they are for the single class cell) are represented by the following equation
L rn = j dl 0 V ndl e0 ; l 2 f1; 2g; (33) j Ldl e is the inverse of B dl . The unknown j is the
are then mapped to (phljk ; B hljk ; e0hljk ) of the hand-off arrival process into the next cell using either a second- or third-order representation according to equations (8) or (11). For each cell j with (n1 ; n2 ) channels such that n1j + n2j = Nj , the probability of blocking for a new arriving call for each class is
jk
jk
jk
where V dljk jk solution to equations (30) and (31) for cell j . These moments
PB 1 j
PNj
=
0
b ea1j n=0 j (Nj n; n)B a1j b PNj PNj n1 b e0a1j n1 =0 n2 =0 j (n1 ; n2 )B a1j b
(34)
4.40E-01
and
0
PB2j
=
b ea2j n=0 j (n; Nj n)B a2j b : PNj PNj n1 b e0a2j n1 =0 n2 =0 j (n1 ; n2 )B a2j b
4.20E-01
(35) 4.00E-01
The hand-off blocking for each class respectively from cell into cell j is
0
PNj
PH 1ij
=
b eh1ij n=0 j (Nj n; n)B h1ij b PNj PNj n1 b e0h1ij n1 =0 n2 =0 j (n1 ; n2 )B h1ij b
Blocking Probability
PNj
i
Sim. New Sim. Handoff 3.80E-01
FPA Erlang B Mom. Match New Mom. Match Handoff
3.60E-01
b hb b ab e0h pbh e0a pba + B B
(36)
3.40E-01
; (0; 1) (0; 2) (0 0)
3.20E-01
; ;
and
1
(1 1)
2
3
4
5
(2 0)
PH 2ij
0
PNj
=
b eh2ij n=0 j (n; Nj n)B h2ij b : PNj n1 b e0h2ij n1 =0 n2 =0 j (n1 ; n2 )B h2ij b
PNj
8
9
;
(1 0)
1; = 1
= 1; =
arrival processes were chosen to provide simple examples of the ability of the moment matching approximation technique to incorporate non-Poisson new call arrival distributions into the performance model where 2 < 1 and 2 > 1.
With the exception of the linear feed-forward model, we allow for multi-directional mobility. Therefore, the hand-off process from cell i to j affects the hand-off process from cell j to i. There must be an initial guess for the moments of the hand-off process to the first cell that will have its departure process analyzed. We then use an iterative process using our enhancement to FPA to update each departure process moment and match the hand-off arrival moments. For the topologies presented in this paper, convergence is very quick, on the order of four or five iterations.
B. Linear Model We start with a restricted mobility model of ten cells allowing only mobility in one direction. We study a system with one channel per cell and five channels per cell. Mobility rate ij is increased from zero to ten times 1=i . Since the state space explosion makes an exact analysis prohibitive, the analytic approximation is compared to a simulation model. Fig. 6 and Fig. 7 show a comparison of the new and hand-off blocking into a ten cell linear wireless network. Simulation results are compared to analytic results obtained using a Poisson hand-off assumption (Erlang B) and using the moment matching technique. In Fig. 6, each cell consists of a single channel. The arrival rate a , hand-off rate , and service rate are all equal to 1. This results in higher blocking than would be experienced in a typical network. This was done on purpose to illustrate that moment matching can provide accurate results in a high blocking
E. Generalization to Two Dimensions For large two dimensional networks, arrival streams can be assumed to be independent. The term independent, used in this context, means that all hand-offs can be viewed as new arrivals. Therefore, we return to the single class model. Solutions to the two dimensional model are the same as that for the single linear model presented in section (IIIA) with two exceptions. The first is that our enhanced FPA is used, since mobility can be in multiple directions. The second difference is that since there is more than one adjacent cell k to each cell i, the following substitution in equations (12)-(15) is made.
where =
7
Fig. 6. Blocking feed-forward 10 cells 1 channel Poisson arrivals
(37)
D. Fixed Point Iteration
W
6 Cell
3.00E-01
j I + I ) ;
= (
P
k6=j jk for all neighboring cells k .
2.50E-01
Blocking Probability
IV. N UMERICAL R ESULTS A. New Call Arrival Processes
2.00E-01 Sim. New Sim. Handoff FPA Erlang B Mom. Matche New Mom. Match Handoff
1.50E-01
In this paper we demonstrate call and hand-off blocking probabilities for several new call arrival processes. We study a B e0 p B e0 p Poisson arrival process, and two non-Poisson arrival processes in which 2 < 1 and 2 > 1. As an example where 2 < 1, we ;; use an Erlang-2 arrival process with mean rate and a squared ;; ; coefficient of variation 2 = 1=2. For another example where ; the 2 > 1, we use a hyper-exponential process with mean Fig. 7. Blocking feed-forward 10 cells 1 channel Poisson arrivals = 1; = 1; = 10 rate and a squared coefficient of variation 2 = 2:8. These b ab b a a
+
b hb b h h
1.00E-01
(0 0) (0 1) (0 2)
(1 1) (2 0)
5.00E-02
1
2
3
4
5
6
Cell
7
8
9
(1 0)
9.00E-02
regime. Fig. 6 shows that hand-off blocking is less than new call blocking. This figure shows that the moment matching approximation with Poisson call arrivals produces results which are indistinguishable from the simulation results, but the Poisson approximation, which provides a close approximation to new call blocking, cannot capture the behavior of the hand-off blocking. In Fig. 7, the mobility rate is increased to ten times the call holding time. The difference between the new call arB e0 p B e0 p rival and hand-off blocking probabilities is more pronounced. The moment matching approximation captures this behavior, ; ; whereas a Poisson hand-off assumption does not. ;
8.00E-02
7.00E-02
Blocking Probability
6.00E-02
b ab b a a
+
Sim. New
5.00E-02
Sim. Handoff FPA Erlang B Mom. Match New
4.00E-02
Mom. Match Handoff 3.00E-02
b hb b h h
2.00E-02
1.00E-02
(0 0) (0 1) (0 2)
; ;
(1 1)
0.00E+00 1
2
3
4
(2 0)
C. Bi-directional Model
5
6
7
8
9
;
(1 0)
Cell
Fig. 9. Blocking bi-directional 10 cells 5 channels Hyper-exp arrivals
We also look at bi-directional mobility under the same conditions as the uni-directional model. Fig. 8 and Fig. 9 compare new call arrival and handoff blocking in both directions for the simulation, Poisson approximation, and moment matching approximation, for some non-Poisson call arrival distributions. Since our example uses homogeneous traffic in both directions, we depict blocking probabilities in one direction only. These figures also show that a fixed point approximation using the moment matching algorithm provides a better approximation than using FPA with a Poisson assumption. D. Generalization of the two dimensional model Fig. 10 depicts a seven cell network with arbitrary mobility throughout the network. Fig. 11 shows new call and handoff call blocking probabilities for Erlang-2 arrivals. Although the Poisson (Erlang B) approximation overestimates new call blocking probabilities for Erlang-2 arrival traffic, it generally underestimates blocking for arrival traffic when 2 > 1. This figure demonstrates the ability of the moment matching method to approximate new call and handoff call blocking for large two dimensional topologies. Although the model is still relatively small, traffic in larger wireless cellular networks is expected to be less dependent.
2:5; 2 = 3:0; = 1; = 10
The moment matching method provides a better approximation for blocking in cellular systems than models using Poisson traffic assumptions. The moment matching method works especially well for small networks with a small number of channels per cell and where traffic arrival distributions are not expected to be Poisson. These characteristics can be expected in micro- or pico-cellular networks. The steady state probability vector for each cell in the network is produced so that performance measures such as handoff blocking vs. new call blocking, probability of forced-termination and dropout, and the distribution of the number of busy circuits in a cell can also be derived. Since this method is based on a single cell decomposition, calculations for the departure moments from each cell can be carried out in parallel and updated at each iteration, pro-
2
7
3
3.00E-02
1
Blocking Probability
2.50E-02
Sim. New Sim. Handoff FPA Erlang B
b ab b hb e0a pba + B B e0h pbh
5
1.00E-02
(0 1) (0 2)
(0 1)
; ;
; ; ; (1; 0) (1; 1) (2; 0) = (0 0)
5.00E-03 (0 0)
(1 1)
4
Mom. Match Handoff
b hb Bb a be0a pba + B e0h pbh
; ; (0; 2)
6
Mom. Match New 1.50E-02
0.00E+00 1
(2 0)
2
3
4
5
6
7
8
9
;
(1 0)
Cell
Fig. 8.
Blocking bi-directional 10 cells 5 channels Erlang-2 arrivals
2:5; 2 = 0:5; = 1; = 10
=
V. C ONCLUSION
3.50E-02
2.00E-02
Fig. 10. Network with general flow in two dimensions
8.00E-02
[4] 7.00E-02
[5] 6.00E-02
Blocking Probability
[6] 5.00E-02
[7]
4.00E-02 Sim. New Sim. Handoff 3.00E-02
FPA Erlang B Mom. Match New Mom. Match Handoff
b hb e0h pbh Bb a be0a pba + B
2.00E-02
[8]
1.00E-02
; ; ; ; (1; 1) (2; 0) (0 0) (0 1) (0 2) (1 0)
[9]
0.00E+00 1
2
3
4
5
6
7
Cell
Fig. 11. Blocking two dimensional 7 cells 5 channels Erlang-2 arrivals
2:5; 2 = 0:5; = 1; = 10
=
[10]
[11]
viding scalability not present in simulation models. The topology is not limited to the models presented here, but can easily be generalized to any topology. The model can be easily expanded to include guard channels and rate dependent arrivals. Further research is being conducted to expand this method to include non-exponential cell handoff processes. R EFERENCES [1] [2] [3]
G. Foschini, B. Gopinath, and Z. Miljanic, “Channel cost of mobility,” IEEE Transactions on Vehicular Technology, vol. 42, no. 4, pp. 1272–80, 1993. W. Gragg and A. Lindquist, “On the partial realization problem,” Linear Algebra and Applications, vol. 50, pp. 277–319, 1983. D. Hong and S. Rappaport, “Traffic model and performance analysis for cellular mobile radio telephone systems with prioritized and nonprioritized handoff procedures,” IEEE Transactions on Vehicular Technology, vol. vt-35, no. 3, pp. 77–92, 1986.
[12] [13] [14] [15] [16] [17] [18]
F. P. Kelly, “Fixed point models of loss networks,” Austral. Math. Soc. Ser. B, vol. 31, pp. 204–18, 1989. L. Lipsky, Queueing Theory: A Linear Algebraic Approach. New York: MacMillan, 1992. L. Lipsky, P. Fiorini, W. Hsin, and A. v. Liefvoort, “Auto-correlation of lag-k for customers departing from semi-Markov processes.” in preparation. D. McMillan, “Traffic modeling and analysis for cellular mobile networks,” in 13th International Teletraffic Congress (A. Jensen and V. B. Iversen, eds.), (Copenhagen, Denmark), pp. 627–632, North-Holland, June 1991. M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach. Baltimore, MD: Johns Hopkins University Press, 1981. M. F. Neuts, Algorithmic Probability. New York: Chapman and Hall, 1995. M. Sidi and D. Starobinski, “New call blocking versus handoff blocking in cellular networks,” in Proceedings of the Conference on Computer Communications (IEEE Infocom), (San Fransisco, California), Mar. 1996. K. Sohraby, “Mobility calculus in cellular wireless networks,” in Conference Record of the International Conference on Communications (ICC), (Dallas, Texas), p. S41, June 1996. T. Stieltjes, “Reserches sur les fractions continues,” Ann. Fac. Sci. Univ. Toulouse, vol. 8, pp. 1–122, 1894. T. Stieltjes, “Reserches sur les fractions continues,” Ann. Fac. Sci. Univ. Toulouse, vol. 9, pp. 1–47, 1894. A. van de Liefvoort, “The moment problem for continuous distributions,” Tech. Rep. CM-1990-02, U. of Missouri-Kansas City, 1990. A. van de Liefvoort, “A note on count processes,” Assam Statistical Review, vol. 5, no. 2, pp. 102–12, 1991. A. van de Liefvoort and L. Lipsky, “A matrix algebraic solution to two k servers in a loop,” JACM, vol. 33, pp. 207–33, 1986. W. Whitt, “Approximating a point process by a renewal process, i: two basic methods,” Operations Research, vol. 30, no. 1, pp. 125–47, 1982. W. Whitt, “Blocking when service is required from several facilities simultaneously,” AT&T Technical Journal, vol. 64, pp. 1807–56, 1985.
m
