Per Connection Performance Analysis of a Frame-based TDMA/CDMA MAC Protocol Containing both Reservation and Contention Slots ∗ Ying-Ju Chen † and Jin-Fu Chang ‡ February 26, 2004 Abstract A mixture of reservation plus contention data slots is now widely implemented in TDMA/ CDMA systems to make a system flexible enough to suit various kinds of packets but preserve the good nature of frame-based protocols. We conducted perfor- mance analysis for an arbitrary connection in such a system under the assumption of MMPP (Markov-modulated Poisson process) arrivals. Accessible slot locations of this connection in a frame is made general. Success probability in accessing a contention slot is also made general. We have obtained the system size distribution which can be used to evaluate the performance of various frame-based MAC protocols. The MMPP arrival pattern can be generalized to the BMAP( batch Markovian arrival process) family to further accommodate a broader set of traffic sources. Keywords: Queueing Analysis; Contention; TDMA; CDMA ∗ Part of the work was performed when the authors were with Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 106. † Corresponding author; Stern School of Business, New York University, 44 W 4th Street, KMC 8-151, New York, NY 10012, USA; [email protected]; Tel: 212-998-0489, Fax: 212-995-4003. ‡ Department of Electrical Engineering, National Chi Nan University, Puli, Nantou, Taiwan 545; [email protected]; Tel: +886-49-2910272, Fax: +886-49-2912569. 1
Transcript
Per Connection Performance Analysis of a
Frame-based TDMA/CDMA MAC Protocol
Containing both Reservation and Contention Slots∗
Ying-Ju Chen† and Jin-Fu Chang‡
February 26, 2004
Abstract
A mixture of reservation plus contention data slots is now widely implemented in
TDMA/ CDMA systems to make a system flexible enough to suit various kinds of
packets but preserve the good nature of frame-based protocols. We conducted perfor-
mance analysis for an arbitrary connection in such a system under the assumption of
MMPP (Markov-modulated Poisson process) arrivals. Accessible slot locations of this
connection in a frame is made general. Success probability in accessing a contention
slot is also made general. We have obtained the system size distribution which can be
used to evaluate the performance of various frame-based MAC protocols. The MMPP
arrival pattern can be generalized to the BMAP( batch Markovian arrival process)
family to further accommodate a broader set of traffic sources.
∗Part of the work was performed when the authors were with Graduate Institute of Communication
Engineering, National Taiwan University, Taipei, Taiwan 106.†Corresponding author; Stern School of Business, New York University, 44 W 4th Street, KMC 8-151,
New York, NY 10012, USA; [email protected]; Tel: 212-998-0489, Fax: 212-995-4003.‡Department of Electrical Engineering, National Chi Nan University, Puli, Nantou, Taiwan 545;
Now we consider the transitions when Xn = 0. In this case, buffer becomes empty at the
beginning of the next accessible slot after the n-th departure, which occurs at the end of
the aq-th slot. Therefore, the next observation point is not necessarily the beginning of
the aq+2-th slot but the next accessible slot following the departure of a new arrival which
arrives after the position associated with Xn. We let Gq(k) represent the probability that
the inspected connection has no packets to transmit during the aq−1-th slot, but does have
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packets to send in the aq-th slot; and has accumulated k packets at the beginning of the
aq+1-th slot.
Gq(k) =
rq∑u=1
P (u, aq − aq−1)P (k, aq+1 − aq)
+
k+rq+cq∑u=rq+1
min(u−rq,cq)∑v=max(u−rq−k,0)
Cmin(u−rq ,cq)v pv
q(1− pq)min(u−rq,cq)−v
P (u, aq − aq−1)P (k + rq + v − u, aq+1 − aq),
(13)
where P (u, aq−aq−1) is them×m probability matrix that u packets have arrived in (aq−aq−1)
slots taking all possible phase changes under consideration.
The first term of Gq(k) denotes the case that the number of packets queued immediately
before the aq-th slot is less than the number of reserved codes so that these packets get
transmitted in this slot, and the contention codes are not used at all. The number of packets
seen at the observation point, the beginning of the (q+1)-th slot, depends only on the arrivals
during (aq − 1, aq+1 − 1].The second term of Gq(k) is more complicated, but still can be explicitly explained.
When the number of packets seen at the beginning of the aq-th slot exceeds rq, the inspected
connection has to seek to use contention codes. The number of contention codes which
need to be acquired by this connection depends on u − rq, the number of packets since rqreserved codes are consumed first. If u − rq < cq, only u − rq packets enter competitionin the aq-th slot; otherwise, all cq contention codes have to be attempted. Let v denote
the number of packets successfully transmitted via the contention codes in the aq-th slot.
The probability P (k + rq + v − u, aq+1 − aq) specifies the number of packets required to
arrive during [aq − 1, aq+1 − 1) so that the inspected connection accumulates k packets atthe beginning of the aq+1-th slot, the next observation point.
Having obtained Gq(k), we are able to derive the transitions for ap+1 ≤ aq−1 and ap+1 >
aq−1, respectively, which are exactly identical to those in [5]. For ap+1 ≤ aq−1,
where []i,j is the element on the i− th row and j − th column of the associated matrix andei = [0 ... 0 1 0 ... 0]
tr where the only 1 appears at the i − th place. Note that the term[I− e(Q−Λ)aM ]−1e(Q−Λ)(aq−1−ap+1) in Eq. (14) is the summation of a geometric series, because
the occurrence of the new arrival may cross one frame boundary. For ap+1 > aq−1, a similar
4.1.3 Construction of the M/G/1-type Transition Matrix
The procedure to construct the transition matrix from probabilities Pr(Xn+1 = k, Jn+1 =
aq, In+1 = j|Xn = l, Jn = ap, In = i) is the same as Sec. 3 with one modification
c = max1≤i≤M(ri + ci). From that we are able to obtain its steady state probability
x = [x0 x1 x2 ...].
4.2 System Size Distribution at an Arbitrary Time
Let t ∈ [aq, aq+1) be an arbitrary time within a frame, where q = 1, ...,M with [aM , aM+1) =
[0, a1). We first obtain the system size distribution for t ∈ [aq−1, aq) and that of t ∈ [aq−1, aq−1) follows Eq. (6), regardless of code allocation policies. Then
where πq−1(z) is the z-transform of the probabilities associated with the aq-th slot in x′is of
Sec. 4.1.3 and πq−i−10 = [π1,q−i−1
0 π2,q−i−10 ... πm,q−i−1
0 ] is the probability that no packet is
awaiting at the beginning of the aq−i-th slot while the MMPP′s phase varies from 1 to m.
Moments can be obtained by Eqs. (7)-(8).
5 A Model with Single-Code Slots
In this section, we consider a system in which all slots contain no more than one code
accessible to the inspected connection. Since a hybrid slot contains at least one reserved code,
in this system no contention codes are accessible. In other words, rq = 1, cq = 0, 1 ≤ q ≤M,and c
′q = 1, 1 ≤ q ≤ N.Restricting to the single accessible code scenario makes us easier to concentrate on the
treatment of multiple successive contention slots. Note that a TDMA system subject to pure
contention in each slot is a subclass of this model; while a hybrid TDMA/CDMA system
may also be thus described if codes accessible by the inspected connection happen to be
mutually non-overlapping.
5.1 State Transition Matrix
In the following we derive the state transition matrix for Xn ≥ 1 and Xn = 0.
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5.1.1 Xn ≥ 1
A. Transitions to a Hybrid Slot other than the a1-th Slot
Let us first consider departures occurring in a hybrid slot except the very first. WhenXn ≥ 1,the next departure can occur at either the next hybrid slot, or a contention slot prior to the
next hybrid slot. In other words, it cannot occur after the time has passed a hybrid slot
since packets have been waiting in the buffer. In particular, when the location associated
with Xn is bq, 1 ≤ q ≤ N , the next departure cannot pass the a1-th slot in the upcoming
frame even if the inspected connection fails to acquire any of the residual contention slots in
5.1.3 Construction of the M/G/1-type Transition Matrix
After completing the derivations of the probabilities of all state transitions, we now are able
to construct the transition matrix as follows.
Q(∞) =
P0,0 P0,1 P0,2 P0,3 P0,4 P0,5 . . .
A0 A1 A2 A3 A4 A5 . . .
0 A0 A1 A2 A3 A4 . . .
. 0 A0 A1 A2 A3 . . .
. . 0 A0 A1 A2 . . .
. . . . . . . . .
, (28)
which is already an M/G/1-type transition matrix. Therefore, we can apply the M/G/1 algo-
rithm on Q(∞) of this format to obtain its steady state probability vector x = [x0 x1 x2 ...].
5.2 System Size Distribution at an Arbitrary Time
We provide the system size distribution when t belongs to an accessible slot here. Observing
that t ∈ [ai−1, ai), 1 ≤ i ≤M is exactly the same as Sec. 4, instead we focus on t ∈ [bi−1, bi),i.e., the time within a contention slot. For convenience we introduce Hi(k, t), a probability
function similar to HCi (k), and its PGF Hi(z, t),
6 AModel with Contention Slots Having Multiple Codes
In this section we treat a model where each contention slot has multiple codes. Mathematical
derivations are much more complicated than previous sections.
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In order to address the complexity brought by such multicode contention slots, we restrict
hybrid slots to contain only reserved codes. Thus the system parameters of this model are:
rq �= 0, cq = 0, 1 ≤ q ≤ M,, and N > 1, c′q ≥ 1, 1 ≤ q ≤ N. In other words, the model
we discuss here is a hybrid TDMA/CDMA system considered in [5] followed by a string
of multicode contention slots. Such framework can easily find its place in many existing
communication protocols.
6.1 State Transition Matrix
Derivations are same as Sec. 4 except transitions corresponding to the a1-th slot and con-
tention slots b1, b2, ..., bN . Thus we concentrate on these two cases in the sequel.
Let us recall Eq. (20) where a pure contention TDMA system is considered. In that
equation, we used only a simple product term [p′r
∏r−1u=q+1(1 − p
′u)] to represent the event
that the inspected connection has failed competitions in the bq+1-th, bq+2-th, ..., br-th slots
because the number of codes available for contention is merely 1.
However, if the number of codes in these contention slots may vary, the derivation is no
longer simple. Elements of the transition matrix may depend on whether or not the number
of codes in the former slot is larger than the latter, and therefore the algorithmic “if-else”
or “switch-case” instructions must be widely employed.
6.1.1 Xn ≥ 1
If there are already packets queued in the buffer when the observation point of Xn is placed
at the beginning of the aq+1-th slot, transmission surely occurs in the aq+1-th slot. In other
words, transitions where there are hybrid slots in between can never occur. Hence
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Pr(Xn+1 = k, Jn+1 = br, In+1 = j|Xn = l, Jn = aq, In = i) = 0, q �=M, 1 ≤ r ≤ N + 1,
P r(Xn+1 = k, Jn+1 = br, In+1 = j|Xn = l, Jn = bq, In = i) = 0, 1 ≤ q ≤ N, r < q + 1.(31)
Now we consider the case when the n-th departure occurs at the end of a contention
slot bq, i.e., the observation point of Xn = l is at the beginning of the bq+1-th slot. We
first consider transitions between two consecutive contention slots, i.e., bq → bq+1. In the
bq+1-th slot, the number of codes available for competition to the inspected connection is
min(l, c′q+1). If the next departure occurs in the bq+1-th slot, the connection will have at least
successfully sent one packet in this slot; moreover, if it has successfully sent less than (l− k)packets in this slot, the number of packets accumulated right before the next observation
point, i.e., the bq+2-th slot of the next frame, will exceed k. Thus, the number of packets
transmitted in this slot shall be lower bounded by max(1, l − k), and we conclude that
We next consider transitions between two non-consecutive contention slots.
Pr(Xn+1 = k, Jn+1 = br,In+1 = j|Xn = l ≥ 1, Jn = bq, In = i)
= etri {
Vq+2∑vq+2=0
...Vs∑
vs=0
...Vr∑
vr=0
Ur∑u=max(1,l+
∑rq+2 vt−k)
CUru p
′ur (1− p
′r)
Ur−u
[r−1∏
w=q+1
(1− p′w)Uw ][r∏
w=q+2
P (vw, bw − bw−1)]P (k + u− (l +r∑
t=q+2
vt), br+1 − br)}ej ,
0 ≤ q ≤ r − 2, 1 ≤ r ≤ N,(32)
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where
Vs = k + c′r − (l +
s−1∑t=q+1
vt), q + 2 ≤ s ≤ r,
Uw = min(c′w, l +
w∑t=q+1
vt), q + 1 ≤ w ≤ r,
vq+1 = 0.
Note that except the one associated with u, there are (r−q+1) summations on the right-handside.
Figure 7: An example of transitions between two contention slots.
In Fig. 7, vs is the number of packets arriving in [bs−1, bs), and u is the number of
packets that are successfully transmitted in the br-th slot. Since the (n+1)-st departure
occurs in the br-th slot, packets competing in slots bq+1, ..., br−1 shall all fail. Therefore, the
number of codes in the bs-th slot available for competition to the inspected connection is the
minimum of the number of accessible codes c′s and the packets accumulated in the queue at
the beginning of that slot, that is, l +∑s
t=q+1 vt. We need not consider the case in which
the number of packets awaiting in the buffer is larger than k + c′r, where c
′r is the maximal
number of packets that can be transmitted in the br-th slot. Consequently, the upper bound
of vs is k + c′r − (l +
∑s−1t=q+1 vt).
Actual value of Eq. (32) can be obtained by running these (r − q) loops in computerprograms. We can further “simplify” the formulas when l +
∑st=q+1 vt ≥ maxs≤i≤r{c′i} –
although it seems even more sophisticated at the first glance– since in this case all codes in the
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bq+1-th,...,br-th slots are for contention. Consequently, we are able to deduce a deterministic
multiplicative term Πs≤i≤r−1(1− p′i)c′i in our formula regardless of the number of arrivals in
[bs − 1, br+1 − 1). In other words, we can eliminate loops associated with vs+1, ..., vr−1 to
significantly reduce computational burden. For simplicity we provide here only the formulas
equivalent to Eq. (32) when k ≥ maxq+1≤i≤r{c′i}.
Pr(Xn+1 = k, Jn+1 = br, In+1 = j|Xn = l ≥ 1, Jn = bq, In = i)
=etri {
Vq+2∑vq+2=Wq+2
c′r∑
u=uq+2
Cc′r
u p′ur (1− p
′r)
c′r−u
(1− p′q+1)Uq+1 [
r−1∏w=q+2
(1− p′w)c′w ]P (vq+2, bq+2 − bq+1)P (k + u− (l +
q+2∑t=q+2
vt), br+1 − bq+2)
+
r−1∑s=q+3
Wq+2−1∑vq+2=0
...
Ws−1−1∑vs−1=0
Vs∑vs=Ws
c′r∑
u=us
Cc′r
u p′ur (1− p
′r)
c′r−u[
s−1∏w=q+1
(1− p′w)Uw ]
r−1∏w=s
(1− p′w)c′w ][
s∏w=q+2
P (vw, bw − bw−1)]P (k + u− (l +s∑
t=q+2
vt), br+1 − bs)
+
Wq+2−1∑vq+2=0
...
Wr−1−1∑vr−1=0
Wr−1∑vr=0
Ur∑u=ur
CUru p
′ur (1− p
′r)
Ur−u
[r−1∏
w=q+1
(1− p′w)Uw ][r∏
w=q+2
P (vw, bw − bw−1)]P (k + u− (l +r∑
t=q+2
vt), br+1 − br)}ej ,
0 ≤ q ≤ r − 2, 1 ≤ r ≤ N, k ≥ maxr≤i≤r
{c′i},
(33)
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where
Vs = k + c′r − (l +
s−1∑t=q+1
vt), q + 2 ≤ s ≤ r,
Ws = maxs≤i≤r
{c′i} − (l +s−1∑
t=q+1
vt), q + 2 ≤ s ≤ r,
Uw = min(c′w, l +
w∑t=q+1
vt), q + 1 ≤ w ≤ r,
us = max(1, l +s∑
q+2
vt − k), vq+1 = 0.
A formula similar to Eq. (32) can be derived for transitions bq → a1, 0 ≤ q ≤ N, as
follows.
Pr(Xn+1 = k, Jn+1 = a1, In+1 = j|Xn = l ≥ 1, Jn = bq, In = i)
=etri {
Vq+2∑vq+2=0
...Vs∑
vs=0
...
VN+1∑vN+1=0
[N∏
w=q+1
(1− p′w)Uw ]
[
N+1∏w=q+2
P (vw, bw − bw−1)]P (k +min(0, r1 − (l +N+1∑
t=q+2
vt)), bN+1 − bN )}ej, 0 ≤ q ≤ N − 1,
(34)
where
Vs = k + r1 − (l +s−1∑
t=q+1
vt), q + 2 ≤ s ≤ N + 1,
Uw = min(c′w, l +
w∑t=q+1
vt), q + 1 ≤ w ≤ N,
vq+1 = 0.
(35)
Note that in this transition, k + min(0, r1 − (l +∑N+1t=q+2 vt)) packets shall arrive during
[a1 − 1, a2 − 1) so that k packets are accumulated in the buffer at the beginning of the a2-th
slot. Finally, transition bN → a1 is the same as that to other hybrid slot.
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6.1.2 Xn = 0
Now we treat the case Xn = 0. We first introduce the probability functions HCr (k), 1 ≤ r ≤
N, and HR1 (k). H
Cr (k) represents the probability that our chosen connection has no packets
at the beginning of the aM -th slot, but has sent packets in the br-th slot and accumulated k
packets at the beginning of the br+1-th slot of the next frame.
HC1 (k) =
k+c′1∑
v=1
min(v,c′1)∑
u=max(1,v−k)
Cmin(v,c′1)
u p′u1 (1− p
′1)
min(v,c′1)−uP (v, b1 − aM)P (k − v + u, a1), (36)
where v denotes the number of arrivals during [aM − 1, b1 − 1), and u denotes the packetssuccessfully transmitted in the b1-th slot. According to Eq. (36), in the b1-th slot the
number of packets competing for the contention codes is min(v, c′1), and the lower bound of
the number of packets transmitted in this slot is max(1, l− k).The probability functions HC
r (k) and HR1 (k) are similar to Eqs. (32) and (34) when we
make l = 0 and start at the aM -th slot. Henceforth, ∀1 ≤ r ≤ N ,
HCr (k) =
V1∑v1=0
...
Vs∑vs=0
...
Vr∑vr=0
Ur∑u=max(1,
∑r1 vt−k)
CUru p
′ur (1− p
′r)
Ur−uI{r∑
t=1
vt �= 0}
[
r−1∏w=q+1
(1− p′w)Uw ][
r∏w=1
P (vw, bw − bw−1)]P (k + u−r∑
t=1
vt, br+1 − br),(37)
where
Vs = k + c′r −
s−1∑t=q+1
vt, 1 ≤ s ≤ r,
Uw = min(c′w,
w∑t=1
vt), 1 ≤ w ≤ r,(38)
and I{} is the indicator function. Note that here ∑rt=1 vt �= 0, otherwise we would not have
any departure in the br-th slot. If∑r
t=1 vt = 0, then Ur = 0. In this case we can also define
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∑Ur
u=max(1,∑r
1 vt−k) F (u) ≡∑0
u=1 F (u) ≡ 0, ∀F (u), to eliminate the indicator function, whichis adopted in the sequel.
We may define the probability function HR1 (k) ≡ HC
N+1(k) to represent the corresponding
function for the a1-th slot. It can be expressed as follows.
HR1 (k) =
V1∑v1=0
...
Vs∑vs=0
...
VN+1∑vN+1=0
[
N∏w=1
(1− p′w)Uw ]
[N+1∏w=1
P (vw, bw − bw−1)]P (k +min(0, r1 −N+1∑t=1
vt), bN+1 − bN ),(39)
where
Vs = k + r1 −s−1∑t=1
vt, 1 ≤ s ≤ N + 1,
Uw = min(c′w,
w∑t=1
vt), 1 ≤ w ≤ N.(40)
With the help of HCr (k) and H
R1 (k), we are ready to obtain the transition probabilities.
Note that in both Eqs. (42) and (44) there is an extra term when comparing with earlier
formulas. This additional term is due to that the n-th departure occurs in less than one
frame before the (n+1)-st departure. Since there are no packets seen at the beginning of the
bq+1-th slot, no codes are attempted and the number of packets transmitted in the br-th slot
depends on the arrivals during [bq+1 − 1, br − 1) in Eq. (42) and [bq+1 − 1, a1+ bN − 1) in Eq.(44).
6.1.3 Construction of the M/G/1-type Matrix
The procedure to construct the transition matrix is exactly the same except c = max{max1≤i≤M{ri+ci},max1≤j≤N{c′j}}.
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6.2 System Size Distribution at an Arbitrary Time
After obtaining the transition matrix and its steady state probabilities x = [x0 x1 x2 ...],
we may go on to obtain the system size distribution. When t ∈ [aq − 1, aq), 2 ≤ q ≤ M ,
derivations are same as those in Sec. 4, where the case t ∈ [b1 −1, b1) can be regarded as the(M + 1)-st hybrid slot since the slot type is irrelevant to the consideration of system size.
Similarly, the a1-th slot can be regarded as the (N + 1)-st contention slot. We shall in the
sequel focus on the cases t ∈ [br − 1, br), 2 ≤ r ≤ N + 1, where t ∈ [bN+1 − 1, bN+1) refers to
the case t ∈ [a1 − 1, a1).
We first introduce probability functions Hbr(k, t), 2 ≤ r ≤ N + 1. Hbr(k, t) represents
the probability that the connection has no packet right before the aM -th slot, and does not
transmit any packet in the contention slots b1, b2, ..., br−1; and has accumulated k packets
up to time t. Since the definition is similar to HCr (k), we immediately obtain the following
equation:
Hbr(k, t) =
V1(k)∑v1=0
...
Vs(k)∑vs=0
...
Vr−1(k)∑vr−1=0
[r−1∏w=1
(1− p′w)Uw ][r−1∏w=1
P (vw, bw − bw−1)]P (k −r−1∑i=1
vi, t− br−1 + 1),
Vs(k) = k −s−1∑i=1
vi, 1 ≤ s ≤ r − 1,
Uw = min(c′w,
w∑i=1
vi), 1 ≤ w ≤ r − 1, 2 ≤ r ≤ N + 1.
(46)
Here we denote k − ∑s−1i=1 vi by Vs(k) to emphasize that it is a function of k.
If k ≥ max1≤q≤r−1{c′q}, we can rewrite Hbr(k, t) in a manner slightly different from Eq.
(33):
31
Hbr(k, t) =
V1(k)∑
v1=W′1+1
[
r−1∏w=1
(1− p′w)c′w ]P (v1, b1 − b0)P (k −
1∑i=1
vi), t− b1 + 1)
+r−1∑s=2
W′1∑
v1=0
...
W′s−1∑
vs−1=0
Vs(k)∑
vs=W′s−1+1
[s−1∏w=1
(1− p′w)Uw ]
r−1∏w=s
(1− p′w)c′w ][
s∏w=1
P (vw, bw − bw−1)]P (k −s∑
i=1
vi), t− bs + 1)
+
W′1∑
v1=0
...
W′r−1∑
vr−1=0
[
r−1∏w=1
(1− p′w)Uw ][
r−1∏w=1
P (vw, bw − bw−1)]P (k −r−1∑i=1
vi), t− br−1 + 1)
2 ≤ r ≤ N + 1, k ≥ max1≤q≤r−1
{c′q},
(47)
where
Vs(k) = k −s−1∑i=1
vi, 1 ≤ s ≤ r − 1,
W′s = max
1≤i≤r{c′i} −
s−1∑i=1
vi, 1 ≤ s ≤ r − 1,
Uw = min(c′w,
w∑i=1
vi), 1 ≤ w ≤ r − 1.
(48)
Note that when vs ≥ W′s + 1,
∑si=1 vi ≥ max1≤i≤r{c′i} + 1 > maxs≤i≤r{c′i}, and conse-
quently all the contention codes in slots bs, ..., br−1 will be attempted. We actually changed
the subscripts and superscripts of those summations associated with vs, 1 ≤ s ≤ r− 1, sincethis simplifies the expression of the PGF Hbr(z, t) =
∑∞k=0Hbr(k, t)z
k.
32
Hbr(z, t) =
cmax∑k=0
V1(k)∑v1=0
...
Vs(k)∑vs=0
...
Vr−1(k)∑vr−1=0
[
r−1∏w=1
(1− p′w)Uw ][
r−1∏w=1
P (vw, bw − bw−1)]P (k −r−1∑i=1
vi, t− br−1 + 1)zk
+∞∑
k=cmax+1
V1(k)∑
v1=W′1+1
[r−1∏w=1
(1− p′w)c′w ]P (v1, b1 − b0)zv1P (k −
1∑i=1
vi, t− b1 + 1)zk−vi
+∞∑
k=cmax+1
r−1∑s=2
W′1∑
v1=0
...
W′s−1∑
vs−1=0
Vs(k)∑vs=W ′
s+1
[s−1∏w=1
(1− p′w)Uw ]
[
r−1∏w=s
(1− p′w)c′w ][
s∏w=1
P (vw, bw − bw−1)]z∑s
i=1 viP (k −s∑
i=1
vi, t− bs + 1)zk−∑si=1 vi
+∞∑
k=cmax+1
W′1−1∑
v1=0
...
W′r−1−1∑
vr−1=0
[r−1∏w=1
(1− p′w)Uw ]
[r−1∏w=1
P (vw, bw − bw−1)]z∑r−1
i=1 viP (k −r−1∑i=1
vi, t− br−1 + 1)zk−∑r−1
i=1 vi ,
(49)
where
Vs(k) = k −s−1∑i=0
vi, 1 ≤ s ≤ r − 1,
W′s = max
s≤i≤r{c′i} −
s−1∑i=0
vi, 1 ≤ s ≤ r − 1,
Uw = min(c′w,
w∑i=1
vi), 1 ≤ w ≤ r − 1,
cmax = max1≤q≤r−1
{c′q}, v0 = 0.
(50)
Observing that Vs(k) is a function of k, whereas Us and W′s are both independent of k, we
can further simplify this PGF by exchanging the orders of summations. This order exchange
is valid since all terms involved are non-negative.
33
Hbr(z, t) =
cmax∑k=0
V1(k)∑v1=0
...
Vs(k)∑vs=0
...
Vr−1(k)∑vr−1=0
[
r−1∏w=1
(1− p′w)Uw ][
r−1∏w=1
P (vw, bw − bw−1)]P (k −r−1∑i=1
vi, t− br−1 + 1)zk
+[
r−1∏w=1
(1− p′w)c′w ][P (z, b1 − b0)−
W′1∑
v1=0
P (v1, b1 − b0)zv1 ]P (z, t− b1 + 1)
+
r−1∑s=2
W′1∑
v1=0
...
W′s−1∑
vs−1=0
[
s−1∏w=1
(1− p′w)Uw ][
r−1∏w=s
(1− p′w)c′w ]
[
s−1∏w=1
P (vw, bw − bw−1)zvi ][P (z, bs − bs−1)−
W′s∑
vs=0
P (vs, bs − bs−1)zvs ]P (z, t− bs + 1)
+
W′1∑
v1=0
...
W′r−1∑
vr−1=0
[r−1∏w=1
(1− p′w)Uw ]
[
r−1∏w=1
P (vw, bw − bw−1)]z∑r−1
i=1 vi [P (z, t− br−1 + 1)−cmax−
∑r−1i=1 vi∑
k=0
P (k, t− br−1 + 1)zk],
(51)
Now we derive Q∗t (z), the PGF of the system size for at t ∈ [br − 1, br).
The first term of Q∗t (z) tells that the latest departure occurs in slots bq+1, ..., br−1 in the
current frame. In this case, we have to discuss case by case the points of arrivals between the
latest point to be observed and time t since they affect the number of contention codes that
are attempted in slots b0, ..., br−1. Consequently, Ibq
br(z, t)’s appearance is similar to Hbr(z, t)
except now we consider only these (r − q − 1) slots and the first term becomes πM+qvq+1, the
steady-state probability of having vq+1 packets at the beginning of the bq+1-th slot given that
a departure occurs at the bq-th slot.
The second term represents the event that the latest departure occurs in a hybrid slot
35
except the aM -th. The third term represents the probability that the latest departure occurs
in slots b0, ..., bN in at least one frame prior to time t.
As before, we use Q∗t (z) where t ∈ [br −1, br) to obtain the system size distribution when
t ∈ (br−1, br−1). After routine differentiations, we compute new termsH ′br(1, t), H”
br(1, t), (Ibr
bq)′(1, t)
and (Ibr
bq)”(1, t) and then obtain the moments of system size.
7 Generalization and Limitations
In this section, we first describe how to integrate the results derived in the previous three
sections to deal with a generally-structured frame-based model, provided that the system
has at least one reserved code for the inspected connection. We then explain why our
methodology does not work for a frame-based pure-contention system. The complexity of
our analysis is also briefly discussed.
7.1 A Model with Alternating Hybrid and Contention Slots
In Sec. 5, we treated a system in which each contention slot contains only one code, and in
Sec. 6 we switched to a system having a succession of multicode slots. Along with a system
containing only hybrid slots, this section provides a procedure to combine characteristics for
a hybrid system with arbitrary slot and code allocations. A hybrid TDMA/CDMA system
may have successive contention slots in which multiple codes are accessible by the inspected
connection, hybrid slots may or may not contain contention codes, and most importantly,
hybrid and contention slots may arbitrarily appear within a frame.
An example of a generally-structured frame is depicted in Fig. 8, where four hybrid
slots and five contention slots exist in a frame for our inspected connection. Since now slot
allocation is made general, the b5-th slot is no longer the last of a frame.
36
.... .... .... .... .... .... .... .... .... ....HH H HC C C C C
a 1 a 2 a 3 a 4b 1 b2 b3 b4 b5
FrameBoundary
FrameBoundary
Figure 8: An example of arbitrarily arranged hybrid and contention slots.
7.1.1 State Transition Matrix
We regard a generally-structured system as a cascade of the models we discussed above.
While deriving the transition matrix or the system size distribution, we first classify a slot’s
type according to the number of contention slots n sitting between the slot and the previous
hybrid slot.
Let us illustrate the classification through the aid of Fig. 8. Suppose that c′1 = c
′2 = 1
but c′4 �= c′5 in Fig. 8. First we classify the types of these four hybrid slots. Referring to
Fig. 8, if the accessible slot prior to our observed hybrid slot is hybrid, i.e., n = 0, then
its derivations should follow Sec. 4. The a2-th slot in Fig. 8 belongs to this category. If
n = 1, e.g., the a3-th slot in Fig. 8, we should follow the treatment of the a1-th slot in Sec.
6, but the derivations are much simpler since the most difficult parts in , HCr (k), Hbr(z, t),
and Ibrbq(z, t) disappear when N = 1. Interested readers are encouraged to derive their own
closed-form formulas of this case. If n > 1 and contention slots prior to this slot all contain
single accessible code, e.g., the a1-th slot in Fig. 8, then derivations in Sec. 5 shall be applied
with only slight modification. Other cases such as the a4-th slot in Fig. 8 are referred to
the general procedure in Sec. 6. Note that n ≤ M +N , where M +N is the total numberof accessible slots in a frame. The inequality holds since we assume that the system has
at least one reserved code, and hence at least one hybrid slot. The cases associated with a
contention slot can also be dealt analogously.
37
The procedure to construct the state transition matrix after having obtained the transi-
tion probabilities is the same as Sec. 6 and the M/G/1 algorithm can be applied to compute
its stationary probability vector, where c = max(max1≤q≤M(rq + cq),max1≤s≤N c′s).
7.1.2 System Size Distribution at an Arbitrary Time
We again combine the derivations in the previous three sections to obtain the system size
distribution for a generally-structured system. First we consider the case that time t belongs
to an accessible slot. The classification is based on the number of consecutive contention
slots prior to the slot to which t belongs. Recall that mathematical derivation is independent
of slot type, we treat hybrid and contention slots in the same manner if they both have the
same number of prior consecutive contention slots.
Referring to Fig. 8, n = 0 for slots b1, b3, b4, and a2. Therefore, we apply to these cases
the derivations in Sec. 4. Since both the a3-th and b5-th slots in Fig. 8 have only one prior
contention slot, the associated system sizes can be derived in the same manner as the b2-th
slot in Sec. 6. Similarly, we obtain system size distribution for t ∈ [a4 − 1, a4) in Fig. 8 as if
we were dealing with the b3-th slot in Sec. 6. Finally, we apply the derivations in Sec. 5 for
the cases t ∈ [a1 − 1, a1) and t ∈ [b2 − 1, b2) in Fig. 8.For t belonging to inaccessible slots, the procedure in obtaining the system size distribu-
tion from the established distribution for accessible slots still works here.
7.2 Limitations
We first illustrate the difficulties we face while applying our methodology to a model with
pure contention scheme. The remaining section is given to discuss the complexity of our
analysis.
38
7.2.1 A Model with Pure Contention Scheme
Now, what if the number of consecutive contention slots prior to a chosen slot approaches
infinity? In other words, what if the system does not have any hybrid slot, i.e., M = 0? In
the sequel we discuss this pure contention model assuming that the number of contention
slots N > 1. The case M = 0, N = 1 can be regarded as a slotted system without the frame
structure, whose performance has been thoroughly investigated in the literature.
Transitions bq → br, where q ≥ r−1, can occur across several frame boundaries regardlessof Xn’s value. They could be regarded as a geometric series and be rewritten as a closed-
form formula such as Eq. (14) if there were hybrid slots. In a model with pure contention
scheme, however, the infinitely many transition probabilities do not follow the geometric
series pattern, and therefore it is invalid to first study the probability function Gq(k) and
then represent all transition probabilities as Gq(k) multiplied by a constant and e(Q−Λ)ubN ,
where u is an integer. The proposed method is fairly inefficient in this case due to summations
involving infinitely many terms.
However, tail terms of this series will vanish as the difference between two consecutive
departures approaches infinity, because the events happen only when no arrival occurs during
a long period of time. In practice, one can compute finite terms to get good approximation
of performance measures for this case.
7.2.2 Complexity Issues
The major portion of computation time is spent in deriving the moments of the system
size. Reexamine these equations we use to obtain the moments in Secs. 4, 5, and 6: Eq.
(15), Eqs. (30), and Eqs. (51)-(52), we find that the complexity of these three models
are respectively O(M), O(N × (M + N)), and O(N × (M + N) × cNmax), where cmax =
max(max1≤q≤M(rq + cq),max1≤s≤N c′s). Note that in the latter two models, although there
are also derivations for the case when t ∈ [aq − 1, aq), the complexity of these equations is
39
O(M +N) and is dominated by the cases when t belongs to a contention slot.
It immediately follows that the models in Sec. 4 and Sec. 5 can be solved in polynomial
time, whereas complexity of the third grows exponentially in N, the number of contention
slots. In words, our analysis may be inefficient when the number of consecutive contention
slots becomes large. In contrast, the number of hybrid slots does not play a dominant role
regarding complexity. Note also that if we regard the problem in Sec. 5 as a special case of
Sec. 6, the problem is polynomially solvable because cmax = 1 and therefore O(N × (M +N)× cNmax) = O(N × (M +N)), which coincides with the result in the above.While reading Sec. 7.1, readers may wonder why we do not regard accessible codes as
contention codes when the success probability that the inspected connection gets to use a
reserved code is 1 and use the methodology developed in Sec. 6, provided that we maintain at
least one hybrid slot. The argument is valid, however to follow that methodology will result
in significant computational burden due to the NP-complete nature of Sec. 6. Henceforth,
it is more efficient to adopt the cascade structure and make classifications.
8 Numerical Results and Discussions
Although we have no intention to less emphasize the importance of the implications conveyed
in the numerical examples provided in this section, the prime purpose is to demonstrate
through these numerical examples that mathematical derivations throughout this paper are
implementable in computer programs. We consider as our traffic source a two-state MMPP
with the following parameters:
Q =
−0.05 0.05
0.05 −0.05
, and Λ =
0.2ρ 0
0 0.1ρ
, (54)
where ρ serves as a measure of the arrival rate since the average arrival rate of this process
In the discussions that follow, we denote slot allocation, reserved code allocation, con-
tention code allocation, and contention probability allocation via row vectors a, r, c, and p,
respectively. If there is no contention code in a hybrid slot aq, i.e., this hybrid slot contains
only reserved codes, we let pq = 1.
In Figs. 9- 10 we consider a system with frame length 48 slots, 6 of which are assigned to
the inspected connection. These six slots all contain reserved codes and are placed uniformly
in time, i.e., a = [8 16 24 32 40 48]. Since all accessible slots are hybrid, we apply results
derived in Sec. 4 to obtain performance measures. In Figs. 9 and 10 we plot the mean and
standard deviation of system size respectively for two different contention probabilities. In
both systems, two reserved codes and one contention code are granted in each accessible slot:
denoted by r = [2 2 2 2 2 2] and c = [1 1 1 1 1 1]. In one system, the contention probabilities
are all 0.5, i.e., p = [0.5 0.5 0.5 0.5 0.5 0.5]. In the other, we use p = [0.2 0.8 0.2 0.8 0.2 0.8].
It follows immediately that the effective number of codes invested per frame in both systems
is 6× (2 + 1× 0.5) = 3× [(2 + 1× 0.2) + (2 + 1× 0.8)] = 15. Since the mean system sizesand their standard deviations almost coincide in these figures, we conclude that queueing
performance is not sensitive to the allocation of probabilities, provided that the effective
number of codes invested is fixed.
In Figs. 11- 12 we switch our attention to the effect of code allocation policies on TDMA
systems. In these figures the frame length is fixed at 40 slots, 3 of which are reserved for the
inspected connection and 7 are for contention slots. These slots are uniformly placed, i.e.,
a = [4 8 12 16 20 24 28 32 36 40]. The contention probability for one system is 0.3 and 0.5
for the other. The system performances in these figures can be obtained from equations in
Sec. 5. According to these figures, the effect of contention probability on the mean system
size is insignificant, regardless of the observation point in a frame.
In Figs. 13- 14, we compare two different code allocation policies. In both systems,
there are 12 reserved codes and 6 contention codes in a frame of 48 slots, and the contention
41
0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 80
0 . 5
1
1 . 5
2
2 . 5
3
3 . 5
4
ρ
Me
an
Sy
ste
m S
ize 6 h y b r i d s l o t s i n a f r a m e o f 4 8 s l o t s
t h e d i s t a n c e b e t w e e n t w o a c c e s s i b l e s l o t s : 8
