A Space-Time Batch-service Queueing Model for Multi-user MIMO

A Space-Time Batch-service Queueing Model for Multi-user MIMO communication systems Boris Bellalta

Miquel Oliver

NeTS - Universitat Pompeu Fabra Roc Boronat 138, 08018 Barcelona Barcelona, Spain

NeTS - Universitat Pompeu Fabra Roc Boronat 138, 08018 Barcelona Barcelona, Spain

[email protected]

[email protected]

ABSTRACT Multi-user MIMO allows the simultaneous transmission of several link-layer frames directed to different users by exploiting the channel’s spatial streams, increasing proportionally the effective channel capacity. Nevertheless, the system performance can be further increased if several frames directed to the same user are combined together on a temporal scale (i.e. packet aggregation), taking benefit from both space and time dimensions. In this paper, a novel finitebuffer space-time general-distributed batch-service queueing model for such a system is presented. Results show the flexibility and accuracy of our queuing model, providing also a first insight on the relation between the space and temporal aggregation parameters and how they can be adjusted to maximise the system performance.

Categories and Subject Descriptors C.2.1 [Network Architecture and Design]: Wireless communication; C.4 [Performance of Systems]: Modelling techniques

General Terms Performance

Keywords Multi-user MIMO, Space-Time Queueing Model

1. INTRODUCTION Multi-user MIMO extends the Single-user MIMO case by allowing the simultaneous transmission1 of multiple frames to single or multiple destinations in both point-to-multipoint

(e.g. broadcast channel) and multipoint-to-point / multipoint (e.g. spatial multiple access) communications [1]. Although Multi-user MIMO relies on recent advances at the physical-layer (i.e. space-time coding), its capabilities must be also considered at higher layers to properly exploit them, specially from the link-layer side, where frames are queued, scheduled and assembled for transmission. In this paper, a finite-buffer space-time batch service queuing model that captures the behaviour of a link-layer transmission queue for general Multi-user MIMO schemes with both spatial multiplexing and temporal aggregation is presented2 . The model is based on batch service queueing models [2], which have received a lot of attention in previous years (for example the GI/M [1,b] /1/K [3], the M/G[a,b] /1/K [4] or the GI/M SC [a,b] /1/K [5]) due to their application to model some features of communication systems such as packet aggregation schemes on a time scale [6]. Examples of use in specific technologies are the works from S. Kuppa et al. [7] and Kejie Lu et al. [8] focusing on WLANs performance analysis. Following the Kendall’s notation, the space-time batch service queuing model is referred as M/G[l,s] /1bd /K, where l  [a, b] are the temporal parameters, with a the minimum and b the maximum number of packets assembled together in a single temporal batch, and s  [smin , smax ] are the spatial parameters, with smin the minimum and smax the maximum number of spatial streams that can used at each transmission. The subscript bd refers to ’batch-dependant’ as the service time of each batch is related to its composition: the number of temporally aggregated packets per spatial stream and the amount of spatial streams used. The queue main features are: • A packet concatenation strategy is used on the temporal scale, where a single header is added to each temporal batch (where a temporal batch is the payload of a group of packets assembled together). Its maximum size is b packets.

1 This is achieved by using multiple antennas at both the transmitter and receiver, as well as, single-antenna receivers are feasible if pre-coding techniques are used at the transmitter [1].

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. This paper is an updated version of the paper presented in MSWiM’09, October 26–29, 2009, Tenerife, Canary Islands, Spain. Copyright 200X ACM X-XXXXX-XX-X/XX/XX ...$10.00.

• Up to smax temporal batches are transmitted simultaneously using multiple transmit antennas. smax can be understood as the maximum number of available channel’s spatial streams. • Next space-time batch is scheduled as soon as previous transmission has finished and there are at least a · smin packets in the queue. Otherwise, it remains idle. 2

In Section 2 both concepts are explained

1111000 0000 111

a,b, smin, smax

Space/Time

s1(t)

MIMO Radio 1

Receiver

A(t)

000 111 111 000

Space/Time MIMO

Radio 2

Transmitter

1111 0000 0000 1111 0000 1111 0000 1111

Radio nt

Space/Time

sn(t)

MIMO Receiver

Figure 1: The Space-Time Multi User MIMO Queue in the Broadcast channel case • The transmitter does not store the packets in transmission. They remain in the queue until the batch is completely transmitted. There are still very few works analysing the queuing implications on MIMO systems. In [9], Zhou et al., propose a queuing model to address multiple modulation and coding schemes in single-user MIMO systems. Taking their work as a general reference, the model presented here is based on a completely different approach, resulting in a more compact and simple analytical representation. Furthermore, it does not totally rely on Markov assumptions (it allows general batch-service distributions). Other works consider also queueing effects on MIMO [10, 11], although they do not focus on a detailed characterisation of the queueing process and rely in high-level features, such as using the effective bandwidth theory. Finally, it’s worth to mention that the presented model is general and does not rely in any technology or specific application. Easily, it can be adapted to each specific case where the Multi-user MIMO with space-time packet aggregation could be considered (e.g. future WLANs). Furthermore, the presented model contributes to fill the still existent gap between link-layer and physical layers, allowing to extend and combine specialised results from both sides.

2. SPACE-TIME PACKET ASSEMBLY In Figure 1, a case example of the Multi-user MIMO queue with space-time packet aggregation is shown. A finite queue length with a maximum capacity of K packets is considered. At each transmission, several packets stored in the queue are assembled together in both temporal (packet aggregation) and spatial (spatial multiplexing) dimensions, achieving simultaneously multiple parallel transmissions and a reduction of unnecessary temporal overheads. Arriving packets are combined in both time and space dimensions, this is: Temporal Aggregation: Packets with the same destination are assembled together, thus reducing the number of required headers, and transmitted over the i spatial stream. The average temporal batch service time at spatial stream i is, Xi,l , which depends on the aggregation factor κ, the packet length (L, bits), the number of packets assembled together in a single batch (l) and the channel capacity (C, bits/second):

L + (l − 1) · κ · L 1 = (1) µi,l C The κ parameter must be understood as the proportional part of useful data (payload) in each packet. For example, a packet of length L bits has a payload length equal to κL bits and a header of length (1 − κ) · L bits. Thus, the aggregation process consists on, given l individual packets, the extraction of the header from each one and assemble their payload together, adding a single header for all of them. This will result in a final temporal batch-length equal to (1−κ)L+l·κ·L bits. Note how the κ parameter has a double effect (given a fixed number of packets l in a batch): first, it causes variable length batches and second, it impacts on the effective traffic load offered to the queue. Along with the benefits of temporal packet aggregation, in [6] it is shown how temporal aggregation in finite-buffers queues can result in unexpected higher blocking probabilities. This is caused by the impact that long batches have on the frequency of the queue departures (there are long time periods where the queue has no departures), increasing the probability that new arrivals fill the queue, even if the total delay to transmit the entire batch of packets is lower than the required to transmit the same number of packets individually. Spatial Aggregation (Multiplexing): The transmission of several frames, each in a single spatial stream, is referred here as spatial aggregation. It is equivalent to spatial multiplexing. It is assumed that all simultaneously transmitted frames start at the same time, allowing the transmitter to encode properly the different spatial streams given that it has some knowledge about the channel state for each destination. Then, the effective duration of a space-time batch is the maximum duration among the individual frames transmitted at each spatial stream, this is max {Xi,l }, ∀i, where i refer to the i spatial stream. Xi,l =

2.1

Space-Time Batch Policy

A new batch is scheduled as soon as it is possible. This is, just after a batch departure and based on the packets already stored in the queue or, if there are less packets than the minimum required to schedule the next batch, just after the arrival of the last required packet. How many packets are scheduled in a batch and how they are distributed be-

tween the spatial and temporal dimensions are defined by the space-time batch policy. Let q denote the queuing state (i.e. the number of packets in the queue) soon after a departure. Next transmission will involve β(q) packets, with: 8 > > >
max smax > > : smax · b

q < a · smin a · smin ≤ q < a · smax a · smax ≤ q < smax · b smax · b ≤ q

(2)

with the following queue state recursion at departure instants qm = qm−1 − β(qm−1 ) + min (vm−1 , K − qm−1 )

(3)

where vm−1 are the packet arrivals between the m − 2 and m − 1 batch departure instants (note that the m batch is scheduled as soon as the m − 1 batch departs, which leaves qm packets in the queue). The number of packets assembled together in the temporal scale (for each spatial stream) is, 8 > >
> : smax b smax · b ≤ q

(4)

and the number of spatial streams used at each transmission is:

s(q) =

8 smin > > < ¨q˝ a

smax > > : smax

q < a · smin a · smin ≤ q < a · smax a · smax ≤ q < smax · b smax · b ≤ q

the steady-state distribution π s based on the PASTA property as Poisson arrivals are assumed3 .

3.1

dm,l =

There are s(q) parallel transmissions, each with an average service time 1/µi,l , i  [1, s(q)], and following a general service time distribution fXi,l (t), where each parallel transmission include l(q) packets temporally aggregated. Notice that, it is assumed that given q packets stored in the queue, they are directed to s(q) different destinations and at least, there are l(q) packets to the same destination. This is an optimistic assumption, specially at low traffic loads and/or with small queues, even packets directed to all destinations arrive with the same probability at the transmission queue. Future works should consider the probability distribution of the number of packets in the queue given a destination and schedule them properly.

3. QUEUEING MODEL A M/G[l,s] /1bd /K queue is used to model the Space-Time Multi-user MIMO system, where l = [a, b] are the temporal parameters and s = [smin , smax ] are the spatial parameters. Packets directed to all destinations arrive to the transmission queue according to a Poisson process with rate λ and are served in batches of β packets, with β  [a·smin , b·smax ]. remain stored in the queue until they are correctly transmitted. Two steps are followed to solve the queuing model: 1) Computation of the departure distribution π d , using the Embedded Markov chain approach and 2) Computation of

Z

∞ 0

e−λt

(λt)m fXl (t)dt m!

(8)

where the first term accounts for the Poisson arrivals, with rate λ, and fXl (t) for the general distributed service times. For the i < a · smin and j[0, K − a · smin ] range, as there are no batches scheduled, the pi,j probabilities are equal to pa·smin ,j . Finally, for all transitions implying to pass through the last state, this is from i ≥ a·smin to j = K−β(i) states, the departure probabilities are simply the sum of previous transitions from state i to j, ∀j, j 6= K − β(i), this P is pi,j = 1 − K−1 u=i pi,u−β(i) .

3.2 (5)

Departure Distribution

The departure probability distribution, π d , is obtained using the Embedded Markov chain approach, solving the linear system π d P = π d , together with the normalisation condition π d 1T = 1. P is the probability transition matrix, where the transition probabilities, pi,j , from state i[0, K] to an state j[0, K] at departure epochs are given by Equation 6. For i ≥ a · smin and j[i − β(i), K − β(i) − 1], all transition probabilities are just related to the arrival of j + β(i) − i packets to the queue, the space-time batch size and its temporal duration at state i (related to l and κ). As both the probability distribution of the space-time batch service duration and the packet arrival distribution are known, the probability of m = j + β(i) − i arrivals during a batch of size β(i) is:

Steady-state Distribution

The steady state probability distribution, i.e., the probability that the queue is at a given state i ∈ [0, K] at any arbitrary time, is computed taking into account the probability that a new frame arrival observes the queue at this target state, that given the PASTA (Poisson Arrivals See Time Averages) property are equivalent. Note that the probability that a new arrival observes the queue at the target state is equivalent to say that the queue has remained at this target state for, in average, 1/λ seconds as the memoriless property of the exponential distribution holds. Figure 2 shows the queue evolution between two consecutive departures, which is related to the pi,j probabilities computed before using Equation 6. This is, the path from state i to j goes through other states and, in average, the queue will remain at the target state during 1/λ seconds. The example depicted is valid for the case of a M/G[l,s] /1bd /K, with l = [1, 2] and s = [1, 2]. Let’s define the period between two space-time batch departures as a cycle, comprising two possible macro-states: non-transmitting (Non-Tx) or transmitting (Tx). As stated before, in each cycle, the queue state changes due to the arrival of new frames, which is done with an average time of 1/λ seconds between two consecutive arrivals. The average duration of each cycle is E[Tc ], which is the sum of both non-transmitting and transmitting periods: E[Tc ] = E[Tnt ] + E[Tt ]. The E[Tnt ] is computed taking into account that the time between two packet arrivals is in average 1/λ and the system is in a non-transmitting state if 3 By steady-state distribition, we refer to the probability distribution at arbitrary times.

pi,j

πis

8 >
:

8 > >
> : 0

·

γtd

1 E[Tt ]

·

1 λ

i < a · smin , j[0, K − a · smin ] i ≥ a · smin , j[i − β(i), K − β(i) − 1] i ≥ a · smin , j = K − β(i) otherwise

P d γnt · E[T1nt ] · λ1 · ij=0 πjd “ “P ”” Pmin(i,K−1) K · j=a·s αjd v=i+1 pj,v−β(j) min PK−1 s 1 − k=0 πk

(6)

0 ≤ i < a · smin a · smin ≤ i ≤ K − 1

(7)

i=K

q={2,3,4,5} Space aggregation q=1 q=3 q=2 q=2 q=3 q=1

q=0

Space aggregation q=0

q=4 q=1

q=5

q=6

q=2

q=0

q=0 q=1

0000 1111 0000000 1111111 0000 00000001111 1111111 0000 1111 1111 11111 00000 0000 1111 0000 1111 1111 0000 0000 000001111 11111 00001111 0000 1111 0000 00000001111 1111111 0000 1111 0000

Cycle Non−tx

Cycle

Cycle

Cycle

Tx

Cycle

Cycle

t

Cycle

Space−Time aggregation Arrival Departure

Figure 2: Temporal evolution (example) of the Space-Time Multi-user MIMO queue previous departure has finished in a lower state than a·smin . This is,

E[Tnt ] =

1 λ

a·smin −1

X

πid · (a · smin − i)

(9)

i=0

E[Tt ] is the duration of the transmitting period, which is equal to the maximum duration among current spatial transmissions and under the requirement, given by the space-time aggregation policy (Equation 2), that the same number of packets are aggregated in all spatial streams4 , this is: K X

E[Tt ] = max(Xi,l ) = max ∀i

∀i

αdj

· Xi,l(j)

!

Performance metrics

Once the π d and π s distributions are obtained, several performance metrics can be computed, such as the packet blocking probability and the probability that the transmitter is empty or non-transmitting:

s Pb = πK ,

X

Pnt =

E[Tnt ] , E[Tnt ] + E[Tt ]

d γtd = 1 − γnt

πqs

(12)

q=0

j=a·smin

(11)

Finally, the steady-state probabilities, π s , are computed following Equation 7. For the states comprised between 0 ≤ i < a · smin , the steady state probabilities are given by the proportional time, in periods of 1/λ seconds in which the system is on that state, which depends on the probability that previous space-time batch had departed in that or in a lower state, as until the queue reaches a · smin packets, there is no batch transmission. For states between 4

3.3

a·smin −1

(10)

P min d where αjd = a·s πk if j = a · smin and πjd otherwise. k=0 Thus, the probability to find the queue in non-transmitting or in transmitting states are, respectively: d γnt =

a · smin ≤ i ≤ K − 1, their steady state probability is computed considering also the possible transitions which require to go through the i state for 1/λ seconds during the service time of current batch. Finally, for the state i P = K, its s s = 1 − K−1 steady-state probability is computed as πK i=0 πi .

How to compute the distribution of the maximum of several random variables is not considered in this work. Those interested in, please refer to [12].

Additionally, the average size of the space-time batches or its counterparts, the average number of spatial streams used and the average batch size per spatial stream, are: K X

E[γ] =

αid β(i)

(13)

αdi s(i)

(14)

αdi l(i)

(15)

i=a·smin

Es [γ] =

K X i=a·smin

El [γ] =

K X i=a·smin

Finally, the average number of packets in the queue, E[N ], and the average packet response time, obtained from E[N ] by applying Little’s Law:

State 0 1 2 3 4 5 6 7 8

Exponential l = [1, 3], s = [1, 2], κ = 0.50 πd πs .5156 (.5158) .3858 (.3858) .2657 (.2656) .2599 (.2599) .1218 (.1215) .1560 (.1557) .0531 (.0530) .0870 (.0868) .0261 (.0261) .0493 (.0492) .0099 (.0097) .0270 (.0269) .0052 (.0052) .0154 (.0154) .0027 (.0027) .0090 (.0091) .0000 (.0000) .0106 (.0107)

Deterministic l = [2, 3], s = [1, 2], κ = 0.25 πd πs .3405 (.3400) .1661 (.1659) .3678 (.3681) .3457 (.3456) .1977 (.1978) .2795 (.2797) .0703 (.0704) .1385 (.1386) .0188 (.0186) .0504 (.0503) .0038 (.0039) .0147 (.0147) .0007 (.0008) .0037 (.0038) .0000 (.0000) .0008 (.0008) .0000 (.0000) .0002 (.0002)

Uniform l = [1, 2], s = [2, 3], κ = 0.75 πd πs .5352 (.5352) .2517 (.2517) .2511 (.2515) .3699 (.3701) .1196 (.1196) .1894 (.1896) .0521 (.0520) .0950 (.0950) .0231 (.0230) .0471 (.0469) .0115 (.0113) .0233 (.0231) .0069 (.0070) .0122 (.0122) .0000 (.0000) .0057 (.0057) .0000 (.0000) .0052 (.0052)

Table 1: Departure and Steady-state distributions for A = 0.8 Erlangs

State 0 1 2 3 4 5 6 7 8

Exponential l = [1, 3], s = [1, 2], κ = 0.50 πd πs .1968 (.1968) .0550 (.0549) .1886 (.1886) .0793 (.0792) .2198 (.2198) .1037 (.1036) .0968 (.0965) .0963 (.0962) .1381 (.1380) .1022 (.1020) .0421 (.0419) .0870 (.0869) .0724 (.0723) .0837 (.0836) .0454 (.0457) .0764 (.0763) .0000 (.0000) .3164 (.3166)

Deterministic l = [2, 3], s = [1, 2], κ = 0.25 πd πs .0206 (.0206) .0055 (.0055) .0813 (.0813) .0276 (.0276) .2371 (.2373) .0890 (.0891) .1861 (.1861) .1282 (.1283) .2953 (.2950) .1835 (.1834) .0799 (.0797) .1693 (.1667) .0993 (.0997) .1481 (.1481) .0000 (.0000) .1035 (.1036) .0000 (.0000) .1473 (.1473)

Uniform l = [1, 2], s = [2, 3], κ = 0.75 πd πs .1974 (.1972) .0554 (.0554) .1683 (.1682) .1028 (.1027) .2140 (.2138) .1200 (.1199) .1105 (.1106) .1113 (.1113) .0814 (.0815) .0098 (.0098) .1353 (.1357) .1009 (.1010) .0928 (.0927) .0960 (.0960) .0000 (.0000) .0727 (.0728) .0000 (.0000) .2415 (.2417)

Table 2: Departure and Steady-state distributions for A = 2.8 Erlangs

E[N ] =

K X

qπqs ,

q=0

E[R] =

E[N ] λ(1 − Pb )

(16)

3.4 Model Validation The M/G[l,s] /1bd /K queuing model provides exact results as it does not relay in any approximations. However, to validate it and show its correctness, numerical results derived from the model are compared with the ones obtained through simulation. The same assumptions are considered for both the model and the simulator. The simulator has been developed in C programming language using the COST simulation libraries [13]. Three different configurations are shown: 1. Scenario 1: a = 1, b = 3 and smin = 1, smax = 2, and κ = 0.50. Exponential batch service time. 2. Scenario 2: a = 2, b = 3 and smin = 1, smax = 2, and κ = 0.25. Deterministic batch service time. 3. Scenario 3: a = 1, b = 2 and smin = 2, smax = 3, and κ = 0.75. Uniform batch service time. Other common parameters are: an average packet length equal to L = 100 bits, a channel capacity equal to C = 100 Kbps and a queue length of K = 8 packets. Regarding the batch service time distribution, it is assumed that it already refers to the entire batch, not to the distribution of individual packets. In fact, it is assumed to be the service distribution of the maximum temporal length among the different spatial streams (see Equation 10).

Tables 1 and 2 show the π d and π s distributions for two traffic loads, A = 0.8 and A = 2.8 Erlangs respectively. Notice the perfect match between the model and simulations, where differences are only caused by rounding errors. Regarding nomenclature, throughout the paper, the traffic load (A) only refers to the the relation between the packet arrival rate and the service time given that no space-time aggregation is done (or equivalently, each batch comprises a single packet and only one spatial stream is used), thus A = λ µ11 Erlangs. These results are only intended to validate the model for different space (smin , smax ) and time (a, b and κ) aggregation parameters, as well as different traffic loads. Thus, comparing them is difficult, although several conclusions can be already derived.

4.

PERFORMANCE EVALUATION

This section focuses on the impact of different temporal (a and b) and space (smin and smax ) values, traffic loads and κ values on the Space-Time Multi-user MIMO queue response. First, four static scenarios are depicted as a first reference about the system performance. They provide a clear relation between the input parameters and the performance achieved. Second, both optimal space and time parameters are computed. Two goals are defined: to minimize the blocking probability (which is equivalent to maximize the throughput) and to minimize the response delay. The considered parameters are a channel capacity of C = 1 Mbps and packets with a fixed length (deterministic) equal to L = 4000 bits. Thus, space-time batches have a deterministic length, only related to l and κ, and also the maximum duration among all spatial streams follows a deterministic

16

0

0.045

10

a=1, b=4; smin=1. smax=3

10

8

6

a=1, b=4; s =1. s min

4

=3

max

a=2, b=4; smin=3. smax=3 a=2, b=4; s =3. s min

0

2

4

6

8

10

−2

10

a=2, b=4; smin=3. smax=5

−4

0.03 0.025 0.02 0.015

10

−6

10

−8

10

a=1, b=4; s =1. s

=3

a=2, b=4; s =3. s

=3

min

−10

10

0.01

min

max max

a=1, b=4; smin=1. smax=5

a=1, b=4; smin=1. smax=5

2

=3

max

Blocking Probability

Average Response Delay

Average Space−Time Batch Size

min

a=1, b=4; smin=1. smax=5

0.035

12

0

a=2, b=4; s =3. s

0.04

14

0.005

=5

−12

a=2, b=4; smin=3. smax=5

10

max

12

0

0

2

Traffic Load (Erlangs)

4

6

8

10

12

0

2

4

Traffic Load (Erlangs)

(a)

6

8

10

12

Traffic Load (Erlangs)

(b)

(c)

Figure 3: Space-Time Multi-user MIMO queue. Basic Scenarios

distribution. The queue size is set to K = 30 packets.

4.1 Basic scenarios Four Space-Time configurations are considered: 1) l = [1, 2] and s = [1, 3], 2) l = [2, 2] and s = [3, 3], 3) l = [1, 4] and s = [1, 5] and 4) l = [2, 4] and s = [3, 5]. In all cases, the same aggregation factor has been considered, κ = 0.5. In Figure 3 the average space-time batch size (Figure 3.a)), the response queuing delay (Figure 3.b)) and the blocking probability (Figure 3.c)) are shown. At low-medium traffic loads the queueing performance is mainly governed for the a and smin parameters and, on the contrary, at high traffic loads, the b and smax parameters become the relevant ones (notice how different minimum parameters do not result in different performances). As expected, between these two regions, both minimum and maximum parameters impact on the overall performance. The relation between the maximum space-time batch size (b · smax ) and the queue size (K) plays also a relevant role on the the E[γ] parameter. For example, in the two first scenarios, E[γ] achieves the maximum batch size value (12 packets) while in the last two, with a maximum space-time batch size of 20 packets, the E[γ] is bounded to 15 packets. The reason is that after the departure of a long space-time batch, the queue suddenly becomes empty or with few packets inside. Thus, resulting in a short next space-time batch. Note how, in steady-state, the system will suffer this pingpong effect between long and short space-time batches. A higher space dimension (higher smax parameter) improve the overall network performance, showing lower blocking probabilities, Figure 3.c), and in general, a lower response delay, Figure 3.b). Considering the delay, notice the importance of the a and smin parameters at low traffic loads. As shown in Figure 3.b), the two cases with a = 2 and smin = 3 show the highest delay until A = 4 and A = 8 Erlangs respectively, as the queue does not start to transmit packets until, at least, there are a·smin packets stored inside. Moreover, as show in Figure 3.c), the use of a · smin > 1 parameters also result in a higher blocking probability, showing no benefits in this specific scenario.

4.2 Optimal parameters Let s∗ and l∗ be the optimal space and time parameters

which: a) minimize the blocking probability (Eq. 17) or b) minimize the response delay (Eq. 18). This is: [s∗ , l∗ ] =

arg min

Pb

(17)

E[R]

(18)

l∈[a,b],s∈[smin ,smax ]

[s∗ , l∗ ] =

arg min l∈[a,b],s∈[smin ,smax ]

Regarding the scenario, the same general parameters as in previous experiments are used. However, now two κ values are considered to study their influence, κ = 0.25 and κ = 0.75. Basically, low κ values make more efficient the temporal aggregation. Figure 4 shows the optimal space and time parameters and its evolution with the traffic load (A) for the two optimization goals and for both κ values: a) minimize the blocking probability, Figures 4.a) and 4.c) and b) minimize the response delay, Figures 4.b) and 4.d). The space and time parameters cooperate together to satisfy the desired goals. For example, in Figure 4.a) with κ = 0.25 and A ≈ 4 Erlangs, it is better to increase a from 1 to 2 and decrease smin from 5 to 4. It means that for this specific traffic load, it is better to transmit a minimum space-time batch of a · smin = 8 packets, using only 4 spatial streams and 2 temporal aggregated packets in each one. If smin was 5, the minimum space-time batch would be equal to 10 packets, requiring a higher initial delay to schedule a batch and thus resulting in a worst performance. Some general observations can be derived: Minimizing the blocking probability: At low traffic loads all space-time parameters remain at its minimum values as increasing them does not result in any gain. Regarding the temporal parameters, for both κ, the b parameter increases as the traffic load grows, although it never reaches its maximum value. It allows longer space-time batches that increase the temporal aggregation efficiency. The a parameter remains constant for high κ values (there is no gain delaying a transmission until several packets can be assembled together, as the overhead reduction does not compensate the time the queue is blocked). However, for low κ values the opposite case is observed, allowing to increase a to benefit from efficient aggregations as the inter-arrival time between packets decreases. About the space parameters, as traffic

5

Time parameters

Time parameters

5

a

4

b

3 2 1 0

0

2

4

6

8

10

a

2

b

1 0

2

4

4

smin

2

s

max

0

2

4

6

8

Traffic Load (Erlangs) − κ=0.25, min(P )

10

s

min

0

2

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grows, the smax rapidly tends to its maximum value. At low traffic, high smin values does not provide any gain, so it remains at its minimum until it jumps to the maximum value. For both κ values, the same results are observed. Minimizing the response delay: With respect to the temporal parameters, a remains at its minimum value (a = 1) for both κ, with the exception with κ = 0.25 and high traffic loads, where it increases to a = 2. Conversely, the b parameter rapidly grows up to its maximum value, specially for low κ, as the temporal aggregation is very efficient and longer batches at low traffic loads do not result in higher queuing delays. At κ = 0.75, the temporal aggregation is not so effective and the system prefers to transmit individual frames until, at medium-high traffic loads, the b parameter moves up to its maximum value. However, the a parameter remains always at its minimum value. Regarding the space parameters, the smax is always at its maximum and the smin follows the same behaviour than b, increasing proportionally as the traffic load grows.

4.3

Optimal Queueing Performance

Using previous optimal space-time parameters, s∗ and l∗ , in Figure 5 the E[γ], E[R] and Pb are shown. The optimisation goals are satisfactorily achieved for each specific case, showing the lowest blocking probability or delay respectively. However, in terms of performance, there are no significant differences between minimising the delay or the blocking probability. Additionally, these small differences are only at low or medium traffic loads, disappearing at high traffic loads as there are enough packets stored in the queue to use the same space-time parameters (the most effective) for both goals (see Figure 4). The same behaviour for both κ values is observed. To minimise the delay, the E[γ] shows the lowest values and increases quasi linearly with the traffic load until the queue starts to saturate, instant at which the temporal aggregation starts to be effective. It means that the queue prefers to transmit the stored frames as soon as possible, even if some efficiency (in terms of aggregation capabilities) is lost. On

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Figure 5: Space-Time Multi-user MIMO queue with optimal parameters

the contrary, to minimise the blocking probability (maximise the throughput) the system prefers to delay some packets in order to achieve a most efficient transmission, allowing to reduce the packet loss due to queue overflow. [4]

5. CONCLUSIONS A M/G[l,s] /1bd /K queuing model for Multi-user MIMO systems with Space-Time frame aggregation has been presented and validated. It allows for the queueing performance analysis of multiple space-time configurations, as well as for the policies governing them. The model presented here can be easily extended to consider prone-error channels, the impact of retransmissions, the trade-off between transmit diversity and/or spatial multiplexing, the consideration of different transmission rates at each antenna, the packet distribution inside the queue based on the destination of each packet, the arrival process (batch arrivals), etc. Moreover, it can be applied to multiple specific wireless technologies such as in cellular networks or WLANs. A prospective use of this model and its future extensions is the evaluation of frame-based scheduling algorithms for multi-user MIMO communication systems. Thus, the performance results presented here provide a reference for further works on this field.

[5]

[6]

[7]

[8]

Acknowledges This work was partially supported by the Spanish Government under project TEC2008-06055/TEC. The author would like to specially acknowledge the contribution of the reviewers to improve the final quality of this paper.

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[12] A. Papoulis. Probability, Random Variables, and Stochastic Processes. McGraw-Hill, 4th edition, 2002. [13] Gilbert (Gang) Chen. Component Oriented Simulation Toolkit. http://www.cs.rpi.edu/ ˜cheng3/, 2004.