Mean-Variance Optimization of Response Time in a Tandem M/GI/1 Router Network with Batch Arrivals Nalan G¨ ulpınar, Peter Harrison, Ber¸c Rustem Department of Computing, Imperial College London South Kensington Campus, London SW7 2AZ, UK. {ng2,pgh,br}@doc.ic.ac.uk
Louis-Francois Pau ∗ Rotterdam School of Management, Erasmus University F 1-19, Po Box 1738, NL 2000 DR Rotterdam, Netherlands.
[email protected]
Abstract The end-to-end performance of a simple wireless router network with batch arrivals is optimized in an M/G/1 queue-based, analytical model. The optimization minimizes both the mean and variance of the transmission delay (or ‘response time’), subject to an upper limit on the rate of losses and finite capacity queueing and recovery buffers. The trade-off between mean and variance of response time is assessed and the optimal ratio of arrival-buffer size to recovery-buffer size is determined, which is a critical quantity, affecting both loss rate and transmission time. Losses may be due to either full buffers or corrupted data. Graphs illustrate performance in the near-optimal region of the critical parameters. Losses at a full buffer are inferred by a time-out whereas corrupted data is detected immediately on receipt of a packet at a router, causing a N-ACK to be sent upstream. Recovery buffers hold successfully transmitted packets so that on receiving a N-ACK, the packet, if present, can be retransmitted, avoiding an expensive resend from source. The impact of the retransmission probability is investigated similarly: too high a value leads to congestion and so higher response times, too low and packets are lost forever, yielding a different penalty.
Keywords: Performance optimization, batch arrivals, queuing theory, router design, wireless network
1
Introduction
Extensive recent advances in communications technology has highlighted the importance of quantitative evaluation of alternate schemes for supporting, at sufficiently high sustainable data rates, packet-based communication wirelessly, with mobility, in public networks. All wireless technologies are driven by an increasing need for mobility with low latency, high availability and the least possible infrastructure and management costs. The so-called MANET – mobile ad-hoc network – is one such development. These networks have no fixed infrastructure, except at the edges, and can be deployed spontaneously over a geographically limited area in a dynamic environment [1]. A MANET is formed by wireless-enabled terminal devices, used to extend the coverage of the access points using multi-hop protocols. Hence these devices can interact as routers, forwarding packets to other devices. Users expect a QoS from MANETs as close as possible to regular fixed networks and there are further constraints, for example relating to energy consumption by the mobile devices, their heterogeneity and the extent to which their location is known. Isolation behavior may occur, ∗
Ericsson, Network Core Products Division, Sweden.
P33/1
where a network node (typically a router) cannot reach any of its neighbors for physical reasons, such as distance, perturbation or radio range; this depends directly on the buffering capabilities at each node. A more precise measure than just isolated node detection can be obtained by estimating the connectivity of a node, i.e. the number of neighbor nodes that it can reach; this is directly related to retransmission rates. Furthermore, the ad-hoc network organization requires a new traffic load balancing mechanism on the network scale. Therefore, the traffic must be differentiated to provide statistics about control packets and routing control packets in order to estimate the resulting overhead; the effect of this overhead can be investigated, and then minimised, in a queuing model. While the focus of this paper is routers, especially those used in wireless communications, similar issues arose in large scale ATM switches. In that case, different buffering techniques were used to solve the contention problems in ATM switch backplanes, and then in QoS studies. In [7], the input, output and combined central and output queuing buffers were studied in an n × n ATM switch, where the cell arrivals followed a Bernoulli process and the destinations were uniformly distributed. As in this paper, analyses of loss and sensitivity to buffer sizes were made, but no moment calculations nor feedback traffic were taken into account. The most significant aspects addressed here are the modelling of recovery buffers in a novel M/GI/1-based network, the computation of higher moments of response times in M/GI/1 queues with batch arrivals and the higher-order optimization of performance – in terms of higher moments than expected values, here just the variance. The paper integrates an analytical performance model for a two-node tandem network into a formal nonlinear optimization framework, which minimizes end-to-end transmission time in an environment of small, realistic network scenarios. This paper describes preliminary research towards an analytical, static, optimization-based design of routers for use in wireless communications. In view of the diverse design options possible, supporting models have many adjustable parameters and choosing the best set for a particular performance objective is a delicate and time-consuming task. In section 2, we describe and model the two-node network of routers with finite capacity and recovery buffers. In section 3, an optimization problem to minimize the mean and variance of the end-to-end delay is introduced. Numerical results are presented in section 4. The paper concludes with a summary of our contributions.
2
A tandem router network
Consider a two-node, tandem M/GI/1 router network with batch arrivals. The end-to-end transmission time (also called response time) in this queueing network model is the basis of the objective function of our optimization and the optimum value of the ratio of arrival to recovery buffer sizes at a node, given the total buffer capacity, is obtained. The network has a sender (source) node and receiver (destination) node at the ends. At each node there are arrival (for queueing) and recovery buffers. The model is depicted in Figure 1. We assume that batches arrive at the server as a Poisson process with mean arrival rate λ 0 batches (not packets) per unit time and batch size with specified probability distribution with mean b. There are many situations where the Poisson approximating assumption is reasonable: for example, a superposition of sparse, independent renewal processes is asymptotically Poisson and the net arriving traffic is indeed likely to come from many sources. An enhancement of the model, which represents more typical Internet workloads with correlated traffic. The majority of traffic that violates the Poisson assumption does so by virtue of burstiness, which is well described by batch arrivals such as we have here. The queuing model allows generally distributed service times, which comprise the time to look up router tables, manage header information and dispatch the packet so that the next can be processed. In a batch-Poisson stream, successive batches arrive after intervals which are independent and exponentially distributed [11]. Customers’ service times, which are independent of each other and of the arrival process, have mean values 1/µ1 and 1/µ2 , respectively; i.e. the service rates are µ1 and µ2 . The departure process from the first queue is approximated as Poisson with unit batches. P33/2
λ r = p( λ − λ") Retransmission N1
N2
N3
LC1 g 1 Sender Node
LC2 g 2
Finite Buffer
λ = b λo + λ r Poisson Batch Arrivals
A1
R1
µ1
g 3
Finite Buffer A2
λ’
Server
µ2
R2
Server
λ" Completed Transmissions
Receiver Node
LF 2 = λ’ − λ"
LF 1 = λ − λ ’ T1
T2
T3
Figure 1: A two-node router network with batch arrivals. In the very special case of unit batch arrivals and exponential service times, this assumption is exact if the node-capacity is infinite, and a good approximation if losses are rare. Conversely, we could easily use a similar M/GI/1 queue with batches at the second node too, but it requires a much deeper analysis to correlate the arrival process there with the departure process at the first queue; this is an open question. Any packet which has not transferred successfully, is lost. The main reasons for losses are either a full buffer or corrupted data on arrival at a buffer. Losses might also occur due to incorrect routing or a wrong entry or late update in the routing table at a node. The effect on the sender (and receiver) is the same as a full-buffer loss. In this model it is assumed that garbage (a packet with corrupted data) is not buffered. In the case of losses due to corrupted data on arrival at each buffer, LC 1 , LC 2 , a negative acknowledgement (N-ACK) is sent upstream, seeking a clean copy of the packet in a recovery buffer, which is used at each node to store any packet which has been successfully sent downstream. If a recover buffer is full on transmitting a packet successfully, then the oldest packet is deleted. Losses due to a full buffer at each node, LF 1 , LF 2 , cause an additional transmission delay by a timeout of duration much greater than the mean per-link transmission delays, for example a prespecified multiple of the estimated end-to-end delay.
2.1
Model structure and parameters
In this section, the model’s design in relation to its main fixed (pre-defined) parameters, control (user defined) parameters and output (performance) parameters, as well as to its constraints, is described; for more detailed information the reader is referred to [12]. The net rate at which packets are resent by the sender node due to losses of all types is defined as the retransmission rate, λr . The total rate of all packet arrivals, including retransmissions, is therefore λ = bλ0 + λr . The probability that a packet is lost because of corruption that leads to garbage data arriving at node 1, node 2 and the receiver are g1 , g2 and g3 , respectively. Such garbage is detected on arrival at a node, before joining the queue, and causes a N-ACK to be sent upstream to the sender. We assume that acknowledgements are processed fast, at high priority and do not incur queueing delays. More accurately, N-ACKs (and ACKs) are generated fast by a process running on a node’s arrival buffer and an ACK packet is sent back to the previous node’s output buffer. Because of the high priority of ACKs and N-ACKs and the fact that they do not have to pass through arrival
P33/3
buffers, they suffer negligible queueing delays, although they may place a small overhead on the nodes. Consequently, the simplified representation of N-ACK transmission in our model should not be unduly inaccurate. N1 , N2 and N3 are the N-ACK delays from node 1 to the sender, from node 2 to the node 1, and from the receiver to node 2, respectively. Hence, referring to Figure 1, the N-ACK delay, i.e. the time for the N-ACK to arrive back at the sender, is N1 , N1 + N2 and N1 + N2 + N3 when corruption is detected at node 1, node 2 and the receiver node, respectively. Losses due to full arrival buffers are influenced by the total, fixed capacities Ci of nodes i = 1, 2. These capacities are partitioned into fixed size arrival buffers (Ai packets) and recovery buffers (Ri packets), defined with the constraints Ai + Ri = Ci at node i = 1, 2. T1 , T2 and T3 are the constant transmission delays for a packet passing over the links between the sender and node 1, node 1 and node 2, and node 2 and the receiver node, respectively. Hence the transmission delay for successful packets, excluding the time spent in the nodes, is T = T 1 + T2 + T3 If a transmission has been successful, then the receiver sends a positive acknowledgement (i.e. an ACK) for the uncorrupted packet to the sender node, according to the transmission protocol. When the arrival buffer at a node is full, the rejection of an arriving packet will result in no ACK being sent. The value of the sender’s timer will reach the pre-specified time-out value, the packet will be deemed lost and a retransmission will be attempted with a given probability. The value of the time-out period is set to a multiple k of the estimated mean transmission time, for some k ≥ 1 set by the user. Let fi define the probability that a packet is rejected at node i due to a full buffer. Now, the probability that a packet is successfully transmitted from the network is the ratio of the net arrival rate to the throughput at the receiver, λ000 /λ. The probabilities of successful packets being transmitted from node 1, node 2 and the receiver are represented by the following constraints, respectively: λ0 λ00 λ000 = (1 − g1 )(1 − f1 ), = (1 − g )(1 − f ), = (1 − g3 ) 2 2 λ λ0 λ00 where λ0 , λ00 are the throughputs of successful packets arriving at the node 2 and the receiver node. The user also defines the probability p of retransmission, 0 ≤ p ≤ 1, which is incorporated in the optimization model. When p = 0, no retransmissions are allowed so losses are higher but congestion is lower. When p = 1, there is always a retransmission attempt and so there are no losses (assuming no catastrophic faults) and the number of retransmissions is unlimited. This causes extra load on the network which might result in heavy congestion and hence significantly longer delays. During transmission, total losses should not exceed a specified maximum loss rate β, measured conventionally as a fraction of the external packet arrival rate. This implies (1 − p)(λ − λ000 ) ≤ βbλ0 Given the input and control parameters, the output parameters are chosen to be: • the utilization of each node; • the throughput of the network, i.e. the rate at which successful transmissions are received; • the mean response time of successful packets, i.e. the mean end-to-end delay from the instant that the packet was first sent to the instant of its successful arrival at the receiver; • the variance of response time of successful packets; • the rate of losses due to either corruption or full buffers.
P33/4
These are determined by our modified queueing model below and defined precisely in the following section. The server utilization at node i = 1, 2 is simply defined to be the ratio of its throughput to its service rate: λ00 λ0 , u2 = u1 = µ1 µ2 The Poisson arrivals have arrival rates λ (batches per unit time) at node 1 and λ 0 (packets per unit time, the throughput of node 1) at node 2. The throughput of successful transmissions from node i is then obtained approximately by direct analysis of the corresponding finite capacity M/M/1/Ai queue (i = 1, 2). This is a plausible assumption since we are dealing with the extreme tail of the queue length probability distribution if losses are very small, the situation usually demanded as a constraint. Large deviation theory then shows that queue length distributions have exponential tails [20], regardless of service time and arrival process characteristics (provided they are independent). Thus in general we approximate the loss probability in queue i = 1, 2 by P (Q = Ai | Q ≤ Ai ) =
P (Q > Ai − 1) − P (Q > Ai ) 1 − P (Q > Ai )
where Q denotes the queue length random variable in the corresponding unbounded queue at equilibrium. At large q, loge P (Q > q) ' −qI where I = sup{θ > 0 | Λ(θ) < 0}, Λ(θ) = loge E[eθA ] + loge E[e−θC ] and A, C are the random variables denoting respectively the number of arrivals and departures in unit time. For an M/M/1 queue, A and C are Poisson with means λ and µ respectively, and we find that I = − log e λ/µ, giving: A1 A2 ρ 1 − ρ µ ρ 1 − ρ µ2 1 1 2 1 2 0 00 λ = , λ = 1 − ρ1A1 +1 1 − ρ2A2 +1 More generally, if arrivals are Poisson batches, where the probability generating function of the batch size is GN (z), we find I is the solution of the equation λ(1 − GN (eθ )) + µ(1 − e−θ ) = 0.
3
Response Times
Let GX (z) denote the probability generating function of a discrete random variable X and let T ∗ (θ) = E[e−θT ] denote the Laplace-Stieltjes transform of the probability distribution of a (continuous) random variable T , where E[.] denotes expectation. The nth moment of any random variable F is denoted Mn (F ) = E[F n ] and, for brevity, F and F represent the mean and variance of F , respectively. We also use F to denote a generic instance of a sequence of independent, identically distributed (iid) random variables F1 , . . . , Fn , i.e. F has the same distribution as Fi for any 1 ≤ i ≤ n. Let the service time random variable for a single customer at a queue be S, with mean 1/µ, and let the batch (Poisson) arrival rate be λ and the batch size random variable be N with mean N = b. Let SB be the service time random variable for batches with size distributed as N . The sojourn time, or waiting time, W , of a customer (in a batch) is the sum of the time it spends waiting to start service and its service time. We consider the last customer in a batch. The last customer enters the queue with its batch and the batch completes its service when the last customer is served. Hence, the service completion of the last customer in a batch represents the completion of the batch, i.e. successful transmission of the message, and the waiting time of the last customer is also the waiting time of the batch. We can therefore obtain the Laplace-Stieltjes transform of the batch response time distribution in such an M/GI/1/∞ queue with batch arrivals of random size N as that in a standard M/GI/1 queue, with unit arrivals but service time random variable SB : WB∗ =
∗ (θ) (1 − ρ)θSB ∗ (θ)) θ − λ(1 − SB
P33/5
where ∗ SB (θ) = E[e−θS ]
= E[E[e−θ(S1 +S2 +···+SN ) |N ]]
= E[(E[e−θS ])N ] = GN (S ∗ (θ))
3.1
Total mean transmission time
The mean waiting time (and arbitrary higher moments) at node 1 is approximated by applying the recurrence formula in [12] to the Laplace-Stieltjes transform of the response time distribution of the 0 M/GI/1/∞ queue with batch arrivals. This yields W 1 = −W ∗ (θ)|θ=0 where 0
W ∗ (0) =
−λM2 (N ) + 3λb − λbµ21 M2 (S) − 2µ1 b 2µ1 (µ1 − λb)
and (recall) ρ1 = µλ1 is the server utilization at node 1. M2 (N ) and M2 (S) are the second moments of batch size and service time distributions, respectively. In numerical experiments, we consider constant (deterministic) and geometric distributions for the batch size. Hence, given mean batch size b, M2 (N ) = b2 , 2b2 − b, respectively. For the service time, we consider constant and exponential distributions and use M2 (S) = µ12 , µ22 , respectively. Clearly the use of the result for an unbounded 1 1 M/GI/1 queue is an approximation since we have finite capacity buffers with losses. However, when loss rates are very small, as is usually required, the error is vanishingly small in view of the exponentially small tails of the queue length distributions – a response time is a sum of service times, one for each waiting task, plus one residual in a non-empty queue. The expected response time at node 2, W 2 , is the M/M/1 response time approximation W2 =
A2 +1 2 1 − (A2 + 1)ρA 2 + A 2 ρ2 2 µ2 (1 − ρ2 )(1 − ρA 2 )
The total expected response time W , i.e. the combined time spent in the two nodes on a successful transmission attempt, is the sum of the expected response times at each node, W = W 1 + W 2 . The variances of the response times at nodes 1 and 2 are computed as 2
W 1 = M2 (W1 ) − W 1 3(1 − ρ1 − λW 1 ) M2 (N ) − b + bµ21 M2 (S) + 3ρ1 µ21 M2 (S) [M2 (N ) − b] = 3µ1 (1 − ρ1 b) 2 ρ1 M3 (N ) − 3M2 (N ) + 2b + bµ31 M3 (S) λ(M2 (N ) − 3b + bµ21 M2 (S)) + 2µ1 b − + 3µ1 (1 − ρ1 b) 2µ1 (µ1 − λb) 2
W 2 = M2 (W2 ) − W 2 i h A2 +1 2 2 −1 + 2(A22 − 1)ρA (1 − ρ2A2 +1 ) 2 − A2 (A2 + 1)ρA 2 − A2 (A2 − 1)ρ2 2 = 2 2 µ22 (1 − ρ2 )2 (1 − ρA 2 ) " # A2 +1 A2 +1 2 2 1 − (A2 + 1)ρA 1 − ρ2A2 +1 1 − (A2 + 1)ρA 2 + A 2 ρ2 2 + A 2 ρ2 + − 2 2 2 2 µ2 (1 − ρ2 ) ρ2 µ2 (1 − ρA µ2 (1 − ρ2 )(1 − ρA 2 ) 2 )
3.2
Overhead
Failed packets, due to either corrupted data or loss at a full arrival buffer, retry a number of times given by the retry-probability p. Because each retry is made independently of previous attempts, this number of attempts is a geometric random variable with parameter p (for a persistently failing transmission). The overhead at any attempt, L, incurred by a failed transmission, i.e. the elapsed time between the start of that attempt and the start of the next attempt, consists of: P33/6
• the time-out delay set to k(W + T ) (k mean successful transmission times) for packets lost due to a full buffer; • the time elapsed between the instant that the failure is detected—i.e. by detection of garbage on arrival at a node—and the instant the re-sent packet is received1 . Note that the second item includes the time required for a N-ACK to be transmitted upstream to the appropriate recovery buffer or to the sender. This overhead is summarised in the following table in terms of types of delays and failures and corresponding probabilities; for a probability of t ∗ , we write t¯∗ = 1 − t∗ . Failure Type Garbage at link 1 Time-out delay due to full buffer 1 Garbage at link 2 (in recovery buffer 1) N-ACK delays (not in recovery buffer 1) Time-out delay due to full buffer 2 Garbage at link 3 (in recovery buffer 2) Garbage at link 3 (in recovery buffer 1) N-ACK delays (not in recovery buffer 1) No failure due to garbage or full buffer
L N 1 + T1 k(W + T ) T 2 + N2 W 1 + T1 + T2 + N1 + N2 k(W + T ) T 3 + N3 W 2 + T2 + N2 + T3 + N3 W +T +N 0
Probability g1 g¯1 f1 g¯1 f¯1 g2 q1 g¯1 f¯1 g2 q¯1 g¯1 f¯1 g¯2 f2 g¯1 f¯1 g¯2 f¯2 g3 q2 0 g¯1 f¯1 g¯2 f¯2 g3 q¯2 q1 g¯1 f¯1 g¯2 f¯2 g3 q¯2 q¯1 0 g¯1 f¯1 g¯2 f¯2 g¯3
Table 1: Overhead for various failures and corresponding probabilities. The mean overhead, L, is expressed in terms of the probabilities of the different causes of failure as follows: L = g1 (N1 + T1 ) + g¯1 f¯1 g2 q1 (T2 + N2 ) + k g¯1 f1 + f¯1 g¯2 f2 (W + T ) + g¯1 f¯1 g2 q¯1 W 1 + T1 + T2 + N1 + N2 + g¯1 f¯1 g¯2 f¯2 g3 q2 (T3 + N3 ) + g¯1 f¯1 g¯2 f¯2 g3 q¯2 q 0 1 W 2 + T2 + N2 + T3 + N3 + g¯1 f¯1 g¯2 f¯2 g3 q¯2 q¯1 0 W + T + N 2
The variance of the overhead is L = M2 (L) − L where
M2 (L) = g1 (N1 + T1 )2 + g¯1 f¯1 g2 q1 (T2 + N2 )2 + k 2 g¯1 f1 + f¯1 g¯2 f2 (W + T )2 + g¯1 f¯1 g2 q¯1 M2 (W1 + T1 + T2 + N1 + N2 ) + g¯1 f¯1 g¯2 f¯2 g3 q2 (T3 + N3 )2 + g¯1 f¯1 g¯2 f¯2 g3 q¯2 q 0 1 M2 (W2 + T2 + N2 + T3 + N3 ) + g¯1 f¯1 g¯2 f¯2 g3 q¯2 q¯1 0 M2 (W + T + N )
q1 , q2 , q10 are the probabilities of finding a clean copy of a corrupted packet in the upstream recovery buffer at node 1 (when the corrupted packet was detected at node 2), at node 2 (when it was detected at the receiver) and node 1 (when it was detected at the receiver), respectively. These probabilities will depend on the size of the recovery buffer—the bigger the buffer, the higher the recovery probability—and the time elapsed between a corrupted packet being detected and the resulting N-ACK arriving at the node holding the clean packet—the longer this delay the more likely the clean packet will have been replaced by a packet transmitted later. For the estimation of these probabilities, the reader is referred to [12].
3.3
Cost Function
In addition to retransmissions, which incur a time-penalty in our model, there is a possibility that a transmission will be abandoned, i.e. ‘lost forever’, in the event that a retransmission is not tried, with probability 1 − p. Such a loss is assigned a more severe time penalty of k 0 where k 0 > k and 1
assumed successful; a second failed transmission is assumed to be a second-order, negligible approximation.
P33/7
00
acquisition time penalty of k > 0. Let Fm be the probability of a loss forever at the mth attempt. This is computed as λ000 λ000 m−1 m−1 Fm = 1 − p (1 − p) 1 − λ λ Let CFm be the cost random variable for transmissions that end after m attempts: i.e. are either successful or lost forever. The response time random variable for successful transmissions and the loss penalty CFm are given in the following table, together with their associated probability of occurrence on the mth attempt. Function
CFm
Response Time Lost Penalty
0
λ000
W + T + (m − 1)L
λ
00
k (W + T ) + k + (m − 1)L
Probability h im−1 000 p 1 − λλ Fm
Table 2: Cost random variable and corresponding probabilities. The mean of the cost function consists of mean response time delays and time penalties and is calculated as CF
= E[E[CFm ]|m] ∞ λ000 X λ000 m−1 = (W + T ) + (m − 1)L p 1− λ λ m=1
+
∞ X
m=1
=
i h 0 Fm k (W + T ) + k 00 + (m − 1)L
h 0 i λ000 (W + T ) + pL(λ − λ000 ) [pλ000 + λ(1 − p)] + (1 − p)(λ − λ000 ) k (W + T ) + k 00 λ(1 − p) + pλ000
Similarly, assuming the constituent delays are independent and recalling that the T and N terms are constant, we obtain an aggregated cost function taking into account second moment (variance) of response time delays and time penalties as CF = M2 (CF ) − CF
2
where 2 M2 (CF ) = E[E[CFm ]|m] ∞ 000 X λ λ000 m−1 = p 1− M2 (W + T ) + 2(m − 1)(W + T )L + (m − 1)2 M2 (L) λ λ m=1
+
∞ X
m=1
0 h i 0 00 00 Fm M2 k (W + T ) + k + (m − 1)2 M2 (L) + 2(m − 1)(k (W + T ) + k )L
Performing the summations and simplifying, we obtain h 02 i 0 00 00 2 000 λ000 M2 (W + T ) + (1 − p)(λ − λ ) k M2 (W + T ) + k + 2k k (W + T ) = (1 − p)λ + pλ000 i h 000 000 0 00 000 000 p(λ − λ ) M2 (L)[(1 + p)λ − pλ ] + 2Lλ (W + T ) + 2L(1 − p)(λ − λ )[k (W + T ) + k ] + [(1 − p)λ + pλ000 ]2
P33/8
3.4
The optimization problem
The optimization model minimizes the mean and variance of the end-to-end delay subject to an upper limit on the rate of losses. For the case of non-zero recovery buffer size, the optimization problem is stated as follows: q min
A1 ,A2 ,R1 ,R2 ,λ,λr ,λ0 ,λ00 ,λ000 ,f1 ,f2
αCF + (1 − α)
CF
subject to λ = bλ0 + λr λ ρ1 = µ1 ρ2 =
λ
0
=
λ00 = λr βbλ0 λ0 λ λ00 λ0 λ000 λ00 Ci
1 ρ1 (1 − ρA 1 )µ1
(1 − ρ1A1 +1 )µ2 1 ρ1 1 − ρ A µ1 1 1 − ρ1A1 +1
2 ρ2 (1 − ρA 2 )µ2
1 − ρ2A2 +1 = p(λ − λ000 )
≥ (1 − p)(λ − λ000 )
= (1 − g1 )(1 − f1 ) = (1 − g2 )(1 − f2 ) = (1 − g3 ) = Ai + Ri 0
i = 1, 2 00
000
0 ≤ λ r , λ , λ , λ , A i , R i , fi
i = 1, 2
where λ0 , g1 , g2 , g3 , β, p, C1 and C2 are pre-determined constants. Notice that 0 ≤ α ≤ 1 and represents trade-off between the mean and variance transmission times. The optimization model minimizes CF when α = 1 and CF when α = 0. Such problems are routine for convex objective functions when the parameters are real numbers and often well approximated when some are integers—by allowing them to be real and choosing the nearest integers as candidate optima. We present our numerical examples in section 4.
4
Numerical results
The optimization model described in the previous section is implemented and integrated with a software package. The software has been written in C++ and uses the nonlinear solver E04UCF, Nag Library [15]. In our computational experiments, the parameters, which are input to the optimization model, are as follows: the service rates at nodes 1 and 2 are µ1 = µ2 = 1.0E + 06, the probabilities of garbage data arriving at nodes 1, 2 and the receiver are g1 = g2 = 0.0002 and g3 = 0.001, 1.0 , and the link transmission delays are the acknowledgement delays are N1 = N2 = N3 = 10.0×µ 1 1.0 T1 = T2 = T3 = 10.0×µ . 1 Partly in order to motivate the optimization model, we have first considered the mean transmission time over the range of retransmission probabilities, 0 ≤ p ≤ 1 in Figure 2. The arrival buffer size at node 2 is fixed at 10 and the arrival rate is chosen to be 90% of the service rate of each node. The graph on the right in Figure 2 is a plot of W (the Z-axis) versus retransmission probability p (the Y -axis) and arrival buffer size at node 1, A1 (the X-axis). P33/9
0.00016
Objective_Function_Value
0.00014
Optimum Value
0.00012
0.0001
8e-05
6e-05
4e-05
2e-05
0.1
0.2
0.3
0.4
0.5 0.6 Probability
0.7
0.8
0.9
1
Figure 2: Mean cost function versus retry probabilities with and without optimization. Although not realistic, this network being overloaded with excessively high accepted loss rate, these parameters are chosen to reveal interesting performance characteristics, in particular optimum operating points, to illustrate our approach. In the left graph, the optimum buffer size at node 1 is obtained as 20. This clearly verifies that the minimum mean transmission time is obtained at the optimum arrival and recovery buffer size. We next carry out experiments to illustrate the performance of the optimisation model and present the results in terms of different parameters, varying the value of the user-defined retransmission probabilities 0 ≤ p ≤ 1. In order to show the impact of batch size distribution on the arrival and recovery buffer sizes, we have considered the optimization model based on an M/G/1 queue with constant (deterministic) and geometric batch size distributions. In the numerical experiments, the mean batch size is chosen to be 3. The results show that the distributions do not have a big impact on the model; i.e. both constant and geometric distributions behave in the similar way in terms of the optimal buffer sizes. For the illustrative purposes we only present the results obtained with constant batch size distribution.
Figure 3: Arrival buffer sizes versus retry probabilities for mean and variance of the cost function. Different attitudes of the variance versus mean of the cost function can be obtained by varying α between the range of 0 and 1. For illustrative purposes, we only present the results of the optimization models which minimize CF and CF when α = 1 (at the left) and when α = 0 (at the right), respectively. In Figures 3 and 4, the optimum arrival and recovery buffer sizes versus retry probabilities, subject to 10% maximum loss rate are plotted. Notice that when retransmission is not permitted, i.e. p = 0, the arrival buffer capacities at nodes 1 and 2 are kept at the minimum values, whereas the recovery buffer capacities are at their maxima. When p > 0, since the number of arrivals is increased due to retransmissions, the arrival buffer sizes are also increased, the recovery buffer sizes being correspondingly decreased so that the constraints on the total node capacities are satisfied. P33/10
Figure 4: Recovery buffer sizes versus retry probabilities for mean and variance of the cost function.
Figure 5: Number of packets transmitted successfully rom nodes 1, 2 and the receiver. Figure 5 displays the number of packets transmitted successfully from nodes 1, 2 and the receiver versus p. This figure shows how, as p approaches 1, the rate of successful transmission increases, since the total arrivals increases due to higher retransmission.
5
Conclusion
We have developed a model of a two-node router network in order to demonstrate the optimization of end-to-end performance, using a Markov model. The numerical results showed how end-to-end delay varies with various parameter combinations, from which optimal operating points could be deduced with considerable, tedious effort. This motivated our formal optimization approach which finds these points automatically. A two-node network was chosen for its simplicity in illustrating our approach, but it is clear that arbitrary sized networks could equally well be accommodated at greater computational effort. The present model is a considerable enhancement of the one described in [12]: the external traffic is bursty (represented by batch arrivals), service times can have any probability distribution, and the penalty incurred by packets that fail completely after a number of retransmissions is accommodated. In the former model, traffic was pure Poisson, service times were exponential and complete failures were not taken into account. Moreover, the present model takes into account the variability in transmission times as well as their mean values, by optimizing variance as well as expected value. These enhancements represent a significant advance, even though we cannot yet represent correlated traffic, as often observed in packet radio networks.
References [1] IETF MANET Mobile Ad-hoc Networks, http://www.ietf.org/htmlcharters-charter.html, Working Group. P33/11
[2] C-K Toh, Ad-hoc Mobile Wireless Networks, Pearson Education, Prentice Hall, New Jersey, 2002 ISBN: 0-13-007817-4. [3] D.M. Blough , M. Leoncini, “The k-neigh Protocol for Symmetric Topology Control in Ad-hoc Networks”, Proc. 4 th Intl Symposium on Ad-hoc Networking and Computing (ACM MOBIHOC), Annapolis, MD,June 2003, pp. 141–152. [4] W. Chen, N. Jain, S. Singh, “ANMP: Ad-hoc Network Management Protocol”, IEEE J. on Selected Areas in Communications (JSAC), Vol 17, No 8, August 1999, pp. 1506–1531 [5] O. Angin, A.T. Campbell, M. Kounavis, R. Liao, “The Mobiware Toolkit: Programmable Support for Adaptive Mobile Networking”, IEEE Personal Communications, Vol 5, No 4, August 1998. [6] F. Eshgi, A. Elhakeem, Y. Shayan, “Performance Evaluation of Multihop Ad-hoc WLANs”, IEEE Communications Magazine, Vol 43, No 3, March 2005, pp. 107–115. [7] M. Guizani, Ala Isam Al-Fuqaha, “A New Queuing Strategy for Large Scale ATM Switches”, IEEE Communications Magazine, Vol 39, No 12, 2001, pp. 142–146. [8] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam,m E. Cayirci, “A Survey on Sensor Networks”, IEEE Communications Magazine, August 2002, pp. 102–114. [9] J. Borje, H.A. Lund, A. Wirkestrand, “Real-time Routers for Wireless Networks”, Ericsson Review Journal, Vol 76, 4, 1999, pp. 190–197. [10] R. Chakka, P.G. Harrison, “A Markov Modulated Multi-server Queue with Negative Customers - The MM CPP/GE/c/L G-Queue”, Acta Informatica, Vol. 37, 2001, pp. 881–919. [11] P.G, Harrison, N.M. Patel, Performance Modeling of Communication Networks and Computer Architectures, Addison-Wesley Publishing, 1992. [12] N. Gulpinar, P.G, Harrison, B. Rustem, L-F Pau, “An Optimization Model for a Two-Node Router Network”, MASCOTS 2004, pp. 147–156. [13] S. Keshav, R. Sharma, “Issues and Trends in Router Design”, IEEE Communications Magazine, May 1998, pp. 144–151. [14] D. Mitra, “Stochastic Theory of a Fluid Model of Producers and Consumers Coupled by a Buffer”, Advances in Applied Probability, Vol. 20, 1988, pp. 646-676. [15] NAG Library, http://www.nag.co.uk, The Numerical Algorithms Group, Oxford. [16] L-F Pau, “Mobile Network Evolution Requirements on Optimization of Routing Function”, Seminar, Department of Computing, Imperial College, London, 2004. [17] L-F Pau, B. Wafae, and A. Bousquet, “Fault Detection Capability Implications for Flow Control in Data Communication Networks”, Proc. 1979 IEEE Int. Conf. AUTOTESTCON’79, Minneapolis, 19-21 Sept. 1979, pp. 42–56. [18] K.C. So, K.E. Chin, “Performance Bounds on Multiserver Exponential Tandem Queues with Finite Buffers”, European Journal of Operations Research, Vol 63, 3, 1992, pp. 463–477. [19] C.S. Sunga, M.H. Parkb, “Performance Analysis of a Queuing System with Facilities Requiring Post-processing”, Computers and Operations Research, Vol 19, 8, 1992, pp. 767–781. [20] A. Ganesh, N. O’Connell and D. Wischik, Big Queues, Lecture Notes in Mathematics 1838, Springer Verlag, 2004.
P33/12
