We study the performance of a statistical multiplexer whose inputs consist of a superposition of packetized voice sources and data. The performance analysis predicts voice packet delay distributions, which usually have a stringent requirement, as well as data packet delay distributions. The superposition is approximated by a correlated Markov modulated Poisson process (MMPP), which is chosen such that several of its statistical characteristics identically match those of the superposition. Matrix analytic methods are then used to evaluate system performance measures. In particular, we obtain moments of voice and data delay distributions and queue length distributions. We also obtain Laplace-Stieltjes transforms of the voice and data packet delay distributions, which are numerically inverted to evaluate tails of delay distributions. It is shown how the matrix analytic methodology can incorporate practical system considerations such as finite buffers and a class of overload control mechanisms discussed in the literature. Comparisons with simulation show the methods to be accurate. The numerical results for the tails of the voice packet delay distribution show the dramatic effect of traffic variability and correlations on performance.
While the "call" or "session" is the basic entity that is set up in many data traffic applications, the performance analysis of data network elements depends on the internal units of traffic into which calls are decomposed.In a packet-switching network, the packet represents the basic internal unit of traffic, and packets from different calls time-share facilities and contend for network resources, giving rise to queuing delays. In this paper, we consider the problem of characterizing the doubly stochastic packet process resulting from a superposition of call types, each type having a stochastically varying number of calls in progress.We obtain statistical properties of the process and use them to obtain an approximating process, based in part upon time constants associated with the packet-rate covariance function. We discuss existing queuing models dealing with this ap proximating class of inputs and present results showing the effect of call and packet traffic parameters on queuing performance.
The optimal filtering equations, as derived by K h a n [I], 121, require the specification of a number of models for a given application. This paper concerns itself with the effect of errors in the assumed models on the filter response. The types of errors considered are those in the covariance of the initial state vector, the covariance of the stochastic inputs to the system, and the covariance of the uncorrelated measurement noise.Presented here is a derivation of a recursive equation for the actual covariance matrix of the estimation error when the filter design is based upon erroneous models. The derived equation can also be used to obtain the covariance matrix of the estimation error when the optimal filter gains are approximated by simple functions of time to be used in a real-time 6ltering application.A numerical example illustrates the use of the derived equations. IETRODUCTIONThis paper is concerned with the determination of the performance of the suboptimal filter obtained when a Kalman [l ] filter design is based upon erroneous noise models and a priori statistics. The performance is specified by the covariance matrix of the suboptimal estimation error.A recursive equation is derived for this matrix which makes possible a quantitative evaluation of the statistical quality of the estimate obtained. SYSTEM MODELThe system model used is as follows:(1) y(tk) ~ll(tk)?.(fJ:) + .L'(lk) ( 4 where 1) r(tt) is the n vector of states at time tk E ( x ( t o ) ) = 0 and E ( x ( t o ) x ' ( f o ) ) = P*(to) 2) + ( f k + l ; a ) is the transition matrix ( n x s ) of the system 3) u(tk) is an n vector of stochastic inputs such that E{,t&)) = 0 and E(zt(t&'(tj)} = St.jQ(tk) 4) y ( f k ) is an m vector of measurements taken a t time t k 5) M(tJ:) is an n t x n matrix (Measurement Sensitivity Matrix) 6) o(tk) is an m vector of additive measurement noise with the following properties: OPTIMAL FILTERThe optimal filter can be represented as a linear dynamic system by the following system of equations [2]:x*(4+1 I fk) = $*(tJ:+l; tk)Z*(tk I fm-1) 4-R * ( t k ! > ' ( t k ) (3) $*(tk+1; tk) = @(tk+1; tk) -R * ( t k ) X ( t J J (4) R*(tk) = q t i + 1 ; a)P*(tl.).l~'(k)[-~~(tn)P*(fk)l~~'(fe) + R(td]-I (5) P * ( f . 4 = $*(ta+l; k)P*(k)+'O~+l; t d + Q(h4 (6)where 1) x * ( t n + l ! t t ) is the optimal estimate of the state vector x ( f t + d , having observed y(to), y(tl), . . , y ( h ) 2) $ * ( f k + l ; ts) is the transition matrix ( n X n ) of the filter 3) R * ( t b )is an n x m matrix of optimal gains 4) P * ( t k + l ) is the covariance matrix of the estimation error. ManuscriptThe author is with Bell Telephone Laboratories. Inc., Whippany. N. J. ANALYTIC RESULTSNow assume that the actual gains employed in the filter design deviate from the optimal gains, i.e., that R ( t k ) = R*(tk) + 6K*(tk) (7) and $ ( k + 1 ; tk) = +*(tk+1; tn) -a K * ( t k ) i ) K ( t k ) .(8)In order to determine the effect of these errors we ask the following question. How close do we come to the optimum by using the following filter equation ?&...
In a variety of, overloaded queueing, systems (e+, an overloaded call processing system), long delays can result either in poor service given to the customer or in customers, unknown to the system, turning "bad." For example, in switching systems, long dial tone delays can result in customers initiating,dialing before receiving dial tone. In this case the system will, not receive ali the digits and an unsuccessful call results. This can lead to the system expending real time on unsuccessful services and; therefore, reduces the effective throughput. Thus, there is a deed for control schemes which reillice the load offered to the processor by selectively refusing service to some customers in such a way as to keep delays,, for those customers which are selected for service, small. This fact has been recognized and has led to improved strategies for local switches. In this paper ,we analyze and compare the performance of various queueing and service disciplines for an M / M / l Queue. We consider LIFO and FIFO schemes with customer rejection mechanisms corresponding to pushing,out or timing out older customers in queue. Delay distributions for served customers are obtained and comparisons based upon throughput-delay tradeoffs are presented. For the situation where customers can turn ribad'' at a random time after their arrival, we compare the throughput of good customers. The results presented are a mixture of classical results, which are briefly stated, and new results which are developed in more detail. The numerical results show a dramatic effect of the queueing and service disciplines on the overload performance and a strong dependence of the throughput of successful services on the meclianism for customers turning "bad." Although results are obtained for a single seii-er queue, they can be used to approximately analyze overload control schemes which control access to distributed systems.
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