Analysis of DAR ( 1 ) / D / s Queue with Quasi-Negative Binomial-II as Marginal Distribution

In this paper we consider the arrival process of a multiserver queue governed by a discrete autoregressive process of order 1 [DAR(1)] with Quasi-Negative Binomial Distribution-II as the marginal distribution. This discrete time multiserver queueing system with autoregressive arrivals is more suitable for modeling the Asynchronous Transfer Mode(ATM) multiplexer queue with Variable Bit Rate (VBR) coded teleconference traffic. DAR(1) is described by a few parameters and it is easy to match the probability distribution and the decay rate of the autocorrelation function with those of measured real traffic. For this queueing system we obtained the stationary distribution of the system size and the waiting time distribution of an arbitrary packet with the help of matrix analytic methods and the theory of Markov regenerative processes. Also we consider negative binomial distribution, generalized Poisson distribution, Borel-Tanner distribution defined by Frank and Melvin(1960) and zero truncated generalized Poisson distribution as the special cases of Quasi-Negative Binomial Distribution-II. Finally, we developed computer programmes for the simulation and empirical study of the effect of autocorrelation function of input traffic on the stationary distribution of the system size as well as waiting time of an arbitrary packet. The model is applied to a real data of number of customers waiting for checkout in an airport and it is established that the model well suits this data.


Introduction
In B-ISDN/ATM network, IP packets or cells of voice, video, data are sent over a common transmission channel on statistical multiplexing basis.The performance analysis of statistical multiplexer whose input consists of a superposition of several packetized sources is not a straightforward one.The difficulty in modeling this type of traffic is due to the correlated structure of arrivals.A common approach is to approximate this complex non renewal input process by analytically tractable arrival process, namely discrete autoregressive process (DAR).The impact of autocorrelation in traffic processes on queueing performance measures such as mean queue length, mean waiting times and loss probabilities in finite buffers, can be very dramatic.
The DAR process, constructed and analyzed by Jacobs and Lewis [1] has developed into one of several standard tools for modelling input traffic in telecommunication networks.The discrete autoregressive process of order 1 [DAR (1)] is known to be a good model for VBR coded teleconference traffic as in Elwalid et al. [2].Kamoun and Ali [3] modeled an ATM multiplexer as a discrete time multiserver queueing system with on-off sources and studied the transient and stationary distribution of the number of packets in the system.
Hwang and Sohraby [4] obtained the closed form expression for the stationary probability generating function of the system size of the discrete time single server queue with DAR(1) input.Hwang et al. [5] obtained the waiting time distribution of the discrete time single server queue with DAR(1) input.Choi and Kim [6] analyzed a multiserver queue fed by DAR(1) input.Kim et.al [7] derived mean queue size in a queue with discrete autoregressive arrival of order p.
In this paper we analyzed a discrete-time multi-server queue with s servers (s > 0) having deterministic service times (specifically, service time is 1) and the following arrival process: Let A m be the number of arrivals at time .Then A m = A m-1 with probability β; otherwise, A m is sampled independently from a quasi-negative binomial distribution-II.The stationary distribution of the waiting time in that queue is calculated numerically with a matrix analytic method.Specifically, the arrival process is first analyzed at embedded times when A m is sampled independently of A m-1 or when A m is less than the number of servers.This analysis reduces to an analysis of a Markov chain of M/G/1 type as presented in Neuts [8].Then the stationary distribution of A m at general m is derived, which in turn gives the stationary distribution of the waiting time.0,1, 2, m   The rest of the paper is arranged as follows.Quasi-Negative Binomial Distribution-II is described in Section 2. Queues with input traffic as DAR(1) with marginal Quasi-Negative Binomial-II is explained in Section 3. Analysis of DAR(1) /D/s queue with marginal Quasi-Negative Binomial Distribution-II is given in Section 4. The stationary distribution of the Markov renewal process is given in Section 5. Deriving the stationary distribution of system size and waiting time of an arbitrary packet is explained in Sections 6 and 7.The quantitative effect of the stationary distribution of system size and waiting time on the autocorrelation function as well as the parameters of the input traffic is illustrated numerically in Section 8.The application to real data set is given in Section 9.

Quasi Negative Binomial Distribution-II
The quasi-negative binomial distribution (QNBD) obtained by Janardan [9], Sen and Jain [10] has the probability mass function as for 0; > 0, > 0, > 0 and 0; < 0; = 0,1, 2 where x be the number of occurrences.When p 2 = 0 the QNBD reduces to negative binomial distribution (NBD) and when n = 1, QNBD reduces to quasi geometric distribution (QGSD) for n = 1.QNBD tends to the Consul and Jain's [11] generalized Poisson distribution.But unfortunately the moments of this distribution appear in an infinite series which is not suitable for summation.The method of moments fails to provide quick estimates of its parameters.Hence Ahmad et al. [12] introduced a new model of quasi negative binomial distribution-II (QNBD-II).This new model has the probability mass function When 2 0 p  this new model reduces to negative binomial distribution.The probabilities of QNBD-II decreases with the successive occurrences.This tendency of probabilities suggests its possible applications in reliability, biometry, and survival analysis.The QNBD-II is uni-model and only its first moment (mean) appears in compact form.The lower and upper bound of Mode M is   Let X be a quasi-negative binomial variate with parameters (n, ) and pmf given by ( 2).If such that 3) Let X be a quasi-negative binomial variate with parameters (n, 1 2 ) and pmf given by (2), then zerotruncated quasi-negative-binomial distribution-II tends to zero-truncated generalized Poisson distribution as .
, p p n  

Queues with DAR(1) Arrivals with Quasi Negative Binomial Distribution-II as Marginal
The input ATM multiplexer with VBR coded teleconference traffic is assumed to be DAR (1) When the input process has quasi-negative binomial distribution-II as marginal we have b x as the pmf of the form (2).

 
Discrete Autoregressive Process of order 1 (DAR (1) is defined by the regression equation as where are i.i.d Bernoulli random variables with and    : 1,2,3, The properties of DAR(1) are as follows 1) is stationary The probability distribution of X(t) is the same as the distribution of Y(t) 3) The autocorrelation function for X(t) at lag t is obtained as t the parameter  is the decay rate of the autocorrelation function.

Analysis of DAR(1)/D/s Queue with Quasi-Negative Binomial-II as Marginal
We assume that the input process is DAR(1) with quasi-negative binomial distribution-II distribution as the marginal distribution and there are s servers (s > 0) whose service occurs at constant rate.In this integer valued time queue, the time is divided into slots of equal size and one slot is needed to serve a packet by a server.We assume that packet arrivals occur at the beginning of slots and departures occur at the end of the slots.Here represents packet arrivals so that X(t) is the number of packets arriving at the beginning of the t slot.
Let N(t) be the number of packets in the system say system size, immediately before arrivals at the beginning of the t th slot.Then is a two dimensional Markov process of M/G/1 queue type.The state space is The number of phases is infinity.So the computation of stationary distribution of is not easy to work out.
In practice by matrix analytical method and using the theory Markov regenerative processes, we compute the stationary distribution of the new process at the embedded epochs   as follows, we have The packet arrivals at and after t  are independent of the information prior to t  given J  .From this, it is The probability transition matrix of the Markov renewal process is computed as follows.

The Stationary Distribution of the
Markov Renewal Process is obtained as above.
Where the elementary matrices are 0 , and

 
We apply matrix analytic method as described below.The transition probability matrix P has infinite order, so that it would have to be truncated before we implement matrix analytic method.We assume that there exists some index N such that for all .That is we assume that the Markov chain does not jump more than N steps at a time so that the matrix is of finite order, see Latouche and Ramaswamy [14].For a numerical illustration , consider the case when s = 5 and N = 14.Then the transition probability matrix P can be obtained as By arranging the transition probability matrix into (sxs) matrices we get In general we can symbolize the transition matrix P as The elements of P can be written as A matrix P of the above structure is said to be of M/G/1 type, which underlines the similarity to the embedded Markov chain of the M/G/1 queue.With respect to the levels , the Markov chain is called skip free to the left, since in one transition the level can be reduced only by one.
By the matrix analytic method we proceed as follows.
Step 1: Find the minimal nonnegative solution G of the matrix equation G can be given by the following iteration See Breuer [15] 0 1 0 G is a stochastic matrix ,and hence we can stop the iteration procedure when Step 3:  and e is the s (s + 1) dimensional column vector whose components are all ones.

Stationary Distribution of
is a discrete time Markov regenerative process with the Markov renewal sequence    , , : =0,1,2 We have The numerator of Equation ( 3) is The denominator of Equation ( 3) is where is the stationary probability vector of the Markov process   whose transition probability matrix is The infinitesimal transition matrix of ( 5) is By solving the balance equations we obtain the stationary distribution of the Markov process By substituting ( 6) into (4) we obtain the denominator of the right hand side of (3) as   The stationary distribution or the limiting probabilities Copyright © 2011 SciRes.AM w i s e j 

Stationary Distribution of Waiting Time of an Arbitrary Packet
Let W denote the waiting time of an arbitrary packet at steady state.Then for = 0,1, 2 . w P(W = w) = (Mean number of arrivals in a slot at steady state whose waiting time is w)/(Mean number of arrivals in a slot) Suppose that there are n packets immediately before arrivals at the beginning of the t th slot and that the number of packet arrivals is j at the beginning of the t th slot, so that N(t) = n and X(t) = j.Then the number of packets whose waiting time is w among the ones who arrive at the beginning of the t th slot is Therefore the mean number of arrivals in a slot at steady state whose waiting time w is Since the mean number of arrivals in a slot is  , the following theorem is obtained from (7).

Theorem 7.1
The distribution of the waiting time W of an arbitrary packet is given by
The parameter β gives the information on the strength of correlation of the input process.Stationary system size is larger for the large β (see Figure 1).Also stationary system size stochastically increases when the parameter p 1 of the input process decreases (see Figure 2).
The complementary distribution function of the waiting time of an arbitrary packet,when λ = 2.5 and β = 0.3,  Stationary waiting time of an arbitrary packet, is larger for large β (see Figure 3).Also stationary waiting time of an arbitrary packet, stochastically increases when the input parameter p 1 decreases (see Figure 4).We assume the number of servers to be 3 Tables 1-3 display the stationary probabilities of the system size for different values of

Analysis and Modeling of a Data Set
In this section we apply the model to a data on the number of initially waiting customers for checking in an airport for a time period of 30 minutes each .The data was collected from morning 8.00 A.M to 11.30 P.M for one week.This includes all the busy periods as well as idle periods.The data is taken from the file customer checkout.xlsxavailable in [17].Table 6 gives the frequency distribution of the corre-sponding data, where x is the number of customers initially waiting for the service.= 0,1, ,30 t  In the present paper we assumed the number of arrivals as DAR(1) with marginal Quasi Negative Binomial II distribution.Thus the data set can be fitted to the the Quasi Negative Binomial II distribution as follows.


To test whether there is a significant difference between an observed distribution and the Quasi Negative Binomial II distribution, we use Kolmogorov-Smirnov [K.S.] test for 0 H : Quasi Negative Binomial II distri- bution with parameter p 1 = 0.021 and p 2 = 0.00513 is a good fit for the given data.Here the calculated value of the K.S. test statistic is 0.017857 and the critical value corresponding to the significance level 0.01 is 0.088924, showing that the assumption for number of arrivals follow Quasi Negative Binomial II distribution is valid (see Figure 5).By applying matrix analytic method we obtain the stationary distribution of system size and waiting time of an arbitrary customer for the Quasi Negative Binomial II/D/s queue.Here the mean = λ =4.3125.To satisfy the stability condition we assume the number of servers as .Also we assume the value of autocorrelation function = 5 s = 0.1  , 1 and 2 .Tables 7  and 8 display the stationary distribution of waiting time of an arbitrary customer and system size.

Figure 2 .
Figure 2. Complementary distribution function of the stationary system size, when λ = 2.5, β = 0.3.0.5, 0.7 & 0.9 and the complementary distribution function of the waiting time when β = 0.3 and p 1 = 0.009, 0.0015, 0.02 & 0.024 (p 2 = 0.0064, 0004, 0.002 & 0.0004) respectively are derived.Stationary waiting time of an arbitrary packet, is larger for large β (see Figure3).Also stationary waiting time of an arbitrary packet, stochastically increases when the input parameter p 1 decreases (see Figure4).We assume the number of servers to be 3Tables 1-3 display the stationary probabilities of the system size for different values of1 2   , ,& p p

Tables 4
and 5 display the stationary probabilities of waiting time of an arbitrary packet for different values of

Figure 4 .
Figure 4. Complementary distribution function of the waiting time of an arbitrary packet, when β = 0.3.

Figure 5 .
Figure 5.The Probability histogram of real data and the Quasi Negative Binomial II distribution with p 1 = 0.021 and p 2 = 0.00513.Table 7. Table showing the stationary distribution of waiting time of an arbitrary customer P(W =ω)when β = 0.1, λ = 4.3125, s = 5.