**ISSN: 0976-3031 **

**Research Article **

**PERFORMANCE ANALYSIS OF A DISCRETE TIME QUEUING MODEL WITH BULK SERVICE POLICY **

**Pukazhenthi.N**

**1**

_{ and Poornima.S}

_{ and Poornima.S}

**2**

1

_{Department of Statistics, Annamalai University, Annamalai Nagar-608 002, Tamilnadu, India }

2

_{Department of Statistics, Krishnasamy college of Science Arts and Management for women, }

### Cuddalore, Tamilnadu, India

**DOI: http://dx.doi.org/10.24327/ijrsr.2017.0806.0391 **

**ARTICLE INFO ABSTRACT **

In this paper we investigate analysis discrete time queue modeling technique which is mainly applied to the analysis of communication systems and it is suitable for the performance evaluation of Asynchronous Transfer Mode (ATM) switches. We assumed that the inter arrival times follows negative binomial distribution and service time is geometric distribution. The service times of successive packets (data) are assumed to be conditionally independent given the bulk size (L,K). The steady state distribution after and before a customer departure at an epoch in the discrete time queuing model are obtained, using embedded Markov Chain technique. Some numerical example are given at the end of this paper.

**INTRODUCTION **

**Motivation**

Discrete-time queuing models have often been used in the performance analysis of ATM multiplexers and switches. In these models, the time axis is assumed to be divided into equal fixed-length of time intervals, referred to as slots. Arrivals and services of customers can start or end at slot boundaries only so that the service time is an integer multiple of slots. These assumptions are suitable for modeling (ATM) networks. A fundamental motive to study discrete-time queues is that they are more appropriate than their continuous-time counterparts for analyzing computer and telecommunication systems, since nowadays these systems are more digital (such as a machine cycle time, bits, and packets) than analogical. In view of this, there have become increasingly important due to their applications in the study of many computer and communication systems such as (ATM) networks.

The packets arrive one by one and their inter arrival times follow negative binomial distribution. The arriving packets are queued in FIFO order. One server transports packets in batches of minimum number and maximum , the service times following geometric distribution. The server accesses new arrivals even after service has started on any batch of initial number ‘j’(L≤ j < ). This operation continues till random service time of the ongoing batch is completed or the maximum capacity of the batch being served attains “ ’ whichever occurs first. Mathias [8] has derived the inter departure time distribution for batches and for single slot, using discrete-time analysis and the coefficient correlation between inter departure times and batch sizes. Mathias and Alexander [7] have shown how discrete-time analysis can be technically were first used to analyze the basic single server systems. Neuts [9] proposed the "general bulk service rule" in which service being only when a certain number of customers in the queue under are available. The service times of successive groups are assumed to be conditionally independent given the bulk size, but may depend on their magnitude. Chaudhry and Gupta [4] has discussed, analysis of the single server discrete-time finite-buffer queue with general inter arrival and geometric service time, 1/ /1/ . Baburaj [3] have discussed, on the transient distribution of a single and batch service queuing system with accessibility to the batches. This studies an deals with the transient distribution of a single and batch service queuing system incorporating accessibility to the batches. Baburaj and Rema [2] have discussed a controllable bulk service

**International Journal of **

**Recent Scientific **

** Research **

*International Journal of Recent Scientific Research *

*Vol. 8, Issue, 6, pp. 17614-17619, June, 2017 *

**Copyright © Pukazhenthi.N and Poornima.S, 2017**, this is an open-access article distributed under the terms of the Creative
Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the
original work is properly cited.

**DOI: 10.24327/IJRSR **
**CODEN: IJRSFP (USA) **

**Article History:**

Received 05th March, 2017 Received in revised form 08th April, 2017

Accepted 10th May, 2017 Published online 28st June, 2017

**Key Words: **

queueing system with accessible and non-accessible batches. Veena Goswami. et.al., [13] analyze in their paper, a discrete-time
multi-server queue in which service capacity of each server is a minimum of *′* *′*_{ and maximum of}_{′ ′}_{ customers. The inter-arrival }

and service-times are assumed to be independent and geometrically distributed. Sivasamy and Pukazhenthi [12] has analyzed the
discrete time bulk service queue with accessible batch with the arrivals and service times as general and negative binomially
distribution. An interesting paper given by Baburaj [1] and he deals with the concept of a discrete time bulk service queue under the
policy '( , , )*′*. The inter-arrival times are assumed to be independent and geometrically distributed. Discrete-time renewal input
bulk service queue with changeover time studies has been discussed by Vijaya Laxmi Pikala. et.al., [14]. In this paper they consider
a model such that a discrete-time infinite buffer renewal input single server queue with changeover time under *′*(a, c, b)′ policy. The
service and changeover time are geometrically distributed. The server begins service if there are at least ccustomers in the queue
and the services are performed in batches of minimum size a and maximum size b(a ≤ c ≤ b). Pukazhenthi and Ezhilvanan [10]
discussed the discrete time queue length distribution with a bulk service rule. Pukazhenthi and Ezhilvanan [11] recently used the
analysis of discrete time queues with single server using correlated times.

The rest of the paper is organized as follows: In section 2, presents the description of the model, the steady state distribution of system occupancy at departure epochs. The join distribution of inter-departure time and number of pockets in a batch are studied in section 3. In section 4, some numerical study for various performance measures of the steady state distribution of the batch server at departure instants are given. A brief conclusion is presented in section 5.

**Queuing Model **

A discrete time single server finite capacity queue with inter arrival time follows negative binomial distribution and service times of
batches follows geometrics distribution. Discrete time queuing model have often been used in the performance analysis of ATM
multiplexes and switches. In these models the time axis is assumed to be divided into fixed length time intervals, referred to as slots
and the servicing of cells synchronosis, (i.e.) the service of a cell can start or end at a slot boundaries only, this implies that the
service time of a cell always consists of an integer number of slots so that it can be considered as a discrete random variable**. **

**Arrival Distribution **

The arrival time of batches are also independent and identically distributed (i.i.d) random variable and have common Pascal Negative-binomial distribute on { ( ; , ) = ( = )∶ = , + 1, + 2 … . } so that

( ; , ) = −1

−1 (1− ) ; = , + 1, + 2 …. … (1)

and is the number of slots required to complete a batch service at the success in a sequence of independent Bernoulli trails
with probability *′ ′* for success and = 1− is the probability of failure. The mean and variance of (1) are as follows

Mean arrival time: ( / )and variance of services time: ( / )

. ( ) = and ( ) = ( / ) … (2)

**Service distribution **

Let the number of customers that service during slot be n. In this model we assume that the service time of customers are independent and identically distributed random variable with probability mass function follows geometric distribution.

{ ( ) = ( = ): = 0,1,2 … . }

. . ( ) = (1− ) ∶ = 0,1,2 … … (3)

Hence, the mean service time (1/ ) and variance is ( / )

. ( ) = 1 , ( ) = (2− )⁄ ( ) = / … (4)

To ensure that the discrete-time queuing system is stable, the assumptions needed for subsequent analyses have been summarized.
Each slot is exactly equal to the transmission time of a batch of size K. The time interval( , + 1) will be referred to as slot
+ 1; = 0,1,2 … Arriving packets are of fixed size and queued in a buffer of infinite capacity until they enter the server. A
packet cannot enter service on its arrival in the slot. The probability of a packet arriving at the close of the slot is ‘ ’ and it's not
arriving at that point is *′ ′* which implies that the inter-arrival time is negative binomial distributed with parameter ‘ ’. A batch of
packets of size ( ≤ ≤ ) starts service at the beginning of a slot, and will end service at success just before the end of a
slot. The probabilities of the batch server completes either a success or a failure at the end of a slot are *′* *′* and *′* = 1− *′*
respectively. For more information see Hunter [5]. Service time of a batch is independent of the number of packets in a batch and
simultaneous occurrence of both arrival of a packet and departure of a batch in a single slot is ruled out. The value of the load ‘ ’of
the server is less than unity (i.e.,).

ρ= E(A)

KE(B)= p

**Distribution of System Occupancy at Departure Epochs **

Let

###

{0, 1, 2, … ,###

} be the number of packets accumulated in the system (queue + service) just after the server has left with nth batch. The steady state distribution { ( ) = lim →*∞*( ) = lim →

*∞*Pr( = ) : = 0,1,2, …

*∞*} of system occupancy at

departure epochs is derived in this section using the embedded Markov Chain (MC) technique.

Let denoting the number of packets that reach the system during the service. Then the distribution { ( ): = 0,1,2 … } of can be derived as follows:

( ) = −

*∞*

(1− ) −1

−1 (1− ) … (5)

for = 0,1,2. . . *∞*

= + −1

−1 (1− )

_{+} + _{(1}_{−} _{)} _{(1}_{−} _{)} _{+} + + 1 _{〖}_{(1}_{− 〗} _{)} _{(1}_{−} _{)} _{⋯}

= + −1

−1 [ (1− )] [1−(1− )(1− )]

= [ (1− )]

(1−[(1− )(1− )])

= [ (1− )]

( + − )

= (1− )

ℎ =

+ −

**Mean **

( ) =

*∞*

+ −1

−1 (1− )

= (1− )

= (1− ) … (6)

**Variance **

( ) =

*∞*

+ −1

−1 (1− ) −

*∞*

+ −1

−1 (1− )

( ) = ( + 1)(1− ) + (1− )+ (1− )

= (1− ) … (7)

The sequence { } of random variables can be shown to form a MC on the discrete state space {0,1,2, … , *∞*} with the following
one step transition probability matrix = where

=

⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪

⎧ _{( )}_{ }_{; 0}_{≤ ≤} _{−}_{1} _{= 0}

( − + ) ; 0≤ ≤ −1 ≥1

( ) ; ≤ ≤ = 0

( − + ) ; ≤ ≤ ≥1 ( + − ) ; ≥( + 1) ≥( − ) 0 ; ℎ

… (8)

The unknown probability (row) vector = ( , , , … ) can now be obtained by solving the following system of equations:

bound N and ‘ ’ and ‘ ’ of unit step probability function have been selected so that the values are very small for all ,

###

and they could be ignored. Thus ∑ = 1 and = 1− ∑ for all 0≤ ≤ and = is a square matrix of order( + 1). The average queue length and average system length applying equation (10) have obtained.

= ( ) + ( − ) ( ) and
*∞*

= ( ) … (10)
*∞*

**Numerical Results **

Some numerical results have been obtained and presented in the form of a table-1 which is self-explanatory. The probability distribution of system occupancy at service completion epochs has been reported in table-1 for a set of values of , , , L and K.

The probability and cumulative probability of the model for the values = 3, = 12, = 0.95 and = 0.10 , = 1 are given in the following table. The average system length and average queue length are also given in the table.

It could be observed that ∑*∞* ( )_{ }_{may not be equal to one in the above case. This is because Transition Probability Matrix P has }
been truncated from its infinite dimensions to the form of a square matrix of the finite order N. Since the distribution { ( )} is now
known, it will be easier to obtain other possible probability distributions of queue length related to various types of transition
epochs.

**Table 1 **Numerical values of {x(n)}

**L = 3 K = 12 p = 0.95 ** **= 0.10 = 1 **

Probability Cum.Probability Probability Cum.Probability x(0)=.74816770 .74816770 x(55)=.00005516 .9954680 x(5)= .01726429 .85823350 x(60)=.00003104 .99974470 x(10)= .00971879 .92019400 x(65)=.00001746 .99985610 x(15)= .00547110 .95507420 x(70)=.00000983 .99991880 x(20)= .00307990 .97470960 x(75)=.00000309 .99995400 x(25)=.00173380 .98576320 x(80)=.00000177 .99997370 x(30)=.00097602 .99198560 x(85)=.00000090 .99999110 x(35)=.00054943 .99548850 x(90)=.00000070 .99999450 x(40)=.00030929 .99746030 x(95)=.00000034 .99999680 x( 45)=.00017410 .99857030 x(100)=.00000017 .99999800

x(50)=.00009800 .99919510

Queue length Lq+ = 0.5551 System length Ls+ = 2.3198

**Table 2 **(For L=4)

**K ** **Lq **

15 0.5111 0.5551 16 0.4792 0.4088 17 0.4509 0.3066 18 0.4259 0.2337 19 0.4035 0.1807 20 0.3833 0.1415

**Fig. 1 **Curve of expected queue length with the change of ρ on Lq

0.5551

0.4088

0.3066

0.2337

0.1807

0.1415

0 0.2 0.4 0.6 0.8 1 1.2

1 2 3 4 5 6

**Ex**

**p**

**e**

**c**

**te**

**d**

** l**

**e**

**n**

**th**

**Lq -value**

Lq

Fig.1 Illustrates the influence of the traffic intensity ρ on the expected queue length for different values of maximum batch size K, arrivals to the system for service in FIFO discipline for fixed values of p=.92, p1=.12, L=4 and α=1.From the fig.1, we observe that

the traffic intensity and expected queue length Lq decrease as the maximum batch size increase.

Fig. 2 Illustrates the influence of the traffic intensity ρ on the expected system length for different values of maximum batch size K,

arrival to the system for service in FIFO discipline for fixed values of p=.92,p1=.18,L=4 and α=1.From the fig.2, we come to know

that as there is decrease in the traffic intensity and there is a decrease in system length also.

**CONCLUSION **

In this paper, we present a discrete time queueing model technique concept. We have obtained the queue length distribution of the system occupancy just before and after the departure epochs. Utilizing this distribution, we have derived the steady-state distributions and some important performance measure. Numerical illustrations in the form of tables and graphs are reported to demonstrate the various performance measure of the parameters on the expected queue length and system length. For future research, one can consider the same model when the arrival time follows hyper geometric distributions.

**References **

1. Baburaj., C. (2010). “An M/G/1 Single and batch service queue under the policy (a,c,d)”. *International Journal of *
*Agriculture and Statistical Sciences, Vol.16, No.2, pp.565-570. *

2. Baburaj., C and Rema., M. (2002). "A controllable bulk service queueing system with accessible and non- accessible batches". International Journal of Inter. information and Management Sciences, Vol.13, (1), pp. 83-89.

3. Baburaj., C. (2000). "On the transient distribution of a single and batch service queueing system with accessibility to the batches". Information and management sciences, Vol. 11, No. 2, pp.27-36.

4. Chaudhry., M.L and Gupta., U.G. (1996). "Performance Analysis of the discrete-time G1/Geom/1/N Queues". *Journal of *

*applied * *Probability, Vol.33, No.1, pp.239-255. *

5. Hunter., J.J. (1983) "Mathematical techniques of applied probability, Vol. II, Discrete time models". techniques and applications. Academic Press, New York.

6. Latouche., G and Ramaswami., V. (1999). "Introduction to matrix analytic methods in Stochastic Modeling". American Statistical Association and the Society for industrial and applied Mathematics. Alexandria, Virginia.

7. Mathias A. D and Alexander K. S. (1998), “Using Discrete-time Analysis in the performance evaluation of manufacturing systems”, Research Report Series. (Report No: 215), University of Wiirzburg, Institute of computer Science.

*8.* Mathias., D. (1998). "Analysis of the Departure process of a batch service queuing system". Research report series. (Report
No: 210), University of Wiirzburg, Institute of computer Science.

9. Neuts., M. F. (1967). "A general class of bulk queues with Poisson input". Annals of Mathematical Statistics, Vol.38, pp.759-770.

**Table-3 **(For L=4)

**K ** **ρ** **Ls **

15 0.5111 2.3198 16 0.4792 1.9094 17 0.4509 1.5848 18 0.4259 1.3243 19 0.4035 1.1128 20 0.3833 0.9394

**Fig. 2 **Curve of expected queue length with the change of ρ on Ls

2.3198

1.9094

1.5848

1.3243

1.1128

0.9394

0 0.5 1 1.5 2 2.5 3

1 2 3 4 5 6

**Ex**

**p**

**e**

**c**

**te**

**d**

** l**

**e**

**n**

**g**

**th**

**Ls- Value**

10. Pukazhenthi, N and Ezhilvanan, M.(2014)"Analysis of the discrete time queue length distribution with a bulk service rule (L.K)". International of mathematics and statistics invention, Vol.2.pp.443-456.

11. Pukazhenthi., N and Ezhilvanan,, M. (2015). "Analysis of discrete time queues single server using correlated times".
*International Journal of Recent Scientific Research, Vol.6.pp.3584-3589. *

12. Sivasamy., R and Pukazhenthi., N. (2009). "A discrete time bulk service queue with accessible batch: Geo/NB (L, K) /1". Opsearch. Vol.46 (3), pp.321-334.

13. Veena Goswami., Umesh., C. Gupta., and Sujit., K. Samanta. (2006)."Analyzing discrete-time bulk-service / / queue". RAIRO Operation Research, Vol.40, No.3, pp.267-284.

14. Vijaya Laxmi Pikala, Goswami., V and Seleshi Demie. (2013). "Discrete-time renewal input bulk service queue with changeover time". International Journal of Management Science and Engineering Management. Vol.8:1. pp.47-55.

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