Single-Server Queueing Model with Non ...

Research & Reviews: Journal of Statistics ISSN: 2278-2273 (Online), ISSN: 2348-7909 (Print) Volume 6, Issue 3 www.stmjournals.com

Single-Server Queueing Model with Non-Homogeneous Compound Poisson Bulk Arrivals with Intervened Poisson Distribution Hemanth Kumar M.1,*, Venkateswaran M.2, Ganesh T.3, P.R.S. Reddy1 1

Department of Statistics, Sri Venkateswara University, Tirupati, Andhra Pradesh, India Department of Science and Humanities, Sri Venkateswara College of Engineering, Tirupati, Andhra Pradesh, India 3 Department of Mathematics, National Institute of Technology Hamirpur, Himachal Pradesh, India 2

Abstract Queuing models are used to analyse the practical situations in communication networks and transportation in a sophisticated manner. In most of the cases, the arrivals are time dependent and can be characterised by a non-homogenous Poisson process. In this paper, for the performance measures of a single-server queuing model with non-homogeneous compound Poisson bulk arrivals have been studied with intervened Poisson distribution. Sensitivity analysis has been carried out. It is observed that the system performance measures are highly influenced by non-homogenous arrivals rate and batch size distribution parameters. Keywords: Non-homogenous Poisson process, bulk arrivals, state dependent service rates, intervened Poisson distribution

*Author for Correspondence E-mail: [email protected]

INTRODUCTION Over the decades, a lot of work has been reported regarding different variations of queueing models, under various practical considerations of which few can be considered as data voice transmission, communication networks, transportation systems, manufacturing etc. In general, we are all aware about the constituent processes of the queueing models, which are: (i) arrival process, (ii) service process, and (iii) queue discipline. To analyse a wide variety of situations, it is customary to consider that the service rate is dependent on the number of customers in the queue. These types of queueing models are referred as queueing models with state dependent/load dependent service rates [1]. Over the past six decades, wide varieties of queueing models were witnessed under bulk arrivals pattern with several distributional assumptions. Following paragraphs provide a glance and coverage of such models. Queueing model of bulk arrivals under a single service line with two distributional assumptions i.e. the arrivals are of Poisson

nature and the service is exponential type, as proposed by Miller [2]. Minh studied single server queuing model with compound Poisson input and time (discrete) dependent parameters in a general service time distribution [3]. Models of two single server queues in series with an assumption of service times are same in both the queues and arrival in terms of Poisson process, was considered by Boxma [4]. The numerical results include conditions of tandem system. The actual waiting time and virtual waiting time at second queue were also studied. A detailed discussion on the vehicle dispatching strategies for bulk arrivals and bulk service queues with vehicle holding and cancellation strategies was done. The length of the queue, relationships between the moments of the waiting time, approximation of variance and queue length distributions were developed [5]. The iterative numerical procedure for the study of queue lengths and waiting times in bulk arrival, bulk service queues with compound Poisson arrivals was developed. A technique imbedded Markov chain has been

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Single-Server Queueing Model with Non-Homogeneous Compound Poisson Bulk Arrivals

used. Further, waiting time distribution for queues with non-Poisson arrivals was also investigated [6]. The analysis of steady state probabilities and moments of the number of customers in the system or queue at three different epochs, post departure, random and pre-arrival for models of the type M/GY/1 was done. The numerical illustration was illustrated for Erlang, hyper-exponential and uniform distributions [7]. Two parallel queuing systems, each of which consists of a single server in continuous time. A closed form expression for the Laplace transform of the ergodic bivariate waiting time distribution and the sum of two waiting times has been analyzed [8]. Chudhry et al. presented an algorithm for multi server queuing system (bulk arrival) with numerical results and various quantities of interest. This procedure was suitable in discrete and continuous to the problems in the theory of dams, inventory control etc. [9]. Later, several authors developed various queueing models for bulk arrivals. An interdependent communication network with bulk arrivals was introduced by Srinivasa et al. [10]. A multi-channel bi-level heterogeneousserver bulk arrival queueing system with an Erlangian service time was studied by Ahmed [11]. M[X]/G/1 queue with bernoulli schedule general vacations times, general extended vacations, random breakdowns, general delay times for repairs to start and general repair times was studied by Khalaf et al. [12]. They assumed that the server has the option to go on the extended vacation of the original vacation completion or rejoins the system to provide service. In all these models, the authors considered the arrivals to be homogeneous and possible to characterise by a compound Poisson process. Some authors have demonstrated that in a TCP connection arrival process, the time between packet arrivals cannot be characterised by an exponential distribution and hence cannot be represented by a Poisson process [13, 14]. Very little work has been reported regarding queueing models with non-homogeneous compound Poisson arrivals, except the works by Fischer et al. and Trinatha et al. who have studied the queueing models in which the arrivals are single and follow a nonhomogeneous Poisson process with a timedependent arrival rate [15, 16].

Kumar et al.

A discrete time single server queuing system with bulking and multiple working vacations was analyzed [17]. Here the inter-arrival times of customers are assumed to be independent and geometrically distributed and obtained the closed form of expressions for the steady state probabilities with some computational experiences. Single-Server queueing model with non-homogeneous compound Poisson bulk arrivals was developed and analysed [18]. It is also assumed that the service completion follows a Poisson process with state dependent service rates. The state dependent service rates have considerable influence on the solution procedure of the model and the performance of the system. A single server queue with finite buffer in a discrete time domain where the packets are transmitted in batches according to minimum and maximum threshold limit, usually known as general batch service rule have been studied [19]. So far the models developed were on Poisson process. However Shanmugam proposed an IPD by highlighting its advantages over conventional Poisson process using an intervention parameter [20, 21]. The present paper is focused on incorporating the IPD into the setup of non-homogeneous bulk arrivals to observe its outcome and performance through sensitivity analysis.

MATERIALS AND METHODS In this section, a detailed discussion is made on the proposed methodology and its performance measures. In continuation of this section, support of numerical illustrations are considered to explain the behaviour of the performance measures of proposed queueing models by keeping the parameters fixed. Intervened Poisson Distribution The Intervened Poisson Distribution (IPD) was proposed by Shanmugam as a replacement for the Zero-Truncated Poisson Distribution (ZTPD) in those instances when some intervention process may alter mean of the rare event generating process under observation [20]. Several possible applications of IPD in the areas of reliability analysis, congestion and queuing studies have been proposed. Further, inferential aspects concerning the Poisson intervention parameter has been discovered by Shanmugam and discussed the medical applications [20, 21].


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Research & Reviews: Journal of Statistics Volume 6, Issue 3 ISSN: 2278-2273 (Online), ISSN: 2348-7909 (Print)

The Probability Mass Function, Mean and Variance of the Intervened Poisson Distribution





1      e 1 1  x   x  x  e      P( X  X )   ,  0 x!    1   V (Y )    e      E  X         1  e  1   and  (e  1)2       



2



Here, ρ is the Intervened Poisson parameter ( 0   <  ) and θ is the Poisson parameter (θ>0). The single-server queueing model with non-homogeneous compound Poisson bulk arrivals has been studied with intervened Poisson distribution. Sensitivity analysis is presented through numerical illustration.

QUEUEING MODEL Let us consider a queue Q and a service station S. It is assumed that the arriving pattern of customers to the queue will be in batches of random size and purely dependent on time, i.e., the actual number of customers in arriving pattern is a random variable X with probability Cx with k customers. In other words, the arrival of customers follows non-homogeneous compound Poisson processes with a mean composite arrival rate λ(t) having bulk size distributions {Cx}. Here, Cx be the probability of arrival pattern observed in a random variable x. The number of service completions in the service station follows Poisson processes with the parameter μ. It is further assumed that the mean service rate in the service station is linearly dependent on the content of the queue connected to it. The queue discipline is first in, first out. The queue capacity is infinite. A schematic diagram representing the queueing model is shown in Figure 1.

Fig. 1: Schematic Diagram of the Queueing Model. In the next section, we observed the performance measures variable x.

PERFORMANCE MEASURES WITH INTERVENED POISSON DISTRIBUTION BULKSIZE ARRIVALS The performance measures of the queueing model are highly influenced by the form of the batch size distribution. It is assumed that the number of customers in any arriving module is random and follows an intervened Poisson distribution with parameters  and ρ. Then the probability mass function of the batch size distribution is:





1      e 1 1  x   x  x  e       P( X  X )  ,  0 x! CK= The probability generating function for the number of customers in the queue is:


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Kumar et al.

 m k e   e  11 1   k   k    k  r t      k  r 1  e P  Z , t   exp     z  1       k! r  r   k 1 r 1   e  e  1 1 1   k   k    k  r t  m k   r tr   1  e      k      , Z  1  z  1    2 k! k 1 r 1  r  r     (1) By expanding P  Z , t  and collecting the constant terms, we obtain the probability that the queue is





empty as:

 e   e  11 1   k   k    k  r t m k   r 1  e      k  P0  t   exp     1   r   k!    r   k 1 r 1   e  e  1 1 1   k   k    k  r t  m k   r tr   1  e      k       1    2 k! k 1 r 1  r  r    





(2)

The mean number of customers in the queue is:

 

L  t    1  e t  

1   t   1  e t       1   e  1   2     

(3)

The variance of the number of customers in the queue is:

e   e  11 1   k   k    k     k  1       V t     .k   1  e2 t   1  e t   k! k 1     2 m

    k  1  2t   1  e2 t    2   t   1  e1t       2  2    The coefficient of variation of the number of customers in the queue is: CV  t  

V t 

L t 

(4)

100 (5)

Where, V(t) and L(t) are as given in Eqs. (4) and (3) respectively (5). However, interest lies in estimating the utility of service station when P0  t  is given. This helps in understanding the service mobility in a queue when a bulk size of random arrival of customers is observed. This intern lead to the measure namely throughput (Thp (t)) using this, it will be convenient to estimate the service pattern of S when these are bulk arrivals. Now utilization of service station is given by:

U  t   1  P0  t 

 m k e  e  1 1 1   k   k    k          1  exp   k!  k 1 r 1 


 r t  k  r 1  e  r   1  r      

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e   e  11 1   k   k    k        k! k 1 r 1 m

k

The throughput of the service station is:

  r t   k  r tr   1  e    r   1  2  r       

Thp  t   U  t 

  m k e  e  1 1 1   k   k    k           1  exp    k!  k 1 r 1    e   e  11 1   k   k    k        k! k 1 r 1 m

(6)

k

 r t  k r 1  e  r   1  r      

  r t   k r tr   1  e     r   1  2  r       

(7) The performance measures L(t) and Thp(t) will help in estimating the average waiting time of a customer in the queue of bulk arrivals, which is denoted by W(t) and given as i.e., the average waiting time of a customer in the queue is:

W t  



L t  Thp  t 

1    t  t        1  e    2 t   1  e       1   e  1      1 e   e  1 1   k   k    k  r t m k  r 1  e      k      1       k! k 1 r 1 r  r 

         1  exp  1 k   m k e   e  1 1      k    k         k!  k 1 r 1 

      r  t   k  r  tr   1  e    r   1  2 r        

(8)

RESULTS AND DISCUSSION In this section, the performance of the proposed queueing model is discussed through a numerical illustration. The customers arrive in batches to the queue, and the arrival of the customers follows a non-homogeneous intervened Poisson process. The composite mean arrival rate is𝜆(𝑡) = 𝜆 + 𝛼𝑡. Each arriving module represents a batch of customers. The number of customers in each arriving module follows an intervened Poisson distribution (ρ, θ). Because the characteristics of the queueing model are highly sensitive with respect to time, the transient behaviour of the model is studied by computing the performance measures with the following set of values for the model parameters: t : 0.05, 0.06, 0.07, 0.08, 0.09 θ : 1.50,2.0,2.5,3,3.5 ρ : 1.50,2.0,2.5,3,3.5 λ : 0.5,0.6,0.7,0.8,0.9 α : 0.02,0.03,0.04,0.05,0.06 μ : 19,20,21,22,23


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Kumar et al.

The performance measures P0(t), L(t), V(t), CV(t), Thp(t), U(t) and W(t) are computed for different values of the parameters t, θ, ρ, λ, α and µ. The results pertaining to the performance measures are reported in Table 1, and shown in Figures 2–7. Table 1: Performance Measures of Non-Homogeneous Bulk Arrivals under IPD. T 0.05 0.06 0.07 0.08 0.09

θ 1

ρ 1

Λ 0.4

α 0.01

μ 18

1.5 2 2.5 3 3.5 1.5 2 2.5 3 3.5 0.5 0.6 0.7 0.8 0.9 0.02 0.03 0.04 0.05 0.06 19 20 21 22 23

P0(t) 0.983 0.980 0.978 0.976 0.974

L(t) 0.034 0.038 0.041 0.044 0.046

V(t) 0.085 0.091 0.097 0.101 0.104

CV(t) 853.526 796.794 755.038 723.345 698.759

U(t) 0.017 0.02 0.022 0.024 0.026

Thp(t) 0.306 0.352 0.393 0.429 0.46

W(t) 0.111 0.108 0.105 0.102 0.1

0.9820 0.9813 0.9809 0.9806 0.9805 0.983 0.9818 0.9814 0.9811 0.9809 0.9788 0.9746 0.9704

0.045 0.057 0.069 0.081 0.094 0.041 0.047 0.054 0.060 0.067 0.043 0.051 0.059

0.146 0.222 0.316 0.428 0.559 0.119 0.157 0.199 0.246 0.298 0.106 0.127 0.148

842.603 828.364 815.861 805.545 797.095 846.652 837.627 828.843 820.93 813.977 763.468 696.978 645.297

0.0180 0.0187 0.0191 0.0194 0.0195 0.0177 0.0182 0.0186 0.0189 0.0191 0.0212 0.0254 0.0296

0.325 0.336 0.344 0.349 0.352 0.318 0.328 0.334 0.340 0.343 0.382 0.458 0.533

0.139 0.169 0.201 0.233 0.276 0.128 0.144 0.161 0.178 0.195 0.1114 0.1117 0.1119

0.9662 0.9621 0.98296 0.98295 0.98294 0.98293 0.98292 0.98316 0.98335 0.98354 0.98374 0.98393

0.068 0.077 0.0341 0.03412 0.03415 0.03417 0.03420 0.03336 0.03267 0.03199 0.03134 0.03071

0.169 0.190 0.085 0.085 0.085 0.085 0.085 0.0821 0.0797 0.0774 0.0752 0.0732

603.634 569.122 853.244 852.961 852.679 852.397 852.115 852.778 864.132 869.585 875.133 880.775

0.0338 0.0379 0.01704 0.01705 0.01706 0.01707 0.01708 0.01684 0.01665 0.01646 0.01626 0.01607

0.608 0.682 0.3067 0.3069 0.3071 0.3073 0.3075 0.3199 0.3329 0.3456 0.3578 0.3696

0.1121 0.1124 0.111 0.111 0.111 0.111 0.111 0.104 0.098 0.093 0.088 0.083

Time vs mean number of customers

Emptiness

0.985 0.980 0.975 0.970 0.965 0.05 0.06 0.07 0.08 0.09 Time

P0(t)

Mean number of customers

Time vs Emptiness

0.0600 0.0400 0.0200

L(t)

0.0000


1

2

3 4 Time

5

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Time vs Utilization

0.12 0.1 0.08 0.06 0.04 0.02 0 0.05 0.06 0.07 0.08 0.09 Time

V(t)

Utilization

Variance

V(t)

0.030 0.025 0.020 0.015 0.010 0.005 0.000 0.05 0.06 0.07 0.08 0.09 Time

Time vs Throughput

Time vs Waiting Time 0.115

0.40 0.30 0.20

Thp(t)

0.10

Waiting Time

0.50

Throughput

U(t)

0.00 0.05 0.06 0.07 0.08 0.09 Time

0.110 0.105 0.100 0.095

W(t)

0.090 0.05 0.06 0.07 0.08 0.09 Time

Fig. 2: Performance Measures with Respect to Time ‘t’.

θ vs Emptiness

θ vs mean number of customers

Emptiness

0.9820 0.9815 0.9810 0.9805

P0(t)

0.9800 0.9795 1.5

2

2.5 θ

3

3.5


0.9825 0.10 0.08 0.06 0.04

L(t)

0.02 0.00 1.5

0.6 0.5 0.4 0.3 0.2 0.1 0

V(t)

1.5

2

2.5 θ

3

2.5 θ

3

3.5

θ vs Utilization Utilization

Variance

θ vs Variance

2

0.0200 0.0195 0.0190 0.0185 0.0180 0.0175 0.0170

3.5


U(t)

1.5

2

2.5 θ

3

3.5

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θ vs Throughput

θ vs Waiting Time

0.350 0.340 Thp(t)

0.320

Waiting Time

Throughput

0.360

0.330

0.310 1

2

3 θ

4

Kumar et al.

0.30 0.25 0.20 0.15 0.10 0.05 0.00

5

W(t)

1.5

2

2.5 θ

3

3.5

Fig. 3: Performance Measures with Respect to θ.

0.9840 0.9830 0.9820 0.9810 0.9800 0.9790

ρ vs mean number of customers

P0(t) 1.5

2

2.5 3 ρ

3.5


Emptiness

ρ vs Emptiness

0.08 0.06 0.04

L(t)

0.02 0

1.5

2

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00

3.5

0.0195 0.0190

V(t)

0.0185 0.0180

U(t)

0.0175 0.0170 1.5

2

2.5 ρ

3

3.5

1.5

0.350 0.340 0.330 0.320 0.310 0.300

Thp(t) 1.5

2

2.5 ρ

3

2

2.5 ρ

3

3.5

ρ vs Waiting Time Waiting Time

ρ vs Throughput Throughput

3

ρ vs Utilization

Utilization

Variance

ρ vs Variance

2.5 ρ

0.25 0.20 0.15 0.10 0.05 0.00

3.5

W(t) 1.5

2

2.5 ρ

3

3.5

Fig. 4: Performance Measures with Respect to Intervened Parameter ρ.


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0.985 0.98 0.975 0.97 0.965 0.96 0.955 0.95

λ vs mean number of customers

P0(t)

0.5 0.6 0.7 0.8 0.9 λ


Emptiness

λ vs Emptinees

0.1 0.08 0.06 0.04

L(t)

0.02 0 0.5

α vs Variance

0.8

0.9

0.040

0.15 0.10

V(t)

0.05

Utilization

Variance

0.7 λ

λ vs Utilization

0.20

0.030 0.020 U(t)

0.010 0.000

0.00

0.5

0.6

0.7 λ

0.8

0.9

0.5 0.6 0.7 0.8 0.9 λ

λ vs Throughput

λ vs Waiting Time 0.1125

0.6 0.4 Thp(t)

0.2

Waiting Time

0.8

Throughput

0.6

0

0.112 0.1115 W(t)

0.111 0.1105

0.5

0.6

0.7 λ

0.8

0.9

0.5 0.6 0.7 0.8 0.9 λ

Fig. 5: Performance Measures with Respect to λ.

0.98297 0.98296 0.98295 0.98294 0.98293 0.98292 0.98291 0.9829

α vs mean number of customers

P0(t)

0.02 0.03 0.04 0.05 0.06

α


Emptiness

α vs Emptiness

0.03422 0.0342 0.03418 0.03416 0.03414 0.03412 0.0341 0.03408 0.03406 0.03404 0.02 0.03 0.04 0.05 0.06


L(t)

α

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α vs Utilization

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.02 0.03 0.04 0.05 0.06

V(t)

Utilization

Variance

α vs Variance 0.01709 0.01708 0.01707 0.01706 0.01705 0.01704 0.01703 0.01702

U(t)

0.02 0.03 0.04 0.05 0.06

α

α

0.3076 0.3074 0.3072 0.307 0.3068 0.3066 0.3064 0.3062

α vs Waiting Time

Thp(t)

Waiting Time

α vs Throughput

Throughput

Kumar et al.

0.02 0.03 0.04 0.05 0.06

α

0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.02 0.03 0.04 0.05 0.06 α

W(t)

Fig. 6: Performance Measures with Respect to α.

μ vs Emptiness

μ vs mean number of customers Mean number of customers

Emptiness

0.9840 0.9835 0.9830

P0(t)

0.9825

18

19

21 μ

22

23

0.0340 0.0330 0.0320 0.0310 0.0300 0.0290 18 19 21 22 23 μ

μ vs Utilization 0.0170

0.084 0.082 0.080 0.078 0.076 0.074 0.072 0.070 0.068

V(t)

Utilization

Variance

μ vs Variance

L(t)

0.0165 0.0160

U(t)

0.0155 18

19

21 μ

22

23


18

19

21 μ

22

23

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μ vs Throughput

μ vs Waiting Time

0.36 0.34 0.32

Thp(t)

0.30 0.28 18

19

21 μ

22

Waiting Time

Throughput

0.38

0.12 0.10 0.08 0.06 0.04 0.02 0.00

23

W(t) 18

19

21 μ

22

23

Fig. 7: Performance Measures with Respect to µ. It is observed that when the other parameters are fixed, time (t) increases, the probability of emptiness of the queue decreases; the mean number of customers and variance in the queue increases; the utilisation of the service station increases; the throughput of the service station increases; the average waiting time of the customer in the queue decreases; the variance of the number of customers in the queue increases; and the coefficient of variation of the number of customers in the queue decreases. As the parameter  increases, the probability of emptiness of the queue decreases; the mean number of customers and variance in the queue increases; the utilisation of the service station increases; the throughput of the service station increases; the average waiting time of the customer in the queue increases; when the other parameters are fixed. With respect to ρ, as ρ value increases, the probability of emptiness of the queue decreases, the mean, variance, utility, throughput and waiting time of the queue are observed to have an increasing pattern. It is also observed that as the parameter (λ) increases, the probability of emptiness of the queue decreases; the mean number of customers and variance in the queue increases; the utilisation of the service station and the throughput of the service station increases; the average waiting time of the customer in the queue increases, by keeping other parameters as fixed. It is also further observed that as the parameter (α) increases, the probability of emptiness of the queue decreases; the mean number of customers in the queue increases, where the

variance is stable. The utilisation of the service station decreases; the throughput of the service station increases; the average waiting time of the customer in the queue is constant, by viewing other parameters as fixed. It is also observed that as the service rate (μ) increases, the probability of emptiness of the queue increases; the mean number of customers and variance in the queue decreases; the utilisation of the service station decreases; the throughput of the service station increases; the average waiting time of the customer in the queue decreases; when the other parameters are fixed. Thus, by regulating the parameters, one can control the congestion in queues and the mean delay in transmission.

CONCLUSION In this paper, a single server queuing model with non-homogeneous bulk arrivals is studied when the distributed type is of intervened Poisson distribution. Here an attempt is made to exhibit the role of intervention parameter and the IPD in estimating the performance measures when there is a random bulk arrival. It is concluded that the queueing model obtained by embedding the characteristics of IPD terms out to explain the various measures of the queue and service station. Another attempt is to claim the advantages of IPD suggested by Shanmugam and the same is incorporated in to the queue system to monitor the pattern of non-homogeneous Poisson bulk arrivals [20].

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2. Miller Rupert G Jr. A Contribution to the Theory of Bulk Queues. J R Stat Soc. Series B (Methodol). 1959; 21(2): 320– 337p. 3. Do Le Minh. The Discrete-Time SingleServer Queue with Time-Inhomogeneous Compound Poisson Inputand General Service Time Distribution. J Appl Probab. Sep 1978; 15(3): 590–601p. 4. Boxma OJ. On a Tandem Queueing Model with Identical Service Times at Both Counters-I, Advances in Applied Probability ,Vol.11 ,No.3, PP 616-643. 5. Powell Warren B. Analysis of Vehicle holding and Cancellation Strategies in Bulk Arrivals, Bulk Service Queues. Transport Sci. 1985; 19(4): 352–377p. 6. Powell Warren B. Iterative Algorithms for Bulk Arrivals, Bulk Service Queues with Poisson and Non-Poisson Arrivals. Transport Sci. 1986; 20(2): 65–79p. 7. Brière G, Chaudhry ML. Computational Analysis of Single-Server Bulk-Service Queues, M/GY/1. Adv Appl Probab. Mar 1989; 21(1): 207–225p. 8. Zhensheng Zhang. Analytical Results for Waiting Time and System Size Distributions in Two Parallel Queue Stems. SIAM J Appl Math. 1990; 50(4): 1176–1193p. 9. Chudhry ML, Medhi J, Templetion JGC. Computational Results of Multiserver Bulk-Arrivals Queues with Constant Service Time. Operations Research Stochastic Processes. 1992; 40(Supplement 2): S229–S238p. 10. Srinivasa Rao K, Prasad Reddy PVGD, Suresh Varma P. Interdependent Communication Network with Bulk Arrivals. International Journal of Management and Systems. 2006; 22(3): 221–234p. 11. Ahmed MMS. Multi-Channel Bi-Level Heterogeneous Servers Bulk Arrival Queueing System with Erlangian Service Time. Mathematical and Computational Applications (MCA). 2007; 12(2): 97– 105p. 12. Khalaf Rehab F, Madan Kailash C, Lukas Cormac A. An M[X]/G/1 Queue with Bernoulli Schedule, General Vacation Times, Random Breakdowns, General Delay Times for Repairs to Start and Genera Repair Times. Journal of Mathematical Research. 2011; 3(4): 8– 20p.

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Cite this Article Hemanth Kumar M, Venkateswaran M, Ganesh T et al. Single-Server Queueing Model with Non-Homogeneous Compound Poisson Bulk Arrivals with Intervened Poisson Distribution. Research & Reviews: Journal of Statistics. 2017; 6(3): 10–21p.


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