Queuing analysis of Geo/Geo/1 queue with catastrophes using Matrix Geometric technique

Introduction

Queuing models with catastrophes have gained importance during last few decades due to their applications in many areas viz. computers and telecommunications, financial management, insurance sector, disaster management and many more areas. Whenever a catastrophe occurs at the system, all the customers are forced to leave the system immediately, the server becomes inoperative instantaneously and it becomes operative as soon as a new customer arrives. Continuous Queuing models with catastrophes have been investigated by many researchers in the past. Indra and Rajan (2017) [4] analysed Markovian queues having two heterogeneous servers with catastrophes using Matrix Geometric technique. Atencia and Moreno (2004) [2] studied Geo/Geo/1 queue with negative customers and disasters. Jeyakumar and Gunasekaran (2017) [3] analysed Geo/G/1 queue with disaster and single vacation. Matrix Geometric technique was conceptualized by Marcel F. Neuts in 1974. Many researchers used this technique in their respective papers. In the current paper, we are presenting a simple analysis of Geo/Geo/1 queue with catastrophe using Matrix Geometric technique. The rest of the paper is organized as follows: In section 2, description of the model is presented. In section 3, Expression for probability generating function (PGF) of system-size for Geo/Geo/1 for LAS-DA is presented followed by computation of the system-size steady-state probabilities. In section 4, Expression for PGF of system-size for Geo/Geo/1 for EAS is presented followed by computation of the system-size steady-state probabilities. Finally, in section 5, numerical study has been done for the analysis of the model for various values of the parameters.

Section 2 Model description

Let the time axis be marked by 0, 1, 2,…, m,… In the queue Geo/Geo/1, it is assumed that inter arrival times I are independent and geometrically distributed as 𝑎𝑎 𝑛𝑛 = 𝑃𝑃(𝐼𝐼 = 𝑛𝑛 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) = (1 -𝜆𝜆) 𝑛𝑛-1 𝜆𝜆 with mean inter arrival time = 1/λ. The inter service times S are independent and geometrically distributed as 𝑏𝑏 𝑛𝑛 = 𝑃𝑃(𝑆𝑆 = 𝑛𝑛 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) = (1 -𝜇𝜇) 𝑛𝑛-1 𝜇𝜇 with mean inter service time = 1/μ. The traffic intensity (𝜌𝜌) is 𝜌𝜌 = 𝜆𝜆 𝜇𝜇 . It is assumed that the customers are served individually according to the first-come, first-served (FCFS) discipline. In late arrival system with delayed access (LAS-DA), the potential arrivals occur in ( m-, m ). After getting service for n slots (n ≥ 1) a customer leaves in (m, m+). While in an early arrival system (EAS), a potential arrival occurs in (m,m+) and a potential departure occurs in (m-,m).

The catastrophes occur at the system with probability 𝜉𝜉 (0 ≤ 𝜉𝜉 ≤ 1). If an arriving customer finds that the server is idle, the service of the arriving customer starts immediately. Otherwise, the arriving customer joins the queue.

Section 3 Derivation of expression of PGF for Geo/Geo/1 with catastrophe with LAS-DA

Let n(t) be the number of customers in the system at time t. Let n be the stationary random variable for the number of customers in the system. We define π 𝑖𝑖 = {𝑛𝑛 = 𝑖𝑖} = lim 𝑡𝑡→∞ 𝑃𝑃{𝑛𝑛(𝑠𝑠) = 𝑖𝑖} where 𝑖𝑖 𝜖𝜖 set of whole numbers and π 𝑖𝑖 represents the stationary probability of 𝑖𝑖 customers in the system. The stationary probability vector Π is given by

to be the probability generating function (PGF) of system-size probabilities.

In Matrix Geometric Method the steady state probabilities π 𝑖𝑖 are related geometrically to each other as

Here R is called rate element.

The infinitesimal generator matrix Q of the Queuing system under consideration is given by

With the help of stationary probability matrix Π and infinitesimal generator matrix Q we get

Multiplying above equations with appropriate powers of s and then adding we get

On further simplification we get

Equation ( 5) represents the expression for PGF of Geo/Geo/1 with catastrophe with LAS-DA. If we put 𝜉𝜉 = 0 in equation ( 5) we get

Equation ( 6) represents the expression for PGF of Geo/Geo/1 with LAS-DA, which is same as obtained by Hunter.

Matrix Geometric Solution of Geo/Geo/1 with catastrophe with LAS-DA

The stationary probability matrix Π is obtained by solving ΠQ = 0

…. (10) Here R < 1 is non-negative solution of equation ( 9) and is given by

The normalizing equation is given by

) R is calculated using equation (11). 𝜋𝜋 0 and 𝜋𝜋 1 are calculated using equations ( 7), ( 8) and ( 12) and are given by

Using equation (10) 𝜋𝜋 𝑖𝑖 for 𝑖𝑖 ≥ 2 are calculated.

Substituting 𝜉𝜉 = 0 in equations ( 11), ( 13) and ( 14) we get 17) Equations ( 10), (15), ( 16) and (17) together represents Matrix Geometric solution of Geo/Geo/1 with LAS-DA. This solution has been obtained from Geo/Geo/1 with catastrophe with LAS-DA by putting 𝜉𝜉 = 0 and match with the solution of Geo/Geo/1 with LAS-DA given by Hunter.

Section 4 Expression of PGF for Geo/Geo/1 with catastrophe with EAS

The infinitesimal generator matrix Q for the above system is given by

Using stationary probability matrix Π and infinitesimal generator matrix Q we get

𝜆𝜆𝜇𝜇̅ 𝜋𝜋 𝑗𝑗-1 + �-𝜆𝜆 ̅ 𝜇𝜇 -𝜆𝜆𝜇𝜇̅ -𝜉𝜉�𝜋𝜋 𝑗𝑗 + 𝜆𝜆 ̅ 𝜇𝜇𝜋𝜋 𝑗𝑗+1 = 0 ; for j ≥ 2 …. (20) Multiplying above equations with appropriate powers of s and then adding we get

On further simplification we get Equation ( 23) represents the expression for PGF of Geo/Geo/1 with EAS, which is same as obtained by Hunter.

Matrix Geometric Solution of Geo/Geo/1 with catastrophe with EAS

The stationary probability matrix Π is obtained by solving ΠQ = 0 26) and is given by

The normalizing equation is given by

) R is calculated using equation (28). 𝜋𝜋 0 and 𝜋𝜋 1 are calculated using equations (24), (25) and (29) and are given by Equations ( 27), (32), ( 33) and (34) together represents Matrix Geometric solution of Geo/Geo/1 with EAS. This solution has been obtained from Geo/Geo/1 with catastrophe with EAS by putting 𝜉𝜉 = 0 and match with the solution of Geo/Geo/1 with EAS given by Hunter.

Numerical Study

The stationary system-size probabilities have been calculated using codes developed in Maple software. All computations have been rounded off to six decimal places. The results have been verified by matching the two sets of means and variances obtained from probability values and from theoretical formulas. The means and variances, as calculated from probability distributions, are given by the following formulas:

Although we are solving infinite capacity model, but for computational purpose we have restricted the system capacity to N. The means and variances formulas for system-size, as derived from probability generating functions, are given by the following:

Mean and Variance for System Size in LAS-DA

Mean and Variance for System Size in EAS

The results for various values of 𝜉𝜉 are shown in the following tables:

References

1. Hunter JJ. Mathematical Techniques of Applied Probability, Volume II, Discrete Time Models: Techniques and Applications. New York: Academic Press, 1983.
2. Atencia I, Moreno P. The discrete time Geo/G/1 queues with negative customers and disasters. Computers & Operation Research. 2004; 31:1537-1548.
3. Jeykumar S, Gunasekaran P. An analysis of Discrete Queue with Disaster and Single Vacation. International journal of Pure and Applied Mathematics. 2017; 113(6):82-90.
4. Indra, Rajan V. Queuing Analysis of Markovian queue having two heterogeneous servers with catastrophes using Matrix Geometric technique. International journal of Statistics and Systems. 2017; 12(2):205-212.