COMPUTATIONAL ANALYSIS OF A MULTI-SERVER BULK ARRIVAL WITH TWO MODES SERVER BREAKDOWN

An extensive Monte Carlo study is performed to analyze a multi-server, bulk arrival (M /M/C; C-1/FCFS) queuing system. The system breakdown limits the system to serve with either C or (C-1) servers. The server breakdown has equal chance over all servers in the system while the arrival process is a Poisson process and the distribution of the bulk size X is a positive Poisson bulk size. Measures of system efficiency including mean queue length, mean waiting time, and blocking probability are introduced. Numerical results are obtained by simulation of the entire system.

M.L. Chaudhry and Nam K. Kim [9] studied a complete and simple solution for the discrete time multi-server queue with bulk arrivals and deterministic service times.Jau-Chanke.[5] studied two thresholds of an (M/G/1) queuing system with server breakdowns and two vacation types.
In this paper a multi-server queuing model (M [x] /M/C; C-1/FCFS) with homogeneous servers is discussed.This system has already been investigated by Chaudhry 8, 12 and Shoukry 2, 3 .The results of these authors are extended and corrected.This system has the following description.Customers arrive in batches according to compound Poisson process with mean batch arrival rateλ.Let N (t) be the number of batches of customers which have arrived by time t where {N (t), t ≥ 0} can be modeled as a Poisson process.Next, we fit a discrete distribution to the size of the successive batches, the size of arrival batch must be positive integers.Thus, for the original arrival process, it is assumed that batches of customers arrive in accordance with the arrival process {N (t), t ≥ 0}.The total number of customers that arrive by time t denoted by Z (t) is given by, where x i is the number of customers in the ith batch.The x i 's are assumed to be IID random variables independent of {N (t), t ≥ 0}, then the stochastic process {Z (t), t≥ 0} is said to be a compound Poisson process.In this case, the bulk size X has a positive Poisson distribution with no bulk size of zero.Now, let {a x } represent a probability sequence that governs bulk size.This probability a x is defined as, a x = P (x; θ ) / [1-P (0;θ )] = where θ is a parameter of bulk size.The average bulk size m in this case will be given by, (3) Customers are served according to exponential service time distribution with parameter µ > 0. The C and C-1 in the model notation above are the number of parallel service channels.At any instant of time only one of the servers breaks down and the system operates with (C-1) servers instead of C servers who the system initially operates with.Thus a system alternates between two modes of a system operation.This is due to breakdown of one of the servers, to scheduling policy, or to one of the servers leaving the system temporarily.Service is on a first come, first served basis.In addition, the queue length in our model is assumed to have no limit.
The paper is organized in 7 sections: Section 2 presents a description of the study design.Section 3 introduces notations and basic assumptions for the model under study.Section 4 describes the methodology used for the analysis of the system together with code and routines used.Section 5 discusses an extensive Monte Carlo results obtained from analyzing the system.The results include different tables and graphs.Section 6 gives an example to show how the method works.Finally, section 7 concludes the paper.
The main intent of this paper is to present numerical results for certain measures of efficiency such as utilization, mean queue length, mean waiting time, average number in the system, the probability of no customer in the system and the blocking probability for this queuing system.Applications of such bulk-arrival, multi-server, and breakdown queuing model appear in different real life systems.Examples of such systems are bank, telecommunication control-facility, classic machine-repair problem, scheduling patients in hospital clinics, telephone exchanges, and taxi stands.

DESCRIPTION OF THE STUDY DESIGN
As indicated by the notation (M [x] /M/C; C-1/FCFS), bulks of customers arrive at random times with mean bulk arrival rateλ.We investigate specifically a distribution of the bulk size, namely positive Poisson bulk sizes.The queuing system under study has homogeneous parallel servers where service time has exponential distribution with mean 1/µ.Our system alternate between two modes of system operation, this is due to breakdown of one of the servers.In one mode of operation all C servers are available and in the other mode only C-1 servers are available.The server is broken randomly.This mean that only one server is allowed to breakdown randomly according to a discrete uniform distribution that assigns one of the servers to be out of service.The mean time that the system operates with C servers and C-1 servers is 1/α and 1/β respectively.
The queue discipline for bulks is FCFS while the service discipline within the bulks is based on randomly choosing one of the customers mentioned earlier.The conditional probability of the customer waiting for d departures before his service commences, given the state of the system n just before the arrival of the customer's bulk, is given by, where, the bulk size X is an r.v. with distribution given by a r =P (X =r), r ≥1, X has mean m , (see [3]).
Customers of a certain bulk are served randomly.If X is the size of a bulk then customer i, i =1,2,…,X has the same chance of joining service.In order to generate an equal chance to all customers in a given batch of size X, either a uniform assignment of the order in which they may be served is used or using a predefined permutation sequence.This permutation routine helps to recognize every customer in the batch.

NOTATION AND ASSUMPTIONS
In this section, the different notation and assumptions used in this article will be introduced.The average waiting time per customer in the queue.Mode-1: All C servers are available for serving the customers in this mode.Mode-0: All (C-1) servers are available for serving the customers in this mode.

Notation
i P , 0 : The probability of having no customers in the system when the system is in mode-i, i= 0,1.β : Transition rate from mode-0 to mode-1.α : Transition rate from mode-1 to mode-0.B P : The blocking probability that defines the probability that all servers are busy.ρ : The traffic intensity.
N : The number of batches.θ : The parameter of the batch size distribution.

Assumptions A1:
The system at any instant of time is in one of two modes of operation mode-0 or mode-1.A2: Groups of customers (bulks) arrive at a multi-server queuing system in accordance with Poisson process with parameter λ .
A3: The probability of two or more events occurring simultaneously is negligible.A4: The queue has C identical exponential servers in parallel with each server having a service rate µ.
A5: The times that system operates with C servers and (C-1) servers have exponential distribution with mean 1/α and 1/β respectively.A6: The bulk size X follows a positive Poisson distribution with parameterθ.A7: The traffic intensity for the system is given by ρ =mλ/Cµ and the condition for existence of a steady state solution is ρ < 1. A9: There is no limit on queue length.A10: Occasionally one of the C parallel servers breaks down and leaves the facility according to a discrete uniform distribution with parameter C. A11: Bulks are served according to FCFS discipline while customers in a bulk are served randomly.

METHODOLOGY
The methodology can be described in the following steps: 1-Generate 10000 interarrival times for 10000 different batches from exponential distribution with mean 1/λ.2-Generate a random batch size for each batch in step 1 using positive Poisson distribution with parameterθ.
3-To determine the service times for each customer in the successive batches, generate random variables from exponential distribution with mean 1/µ for each server.4-Determine the intervals of time that the system operates with C servers by generating random variables from exponential distribution with mean 1/α.5-To determine the intervals of time that the system operates with (C-1) servers, generate random variables from exponential distribution with mean 1/β.6-To determine which server will be down, generate random variables from discrete uniform distribution with parameter C.Where the integers 1, 2, 3,…,C occur with equal probability.7-Calculate the event time of breakdown and repairing for each server based on the intervals of time during which the system works with C-1 and C servers respectively.(See step 4&5).8-Calculate the traffic intensity ρ from equation ρ = mλ/cµ.9-Find the number of all customers that enter the system as a sum of sizes of batches that arrive to the system.10-Find the arrival time of a batch.11-Determin the mode at which an arriving batch will find the system as follows: • If the arrival time of a batch is greater than the event time of breakdown and less than the event time of repairing, the system will be in mode-0.• If the arrival time of a batch is greater than the event time of repairing and less than the event time of breakdown, the system will be in mode-1.12-An arriving customer will immediately start service if one of servers is free or wait until any server becomes free.13-Calculate the departure time of a customer from system as the sum of arrival time plus service time.14-Calculate the waiting time of a customer in the queue.15-Calculate the waiting time of a customer in the system.16-Calculate the busy time for each server.17-Repeat steps 12 to 16 until all customers in a given batch are served.18-Calculate the departure time of a batch.19-Calculate the sum of waiting times of customers in queue.20-Calculate the sum of waiting times of customers in system.21-If a server is broken down during serving a certain customer this customer will quit service and will start service at the first server available.As a result, repeat the steps 12 to 16 and 19 to 21 again.22-Define the different time intervals during which system operates in the two modes.23-Repeat the steps from 10 to 22, 10000 times.24-Calculate the probability that the system is in mode-i , i= 0,1while there is no any customer in the system where T is the length of time we observed the system (which equals to the master clock time since it starts at zero).

RESULTS
The program was tested extensively for values of (ρ, C, X) with 0.1≤ρ≤0.9,1≤C≤100 and batch size X≤100 and it did not encounter any problems.We note that tables 1, 2, 3, 4, 5 and 6 contain two variables.The first variable represents the traffic intensity ρ and the second variable represents the relative transition rate α/(α+β).While the different values inside each table represent performance measures (output data).These measures are Based on the methodology explained in the previous section, the input data includes the number of servers C, the number of batches N, the parameter of size of batches θ, mean service time 1/µ, mean interarrival time 1/λ, the mean time that the system operates with c servers 1/α and the mean time that the system operates with (C-1) servers 1/ β.Different performance measures are calculated.These measures include: 1-The expected number of customers in the queue and in the system be Q L and S L respectively.2-The expected waiting time per customer in the queue and in the system are Q W and S W respectively.3-The probability of having 0 customer in the system 0 P .

4-
The blocking probability B P .We note that the traffic intensity ρ changes with variation the value of 1/µ.So the value of relative transition rate α/(α+β) is varied due to the changes in values 1/α and 1/ β. Results shown in table 1, 2, 3, 4, 5 and 6 are obtained by inserting these input data and running the program.
For example, figure 1 indicates the relation between traffic intensity ρ and average number of customers in the queue L Q with the increase of relative transition rate REL which equals α/(α+β).

EXAMPLE
This example considers a system that operates initially with three homogenous servers.When one of the servers breaks down, the system operates with two homogenous servers (system is in mode-0).After the repair of the broken server and putting it back into service the system re-operates with three homogenous servers again.Hence, the system alternates between two modes of system operation.There is no limit on system capacity.Now, consider five bulks that arrive at times 0.344, 13.787, 16.449, 22.082, and 23.768.The sizes of bulks are 8, 3, 6, 4, and 6.Thus the three servers are to serve 27 customers that arrive in bulks according to exponential distribution with means 2.985.Service times for each server are 0.103, 4.033, 0.799, 0.506, 9.353, 0.039, 1.526, 0.328, and 2.896.The servers are broken down based on discrete uniform distribution.The order of which servers break down is server 3, server 1, server 3 and finally server 2. The time intervals that the system operates with three servers have an exponential distribution with mean 5.190.These times are 3.302, 1.001, 3.777, and 4.957.The time intervals that the system operates with two servers have an exponential distribution with mean 15.567.These times are 9.905, 3.026, 11.331, and 14.872.The first bulk consists of 8 customers of which 3 customers start the service immediately without waiting and take service time equals 0.103 for each one of them then depart the system at 0.447.The remaining 5 customers of the first bulk will be waiting until any server becomes available.At time 0.447 all servers are available.So 3 customers will enter the service and each one of them takes service time equals 4.033 and departs at 4.480 except customer number 6.During serving customer 6 at server 3, this server breaks down at time equals 3.302.Now, the system operates with server 1 and server 2 (i.e. the system is in mode-0).The customer number 6 will start the service again at the first server available.The server 1 and server 2 are ready for serving at time 4.480.So the customer 6 and customer 7 will start the service at time 4.480 and each one of them takes service time equals 0.799 then depart at time 5.278.At time equals 13.207, the server 3 will put back into the service in the system (i.e. the system is in mode-1).Other customers in the system and new arriving batches will be similarly served.We note that the last bulk will depart the system at time equals 43.792.The corresponding performance measures for

CONCLUSION
Based on the previous results, this investigation can be concluded by: -1-Tables 1 and 2 show that the increase of relative transition α/(α+β) has remarkable effect on the average number of customers in the queue and in the system.While the increase of the traffic intensity ρ has a noticeable effect on the same.2-The average number of customers in the queue and in the system increase as the average bulk size increases.So the increase of relative transition α/(α+β) has noticeable effect on Q L and S L when the average bulk size increases.
3-The most effect on the average number of customers in the system and in the queue is happening when the traffic intensity ρ is very close to unity.The absence of one server affects on the average number of customers as traffic intensity near the unity (heavy traffic).4-Tables 3 and 4 show that the average waiting time in the queue and in the system are highly affected by the absence of one of the servers for both low and high traffic intensity.5-Table 5 shows that the probability of having no customers in the system p 0 when a batch arrives i.e. the percentage of time the system is idle is not highly affected by the absence of one of the servers.While it decreases as the relative transition α/(α+β) increases for low traffic intensity and decreases noticeably as the relative transition α/(α+β) increases for high traffic intensity.Also we note that it decreases as the traffic intensity increases.6-Table 6 shows that the blocking probability B P is highly affected by the increase of traffic intensity.While it increases as the traffic intensity increases.So the blocking probability has remarkable effect resulting from absence of one of the servers.Where it increases as the relative transition α/(α+β) increases.
Finally, this paper was to come up with easily implementable algorithms to solve problems involving bulk-arrival queues with a breakdown of one server in case of steady state.This approach was preferred to producing large tables of exact results, varying the queuing parameters because of the endless list of possible combinations when applied to bulk queues and one of the servers breaks down.The performance measures are changed in response to the changes on the operating parameters.We documented the behavior of the system when one of the servers temporarily leaves the system with useful graphical representation to give the reader an opportunity to watch the system behavior over the traffic intensity and the relative transition rate.
C : Number of parallel servers.λ : Mean bulk arrival rate.µ : Constant service rate of a server.m : The average bulk size.Q L : The average queue length.S W : The average waiting time per customer in the system.S L : The average number of customers in the system.Q W : the probability that no customers are in system 0 P from relation, the blocking probability P B, i.e. the probability that any customer in an arriving batch must wait.the average waiting time in system S W from relation, the average number in the system L S from relation,