2nd National Conference on Data Envelopment Analysis August 4-5, 2010
Faculty of Sciences, Islamic Azad University of Rasht
1
Evaluating the performance of multiple comparable queuing by using DEA
Y. Jafari, Department of Mathematics, Semnan branch, Islamic Azad University, Semnan, Iran, Email: [email protected]
R. Madahi, Department of Mathematics, Najafabad branch, Islamic Azad University, Najafabad, Iran, Email: [email protected]
S. Khodayifar, Department of Mathematics, Tehran University, Tehran, Iran.
Email: S[email protected]
Abstract: Queuing models play a very important role in real life situations. In this paper, we Evaluates the performance of multiple Queuing by using of Data Envelopment Analysis. Goal find the best queuing from a source to a destination in the presence of additive attributes such as, interarrival rate, service rate, the average length queue, the average waiting time, the average serviced customers and busy period density such that it possess at least cost and at most profit. It is clear, decision makers can have several different goals and these goals may be in conflict with each other. A numerical example is solved by our proposed method.
Keywords: Data envelopment analysis; Finite queues; Infinite queues.
1. INTRODUCTION
A. DEA Models
Data envelopment analysis (DEA) is a method for evaluating efficiency of decision making units (DMUs).
Consider n decision making units (j 1, ,n)
DMUj = ,
each
DMUj consuming input levels (i 1, ,m) xij = to produce output levels (r 1, ,s)
yrj = . The relative efficiency score of
DMUo under the CCR model is given by the following optimization problem:
0 v , u
) 1 ( n
, , 1 j j 1
Tx j v Ty u . t . s
xo vT yo uT Max
≥
=
≤
Where u and v represent vectors for the output and input weights, respectively.
We point out that the DEA model (1) is equivalent to the following linear program which is called the output- oriented formulation for the CCR model:
0 v , u
) 2 ( 0
Y u X v
, 1 y u . t . s
x v Min
T T
o T
o T
≥
≤ +
−
=
B. Queuing models
This section will explore some elements of queuing theory that are needed to create the proposed method. It is not intended to explain queuing theory nor its background, only the elements and formulas needed for this paper will be explained briefly. For a concise description of these methods, the reader should refer to an operations research handbook (e.g. Hillier and Liebermann [2], Gross and Harris [1], Takagi [3]).
A commonly used queuing model is the M/M/s model, where the first M stands for Markovian or pure random arrivals, the second M means Markovian service times and s equals the number of servers. The elapsed time from the start of service to its completion is called service time. The M/M/s model assumes that all interarrival times and service times are independently and identically distributed according to two (distinct) exponential distributions.
Using the M/M/s model with s parallel servers, queuing theory predicts that the expected waiting time in queue (excluding service time) for each in dividual customer equals:
) 3 ) (
1 (
! s
W P 2 s
1 s q o
q −ρ µ
ρ
= λ
−
with
) 4 ( 1
1
! s
) / (
! n
) / ( P 1
1 s
0 n
s q n q
o =
∑
−= λ µ + λ µ −ρ.
2 and
) 5 s (
q
µ
=λ ρ
with λq the mean arrival rate (i.e. expected number of arrivals per unit of time) and µ the mean service rate (i.e. expected number of customers completing service per unit of time). Consequently 1/λq and 1/µ can be interpreted as the expected interarrival time and the expected service time, respectively. The number of customers already in the system (i.e. being served and waiting) upon a random arrival is denoted by n. P0 is the steady-state probability of zero customers in the system.
Finally, _ (the expected utilization rate of the system), can be interpreted as the proportion of time that each server is busy. In order to ensure that the queue does not grow without limit, ρ must be smaller than one.
Also, the average length queue equals:
) 6 ) (
s ( )!
1 s (
/ ) / (
L P 2
q q s q o
q − µ−λ
µ λ µ
= λ
For description of other methods, the reader should refer to an operations research handbook (e.g. Hillier and Liebermann [2], Gross and Harris [1], Takagi [3]).
2. THE PROPOSED METHOD
Consider a set of n queuing, with each queuing satisfies in a queuing model and consider j-th queuing as (j 1, ,n)
DMUj = . A given DMU is not efficient if some other DMU can produce the same amounts of output with less of some resources and not more of any other. Goal finds the best queuing from a source to a destination in the presence of additive attributes such as, interarrival rate, service rate, the average length queue, the average waiting time and busy period density such that it possess at least cost and at most profit. Hence, we can denotes inputs and outputs for each DMU, instance, interarrival rate, service rate, the average length queue, the average waiting time and busy period density are inputs for queuing and the number server and the average serviced customers are outputs for queuing. All DMUs, ( (j 1, ,n)
DMUj = use the same number, m, of inputs )
m , , 1 i ij(
x = to produce the same number, s, of outputs (r 1, ,s)
yrj = . Note that the inputs of DMUs only differ in the quantity. This is also true to the outputs.
Further suppose )
yj j, x j (
DMU = where
mj) x , j, x1 j (
x = and )
ysj , j, y1 j (
y = . The efficiency of j-th queuing can be determined by using the DEA technique. The set toward this end, the following linear fractional program is solved:
0 v , u
) 7 ( n
, , 1 j j 1
Tx j v Ty u . t . s
xo vT yo uT Max
≥
=
≤
That the DEA model (3) is equivalent to the following linear program which is called the input-oriented formulation for the CCR model:
T o T
o
T T
Min v x s.t. u y 1,
v X u Y 0 ( 8)
u, v 0
=
− + ≤
≥
3. NUMERICAL EXAMPLE
Here, we present one example. Suppose we have eight queuing as DMUs as M/M/2 queue whit λ the mean arrival rate, µ the mean service rate which are presented in Table 1.
TABLE 1- THE EIGHT QUEUING AS DMUS AS M/M/S QUEUE
Queuing λ µ N
1 1.5 2 3
2 4 7 7
3 3.2 4.2 4.7
4 .2 2 6
5 3.5 4.2 5.5
6 3.2 5.7 2
7 6.5 7.2 5
8 3.2 3.7 3
Table (2) presents the results using of formulations (3) and (6) for obtaining values of the average waiting time and the average length queue as the follow:
TABLE 2- THE RESULTS ABOUT VALUES OF WAITING TIME AND LENGTH QUEUE.
Queuing
Wq Lq
1 0.081818182 0.030681818 2 0.012698413 0.001036605 3 0.040419351 0.007332309 4 0.001253133 0.000062656 5 0.050020008 0.009924605 6 0.015005774 0.001477946 7 0.03554028 0.004456246 8 0.06216459 0.014530803 The results of model (8) are given in Table (3).
TABLE 3- THE RESULTS OF MODEL (8) FOR DMUS OF TABLE (3) .
1 2 3 4 5 6 7 8
.50 .33 .37 1.00 .43 .11 .23 .27 In this example, we obtain the efficiency each queuing.
For the example, the CCR model shows that the First queue must the average waiting time decreased of 0.081818182 to 0.040909091.
4. REFERENCES
[1] D. Gross, C. M. Harris, Fundamentals of Queueing Theory, 2nd Edition. Wiley, New York, 1985.
[2] F. S. Hillier, G. J. Lieberman, Introduction to operations research, 6nd Edition. Industrial Engineering Series, McGraw- Hill, New York 1995.
[3] H. Takagi, Queuing Analysis, 2: Finite Systems. North-Holland, Amsterdam 1993.
