The potentials method for the M/G/1/m queue with customer dropping and hysteretic strategy of the service time change

(1)

DOI: 10.17512/jamcm.2016.1.20 e-ISSN 2353-0588

THE POTENTIALS METHOD FOR THE M/G/1/m QUEUE WITH CUSTOMER DROPPING AND HYSTERETIC STRATEGY

OF THE SERVICE TIME CHANGE

Yuriy Zhernovyi¹, Bohdan Kopytko²

1 Ivan Franko National University of Lviv Lviv, Ukraine

2 Institute of Mathematics, Czestochowa University of Technology Częstochowa, Poland

yu.zhernovyi@lnu.edu.ua, bohdan.kopytko@im.pcz.pl

Abstract. We propose a method for determining the probabilistic characteristics of the M/G/1/m queueing system with the random dropping of arrivals and distribution of the service time depending on the queue length. Two sets of service modes, with the service time distribution functions F x_n( ) and F xɶ_n( ) respectively, are used according to the two- threshold hysteretic strategy. The Laplace transforms for the distribution of the number of customers in the system during the busy period and for the distribution function of the length of the busy period are found. The developed algorithm for calculating the stationary characteristics of the system is tested with the help of a simulation model constructed with the assistance of GPSS World tools.

Keywords: single-channel queueing system, random dropping of customers, hysteretic strategy for service time, potentials method, stationary characteristics

1. Introduction

Studies show [1-3] that the random dropping of arrivals is a powerful tool for parameter control of a queueing system. The dropping can not only regulate the queue length, loss probability of customers, waiting time, and queue length variance, but also regulate several of these parameters simultaneously.

In order to increase the system capacity, threshold strategies of the service intensity (service time) change are used in queueing systems. In the general case, the essence of this strategy is that the service time distribution depends on the number of customers in the system at the beginning of each customer service [4].

With the help of the potentials method, we have developed an efficient algorithm for computing the stationary distribution of the number of customers in the systems with threshold functioning strategies [4-8].

(2)

In this paper we consider the M/G/1/m queueing system with random dropping of customers and distribution of the service time depending on the queue length.

Two sets of service modes (the main mode and overload mode), with the service time distribution functions F x_n( ) and F xɶ_n( ) respectively, are used according to the two-threshold hysteretic strategy. The overload mode with the functions F xɶ_n( ) starts functioning if at the beginning of service of a customer, the number of customers in the system satisfies the condition n>h₂. The return to the main mode with the functions F x carried out at the beginning of service of the customer, _n( ) for which n=h₁, where 1≤h₁<h₂ <m.

Each arriving customer can be accepted for service with a probability depending on the queue length. We assign this probability according to the rule: if at the time of the arrival of a customer n customers are in the system, then the customer is accepted for service with probability β_n and leaves the system (is discarded) with probability 1−β_n. Fix a threshold value h (1≤ <h m) and suppose that β_n = for 1 1≤ ≤ andn h, β_n =βɶ (0< <βɶ 1) for h+ ≤ ≤1 n m. In paper [9] we also studied the queueing systems with random dropping of arrivals. In contrast to this article, in [9] the probability β_n does not change during the time interval from the beginning to the completion of service of each customer.

2. Basic random walks

We consider the M/G/1/msystem,wherem is the maximum number of customers in the queue. Let λ be a parameter of the exponential distribution of the time intervals between moments of arrival of customers. Suppose that, if at the beginning of service of a customer the number of customers in the system is equal to

{1, 2, , 1},

n∈ … m+ then the service time of this customer is a random variable with distribution function F x_n( ) (x ≥0) for the main mode and F xɶ_n( ) (x ≥0) for the overload mode.

Denote by P_n the conditional probability, provided that at the initial time the number of customers in the queueing system is equal to n∈{0,1, 2,…,m+1}, and by E (P) the conditional expectation (the conditional probability) if the system starts to work at the time of arrival of the first customer. Let ( )η x be the number of customers arriving in the system during the time interval [0; ).x Let

(0) ( ) ( )

0 0

(0) ( ) ( )

0 0

( )

( ) ( ) ( ), ( ) ( ), 0,1, 2, ;

! ( )

( ) ( ) ( ), ( ) ( ), 0,1, 2, ;

!

i

sx i s x

n n n n n

i

sx i s x

n n n n n

f s f s e dF x f s x e dF x i

i

f s f s e dF x f s x e dF x i

i

λ

λ λ

∞ ∞

− − +

∞ ∞

− − +

= = + = =

∫ ∫

…

ɶ ɶ ɶ ɶ ɶ …

(3)

0 0

( ) < , ( ) < , ( ) 1 ( ), ( ) 1 ( ).

n n n n n n n n

M xdF x M xdF x F x F x F x F x

∞ ∞

=

∫

∞ ^ɶ =

∫

^ɶ ∞ = − ^ɶ = − ^ɶ

For Re s≥ and 0 n∈{1, 2,…,h₂} consider the sequences π_ni( ),s q_ni( )s and

n( ),

R s defined by the relations:

0

,

0

, 1

0

( ) 1 { ( ) 1} ( ), { 1,0,1, , 1};

( )

( ) 1 { ( ) 1} ( );

( )

( ) { ( ) } ( ) , {0,1, 2, , };

( ) { ( ) 1} ( ) ; ( ) 1

sx

ni n n

n

sx

n m n n n

n

sx

ni n n

sx

n m n n n n

n

s e x i dF x i m n

f s

s e x m n dF x

f s

q s e x i F x dx i m n

q s e x m n F x dx R s

f

π η

η

∞

−

∞

−

∞

−

∞

−

− +

= = + ∈ − − −

= ≥ − +

= = ∈ −

= ≥ − + =

∫

P

…

, 1

( )s π_n₋ ( )s .

(1)

Similarly, for Re s≥ and 0 n∈{h₁+1,h₁+ …2, , }m we set the sequences

ni( ),s

π^ɶ qɶ_ni( )s and R sɶ_n( ) :

0

,

0

, 1

0

( ) 1 { ( ) 1} ( ), { 1,0,1, , 1};

( )

( ) 1 { ( ) 1} ( );

( )

( ) { ( ) } ( ) , {0,1, 2, , };

( ) { ( ) 1} ( )

sx

ni n n

n

sx

n m n n n

n

sx

ni n n

sx

n m n n n

s e x i dF x i m n

f s

s e x m n dF x

f s

q s e x i F x dx i m n

q s e x m n F x dx

π η

η

∞

−

∞

−

∞

−

∞

−

− +

= = + ∈ − − −

= ≥ − +

= = ∈ −

= ≥ − +

∫

P

ɶ ɶ …

ɶ ɶ ɶ

ɶ

ɶ ɶ …

ɶ ɶ

, 1

; ( ) 1 .

( ) ( )

n

n n

R s ⁼ f s π ₋ s ɶ

ɶ ɶ

(2)

Let T and Tɶdenote the exponentially distributed random variables with parameter λ and λ λβɶ= ɶ respectively, and Z is a random variable distributed according to the law of Pascal, that is, {P Z =k}=βɶ^k(1−βɶ)^{n k}⁻ , k= 1, 2,…. It is known [10], that ZT = ɶ that is, as a result of a random decimation of the simplest flow we T, obtain a simplest flow.

(4)

Given the above, for h+ ≤ ≤ we find 1 n m

0 0

( )

{ ( ) } , 0 ;

!

( )

{ ( ) 1} 1 { ( ) } 1 .

!

j x n

m n m n j

x

n n

j j

x j x e j m n

j

x m n x j e x

j

λ

η λ

η η λ

−

− −

−

= =

= = ≤ ≤ −

≥ − + = −

∑

= = −

∑

P

P P

ɶ

For 1 n≤ ≤ we obtain h

{ }

1, 1 1,

0

1,

( )

{ ( ) } , 0 ;

!

{ ( ) } ( 1) ( 1) ( 1) ( )

( ) ( ), 1 ;

{ ( ) 1} 1 { ( ) }

( 1) ( ) (

j x n

n

h n n h j h n n h j

m n

n n

j

h n m h

x j x e j h n

j

x j h n T n h j T x h n T n h j T

G x G x h n j m n

x m n x j

h n T m h T x G x

λ λ

η η

−

− + − + − − + − +

−

=

− + −

= = ≤ ≤ −

= = − + + − + − < < − + + − + =

= − − + ≤ ≤ −

≥ − + = − = =

= − + + − < =

∑

P

P P

P

ɶ ɶ

ɶ

{ }

,

1 1 1

0 2

1

1 1

0 0 0

,0

);

( )

1 ( 1) ( 2 )!( )

( 1)!( 1)!( )

( )!( ) ( ) ! ( )

( 1) ,

! !

!

1 ;

( )

n r

n n r r

k k

n r r k

k j

j i i

n r k k

j x x

k j k

j i i

n

G x nT rT x

C n r k

n r

k j x k x

e e

i i

j

r m h

G x nT x

λ λ

λ λ λ λ

λ λ

λ λ λ λ

λ λ

− + − −

= + − − +

+ − −

+ + +

= = =

= + < =

= − − + − − − ×

− − −

 + − 

× − + 

≤ ≤ −

= <

∑

∑ ∑ ∑

P

ɶ

ɶ ɶ

ɶ

1

0

( )

1 .

!

k n x

k

e x

k

λ ⁻ λ

−

=

= −

∑

Taking into account the expressions for P_n{ ( )η x = j} and equalities

( ) ( )

1 0 0

( ) ( )

1 0 0

( )

( ) ( ) 1 ( ) ,

! ( )

( )

( ) ( ) 1 ( ) ,

! ( )

k k k i

s x i

n nk k n

i

k k k i

s x i

n nk k n

i

x s

e F x dx g s f s

k s

x s

e F x dx g s f s

k s

λ

λ λ λ

λ λ

λ λ λ λ λ

λ λ

∞

− +

+ =

∞

− +

+ =

  +  

 

= + = +  −   + 

  +  

 

= + = +  −   + 

∫ ∑

∫

^ɶ ^ɶ

∑

^ɶ

by (1), (2) calculate the terms of the sequences π_ni( ),s q_ni( ),s π^ɶ_ni( )s and qɶ_ni( ).s

(5)

For h+ ≤ ≤1 n h₂ we find

( ) , 1

( )

, , 1

0 0

( )

( ) , ( ) ( ), 0 ;

( )

1 ( )

( ) 1 1 ( ), ( ) ( ).

( )

j n

n j n j n j

n

m n m n

j n

n m n n n m n n j

j j

n

f s

s q s g s j m n

f s

s f s q s f s g s

f s s

π λ λ

π λ λ λ

λ

−

− −

− − +

= =

= + = + ≤ ≤ −

− +

= − + = − +

∑

+

∑

ɶ ɶ

ɶ

ɶ ɶ

ɶ

(3)

For 1 n≤ ≤h₂≤ and h 1 n≤ ≤ ≤h h₂ we obtain

( ) , 1

( )

( ) , ( ) ( ), 0 ;

( )

j n

n j n j n j

n

f s

s q s g s j h n

f s

π ₋ = ⁺λ = +λ ≤ ≤ −

( )

, 1 1, 1 1,

0

1, 1 1,

0

, 1, , 1 1,

0

( ) 1 ( ) ( ) ( ),

( )

( ) ( ) ( ) ( ) , 1 ;

( ) 1 ( ) ( ), ( ) (

( )

sx

n j h n n h j h n n h j n

n

sx

nj h n n h j h n n h j n

sx sx

n m n h n m h n n m n h n m h

n

s e G x G x dF x

f s

q s e G x G x F x dx h n j m n

s e G x dF x q s e G

f s π

π

∞

−

− − + − + − − + − +

∞

−

− + − + − − + − +

∞

− −

− − + − − + − + −

= −

= − − + ≤ ≤ −

= =

∫

0 1

, 1 1

0 0

2

1 ( ) ( )

1 1

0 0 0

) ( ) ;

( ) ( ) ( ) ( 1) ( 2 )!

( 1)!( 1)!( )

( )!( ) !

( ) ( 1) ( ) ( ) ,

!

1

n

n n r r

sx k

n r n n n r r

k k j

j

n r k k

k j i i

n n

k j k

j i i

x F x dx

e G x dF x f s C n r k

n r

k j k

f s f s

j

r m h λ λ

λ λ

λ λ λ λ λ λ

λ λ

∞

∞ −

−

+ − −

= + − − +

+

+ + +

= = =

= − − + − − ×

− − −

 + − 

× −  − + + + 

≤ ≤ −

∫

∫ ∑

∑ ∑ ∑

ɶ ɶ ɶ

ɶ ɶ

ɶ

;

1 ( ) ,0

0 0

( ) ( ) ( ) ( );

sx n k

n n n n

k

e G x dF x f s f s λ

∞ −

−

=

= −

∑

+

∫

⁽⁴⁾

1

, 1 1

0 0

2

1

1 1

0 0 0

,0 0

1 ( ) ( 1)

( ) ( ) ( 2 )!

( 1)!( 1)!( )

( )!( ) !

( ) ( 1) ( ) ( ) ,

!

1 ;

( )

n n r r

sx n k

n r n n r r

k k j

j

n r k k

k j

ni ni

k j k

j i i

sx n

e G x F x dx f s C n r k

s n r

k j k

g s g s

j

r m h

e G x

λ λ λ λ

λ λ λ λ λ λ

λ λ

∞ −

−

+ − −

= + − − +

+

+ + +

= = =

−

− −

= − + − − ×

− − −

 + − 

× −  − + + + 

 

≤ ≤ −

∫ ∑

∑ ∑ ∑

ɶ ɶ

ɶ ɶ ɶ

ɶ

1

0

1 ( )

( ) ( ).

n n

n nk

k

F x dx f s g s

s λ

∞ −

=

= − −

∑

+

∫

(6)

Expressions for π^ɶ_ni( )s and qɶ_ni( )s are similar with presented in (3) and (4) for

1 1 1

h + ≤ + ≤ or h n h+ ≤1 h₁+ ≤1 n, and for h₁+ ≤ ≤ respectively, if we 1 n h replace f_n, f_n^{( )}^j , g_{n j} and F_n by fɶ_n, fɶ_n^{( )}^j ,gɶ_{n j} and Fɶ_n.

Note that

0 0 0 0

1 1

lim ( ) lim ( ) 1, lim , lim .

1 ( ) 1 ( )

n n n n

s s s s

n n

f s f s M M

f s f s

→+ = →+ = →+ = →+ =

− −

ɶ ɶ

ɶ (5)

Introduce the notation:

0 0 0

lim ( ), lim ( ), lim ( ),

lim ( ), lim ( ), lim ( ).

ni ni ni ni n n

s s s

ni ni ni ni n n

s s s

s q q s R R s

π π

→+ →+ →+

= = =

= = ɶ = ɶ

ɶ ɶ ɶ ɶ

With the help of equalities (1)-(5) we can obtain expressions for the members of the sequences π_ni, q_ni, R_n,π^ɶ_ni, qɶ_ni and Rɶ_n.

3. Distribution of the number of customers in the system during the busy period

Let ( )ξ t be the number of customers in the system at time .t Denote by P_{F n}_, (P_{F n}_, ) the conditional probability, provided that at the initial time the number of customers of the system is equal to n and the service begins with the service time distributed according to the law F x_n( ) (F xɶ_n( )).

Let = inf{τ t≥0 : ( ) = 0}ξ t denote the length of the first busy period for the considered queueing system, and for k∈{1, 2,…,m+1}

, 2

, 1

2 2

0 0

( , ) { ( ) , }, 1 ;

( , ) { ( ) , }, 1 1;

( , ), 1 ;

( , )

( , ), 1 1;

( , ) ( , ) , ( , ) ( , ) , R 0.

n F n

n n

n

st st

n n n n

t k t k t n h

t k t k t h n m

t k n h

t k t k h n m

s k e t k dt s k e t k dt e s

ψ ξ τ

ϕ ψ

ψ

ϕ ψ

∞ ∞

− −

= = > ≤ ≤

= = > + ≤ ≤ +

 ≤ ≤

=  + ≤ ≤ +

Φ =

∫

Φ =

∫

>

P ɶ P

ɶ

ɶ ɶ

It is evident that ϕ₀( , )t k = 0,

1( , ) 1( , ).

h t k h t k

ψɶ =ϕ With the help of the formula of total probability we obtain the equalities:

(7)

1 0 0

0

2

1 0 0

0

( , ) { ( ) } ( , ) ( )

{ ( ) 1} ( , ) ( )

{ 1} { ( ) } ( ), 1 ;

( , ) { ( ) } ( , ) ( )

{ ( ) 1} ( , )

m n t

n n n j n

j t

n m n

n n

m n t

n n n j n

j t

n m

t k x j t x k dF x

x m n t x k dF x

I n k m t k n F t n h

t k x j t x k dF x

x m n t x k dF

ϕ η ϕ

η ϕ

η

ψ η ψ

η ψ

−

+ −

=

−

+ −

=

= = − +

+ ≥ − + − +

+ ≤ ≤ + = − ≤ ≤

= = − +

+ ≥ − + −

∑ ∫

∫

∑ ∫

∫

P

P P

P

ɶ ɶ ɶ

ɶ ɶ

1

1 1 1

0

( )

{ 1} { ( ) } ( ), 1 ;

( , ) ( , ) ( ) { 1} ( ).

n

n n

t

m m m m

x

I n k m t k n F t h n m

t k t x k dF x I k m F t

η

ψ ₊ ψ ₊ ₊

+

+ ≤ ≤ + = − + ≤ ≤

=

∫

− + = +

P ɶ

ɶ ɶ

(6)

Here { }I A is the indicator of a random event A ; it equals 1 or 0 depending on whether or not the event A occurs.

( )_n ( , ) { 1} _{n k n}, ( ), ( )_n ( , ) { 1} _{n k n}, ( ).

f s k =I n≤ ≤ +k m q ₋ s fɶ s k =I n≤ ≤ +k m qɶ ₋ s Taking into account the relations (1) and (2), from (6) we obtain the system of equations for the functionsΦ_n( , )s k and Φɶ_n( , )s k :

1

, 1 1 ( ) 2

=0

( , ) = ( ) ( ) ( , ) ( , ), 1 ;

m n

n n n j n j n

j

s k f s ^{+ −} π ₋ s _{+ −} s k f s k n h

Φ

∑

Φ + ≤ ≤ ⁽⁷⁾

1

, 1 1 ( ) 1

=0

( , ) ( ) ( ) ( , ) ( , ), 1 ;

m n

n n n j n j n

j

s k f s ^{+ −}π ₋ s _{+ −} s k f s k h n m

Φ^ɶ = ^ɶ

∑

^ɶ Φ^ɶ + ^ɶ + ≤ ≤ ⁽⁸⁾

1

1 1

1 ( )

( , ) ( ) ( , ) { = 1} ^m

m m m

f s s k f s s k I k m

s

+ + +

Φ = Φ + + − ɶ

ɶ ɶ ɶ

with the boundary conditions

1 1

0( , )s k 0, _h ( , )s k _h ( , ).s k

Φ = Φɶ = Φ (9)

(8)

For solving the systems of equations (7)-(9) we will use the functions R_ni( )s and Rɶ_ni( ),s defined by the recurrence relations:

1

1 1 , 1 1 1, 1, 1 ,

=0 2

1

1 1 , 1 1 1, 1, 1 ,

=0 1

( ) ( ); ( ) ( ) ( ) ( ) ( ) ( ) ,

0 1, 1 1;

( ) ( ); ( ) ( ) ( ) ( ) ( ) ( ) ,

j

n n n j n n j n n i n i j i

i

j

n n n j n n j n n i n i j i

i

s R s s R s s f s s s

n h j m n

s R s s R s s f s s s

h n m

π

−

+ + + + + + + −

−

+ + + + + + + −

 

= =  − 

 

≤ ≤ − ≤ ≤ − −

 

= =  − 

 

≤ ≤ −

∑

ɶ

ɶ ɶ ɶ ɶ ɶ ɶ ɶ

R R R R

1, 1≤ ≤j m− −n 1.

(10) Introduce the notation:

2

2 2 2 2

2

, , ,0

=1

1

, ,

=1 = 1

, ,

=1

( ) ( ) ( ) ( ) ( ), ( ) = ( ) 1;

( ) ( ) ( ) ( ) ( ) ( ) ;

( ) ( ) ( ) ( )

h n

n n h n ni n i n i h n i h h

i

h n m

n ni n i n i m n i n i j n i j

i j h

m n

n n m n ni n i n i m n

i

C s s s f s s C s s

D s s f s s s A s

A s s s f s

π

π π

π

−

− + + − −

− −

+ + − − + − −

+

−

− + + − −

= − ≡

 

=  + 

= −

∑

∑ ∑

∑

ɶ ɶ

ɶ ɶ ɶ ɶ

R R R

R

R R

2 2

2

2 1

( ) , ( )

=1 = 1 =1

( ); ( ) ( , ) 0;

( , ) ( ) ( , ) ( ) ( ) ( ) ( , ) .

i h h

h n m m j

n ni n i n i n i j n i ju j u

i j h u

s D s D s k

D s k ⁻ s f ₊ s k f ₊ s ⁻ π ₊ _{− −} s ⁻ s f ₊ s k

+

= ≡

 

=

∑

 −

∑ ∑



ɶ

R Rɶ

Reasoning as in the proof of Theorem 1 of [7], we obtain the following statement.

Theorem 1. For all k∈{1, 2,…,m+1} and Re s> the equalities 0

( )

(

⁰ ⁰ ⁰

)

0

2

( , ) 1 ( ) ( , ) ( ) ( ) ( ) ( ) ( , )

( )

( , ), 1 ;

n n n n m

n

s k C s D s k C s D s C s D s s k

C s

D s k n h

Φ = + − Φ −

− ≤ ≤

ɶ ɶ

( ) 2

=1

( ) 1 2

=1

1

1 1 1

( , ) ( ) ( , ) ( ) ( , ), 1 1;

( , ) ( ) ( , ) ( ) ( , ), 1 ;

1 ( )

( , ) ( , ) ( ) ( , ) { = 1}

m n

n n m ni n i

i m n

n n m ni n i

i

m

m m m m

s k A s s k s f s k h n m

s k A s s k s f s k h n h

f s

s k s k f s s k I k m

s

−

+

−

+

+ + + +

Φ = Φ − + ≤ ≤ −

Φ = Φ − + ≤ ≤

Φ = Φ = Φ + + −

∑

ɶ

ɶ ɶ

ɶ

ɶ ɶ

ɶ

ɶ ɶ ɶ

R

(9)

are fulfilled, where

( )

1

1 1 1 1

1 1 1

0 ( ) 0

=1

0 0

( ) ( , ) ( ) ( , ) ( ) ( , )

( , ) .

( ) ( ) ( ) ( ) ( )

m h

h h i h i h

i m

h h h

C s D s k s f s k C s D s k

s k

C s D s C s D s A s

−

+

 

− −

 

 

Φ =

− +

∑

^ɶ ^ɶ

ɶ ɶ ɶ

R

4. Busy period and stationary distribution

If the system starts functioning at the moment when the first customer arrives, then

( )

1 0

1 0 1 0 0 1 1

0

{ ( ) , } ( , ) =

1 ( ) ( , ) ( ) ( ) ( ) ( ) ( , ) ( , ).

( )

st

m

e t k t dt s k

C s D s k C s D s C s D s s k D s k C s

ξ τ

∞

− = > = Φ

= + − Φ −

∫

^P

ɶ ɶ

(11)

To obtain a representation for

0

{ }

e st τ t dt

∞

− >

∫

^P we sum up equalities (11) for k running from 1 to m+ . Given the definitions of 1 f_{( )}_n ( , ),s k fɶ_{( )}_n ( , ),s k q_{n j}( )s and

n j( ),

qɶ s it is not difficult to ascertain that

1 1 1

( ) ( ) 2

=1 = =0

1 1 1

( ) ( ) 1

=1 = =0

1 ( )

( , ) ( , ) ( ) , 1 ;

1 ( )

( , ) ( , ) ( ) , 1 .

m m m n

n

n n n j

k k n j

m m m n

n

n n n j

k k n j

f s k f s k q s f s n h

s

f s k f s k q s f s h n m

s

+ + + −

= = = − ≤ ≤

= = = − + ≤ ≤

∑ ∑ ∑

∑

^ɶ

∑

^ɶ

∑

^ɶ ^ɶ

2

2 1

,

=1 = 1 =1

1 ( )

( ) ( ) ( ) ( ) ( ) ;

h n m m j

n i j u

n ni n i n i j n i ju

i j h u

f s f s

D s s f s s s

s π s

− − −

+ +

+ + − −

+

 − − 

=

∑

^R  −

∑ ∑

^R^ɶ ^ɶ 

( )

1

1 1 1

0 0

=1

0 0

1 ( )

( ) ( ) ( ) ( ) ( )

( ) .

( ) ( ) ( ) ( ) ( )

m h h i

h h i h

i m

h h h

f s

C s D s s C s D s

s s

C s D s C s D s A s

− +

 − 

 − −

 

 

Φ =

− +

∑

^ɶ ^ɶ

ɶ

ɶ ɶ

R

Thus, (11) confirms the following statement.

(10)

Theorem 2. The Laplace transform of the distribution function of the length of the busy period is defined as

( )

0

1 0 1 0 0 1 1

0

{ }

1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ).

( )

st

m

e t dt

C s D s C s D s C s D s s D s C s

τ

∞

− > =

= + − Φ −

∫

^P

ɶ ɶ

(12)

To find

0

{τ t dt} ( )τ

∞

> =

∫

^P ^E we need to pass to the limit in (12) as s → + 0.

We use the sequences π_ni, R_n, π^ɶ_ni and Rɶ_n, as well as sequences R_ni and Rɶ_ni, obtained by limit passages:

lim0 ( ),

ni ni

s s

= →+

R R

lim0 ( ).

ni ni

s s

= →+

ɶ ɶ

R R For R_ni and Rɶ_ni (10) implies the recurrence relations:

1

1 1 , 1 1 1, 1, 1 ,

=0

2

1

1 1 , 1 1 1, 1, 1 ,

=0

1

; ,

0 1, 1 1;

; ,

1, 1 1.

j

n n n j n n j n i n i j i

i

j

n n n j n n j n i n i j i

i

R R

n h j m n

R R

h n m j m n

π

−

+ + + + + + + −

−

+ + + + + + + −

 

= =  − 

 

≤ ≤ − ≤ ≤ − −

 

= =  − 

 

≤ ≤ − ≤ ≤ − −

∑

ɶ ɶ ɶ ɶ ɶ ɶ ɶ

R R R R

(13)

Note that

2 1, 1 2 1 2,1 1.

h Rh h

π _{+ −} = ₊ =R =

Using the relations (13) and taking into account the equalities

2 1

1 1

1, 1 ; 1, 1 ;

m n m n

n j n j

j j

n h h n m

π π

− −

=− =−

= ≤ ≤ = + ≤ ≤

∑ ∑

^ɶ

we can prove that

, , 2

1

, , 1

1

1, 1 ;

1, 1 1.

m n

n m n ni n i m n i

i m n

n m n ni n i m n i

i

n h

h n m

π π

−

− + − −

=

−

− + − −

=

− = ≤ ≤

− = + ≤ ≤ −

∑

ɶ ɶ ɶ

R R

(14)

Given (5) and (14), using (12) we obtain the following statement.

(11)

Theorem 3. The mean length of the busy period is determined in the form

1

1 1 1

0 1 1 2

1

( ) ( , ) ,

m h

h h i h i

i

D D R h h D M

τ

−

+

=

 

= − −  −

∑



E Rɶ ɶ (15)

where

2

2 2

2 1 2 1

1 0, 1 1,

, 1 2

=1 = 1 =1 , 1

; ( , ) .

h n m m j

h h

n n i n i n i j n i ju j u

i j h u h h h

D M π M R h h

− − −

+

+ + − − +

+ − +

  −

=

∑

^R  −

∑ ∑

^R^ɶ ^ɶ  =^R_R ^R

Introduce the notation: lim { ( ) } _k,

t ξ t k p

→∞P = = k∈{0,1, 2,…,m+1}. Reasoning as in the paper [4], from (11) we obtain formulas for the stationary distribution of the number of customers in the system.

Theorem 4. The stationary distribution of the number of customers in the system is given by

( )

1

1 1 1 1 1 1

0

1

0 0 0 0 , 1 1, 1 1

=1

0 0 , 1 2 , ,

=1 =1

1

1 1, 1 1 2

=1

0 0 1

1 ;

1 ( )

, 1 ;

( , )

, 1 ;

( ) ( )

k

k k k i i k i i i k i

i k k h

k i i k i h i h i k h i h i h i k h i

i i

k

i i k i i

k

p

p p q q q k h

p p q R h h q q

q h k h

p p D k D k R

λ τ λ

λ

−

− + − −

−

− + − − + − −

−

+ − −

= +

 

=  + −  ≤ ≤

 

=  + − −



−  + ≤ ≤



= − +

∑

∑ ∑

∑

E

ɶ ɶ

R R R

R

1

1 1 1 1

1

1 1 1 1

1 2 ,

=1

2 1

0 0 1 1 2 , 1

=1

( , ) ( ) ,

1 ,

( 1) ( 1) ( , ) ( 1) ,

k h

h i h i k h i h i

m

m h

h i h i m h i h

i

h h q D k

h k m

p

p D m D m R h h q D m

λ

−

+ − −

+

−

+ + − −

  

  − 

  

 

+ ≤ ≤

=

  

 

=  + − + +  − + 

∑

ɶ ɶ

ɶ ɶ R

R

(16)

where

2

2 1

, 2 , ,

=1 = 1 =1

( ) { 2 } .

h n k k j

n ni n i k n i n i j n i ju j u k j u

i j h u

D k ⁻ q ₊ _{− −} I h k m ⁻ π ₊ _{− −} ⁻ q ₊ _{− −}

+

 

=  − + ≤ ≤ 

 

∑

^R

∑ ∑

^R^ɶ ^ɶ

(12)

Using (15) we find the ratio of the mean number of customers served per unit of time to the mean number of all arriving customers per unit time and obtain the formula for the stationary service probability

1

1 1

sv 0 0 1 1 2

=1

( , ) ,

m h

h i h i

p T T R h h T

  − 

 

=  − + 

∑

− 

P Rɶ (17)

where

2

2 1

,

=1 = 1 =1

1 .

h n m m j

n ni n i j n i ju

i j h u

T ⁻ ⁻ π ₊ _{− −} ⁻

+

 

=  − 

 

∑

^R

∑ ∑

^R^ɶ

We find the stationary queue characteristics - the average queue length E( )Q and average waiting time (EW) - by the formulas

1

1 sv

( ) , ( ) ( ).

m k k

Q kp W Q

+ λ

=

∑

=^E

E E

P (18)

5. Example for calculating of stationary characteristics

Assume that m=6, λ=10, h₁=2, h₂= =h 4, βɶ =0.4, the uniform distribution on the intervals (0; 0.5] and (0; 0.25] corresponds to the distribution functions of the service time F x_n( )=F x( ) 1

(

≤ ≤n h2

)

and F xɶ_n( )=F xɶ( )

(

^h¹^{+ ≤ ≤ +}¹ ⁿ ^m ¹

)

respectively. Thus,

(

^0.5

) (

^0.25

)

2 4

0.25, = 0.125, ( ) 1 ^y , ( ) 1 ^y .

n n

M M M M f y e f y e

y y

− −

= = ɶ = ɶ = − ɶ = −

The row "p_k" of Table 1 contains steady-state probabilities p_k, calculated by the formulas (16). For the sake of comparison, the same table contains the corre- sponding probabilities evaluated by the GPSS World simulation system [11, 12]

for the time value t =10 .⁶ The values of the stationary characteristics found by the formulas (15), (17) and (18), and calculated with the help of GPSS World, are shown in Table 2.

(13)

Table 1 Stationary distribution of the number of customers in the system

Number

of customers k 0 1 2 3 4 5 6 7

p_k 0.00373 0.01504 0.05757 0.13005 0.24539 0.33756 0.15697 0.05369 p_k

(GPSS World) 0.00371 0.01506 0.05768 0.13005 0.24527 0.33713 0.15693 0.05418

Table 2 Stationary characteristics of the system

Characteristic E(τ) E(Q) E(W) P_sv

Analytical value 26.724 3.511 0.541 0.650

Value according to GPSS World 26.894 3.512 0.541 0.649

Calculations show that if the random dropping of customers is not used, then for the considered data the value of the capacity of the system P_sv is increased by 13.1%, but ( )EQ and (EW) are also increased by 25.6% and 11.1% respectively.

6. Conclusions

With the help of the potentials method, we have obtained simple and suitable formulas for numerical realization for finding the stationary characteristics of the M/G/1/m queueing system with the random dropping of customers and hysteretic change of the service time. We have examined a fairly general statement of the problem, because we assume that the service time depends on the number of customers in the system and the dropping probability is a function of the queue length at the time of arrival of a customer.

Our approach, unlike most of the methods used to study the semi-Markov models of queueing, allows to investigate not only stationary, but also the transient regime of the system, in particular, to find the Laplace transforms for the distribution of the number of customers in the system during the busy period and for the distribution function of the length of the busy period.

References

[1] Chydziński A., Nowe modele kolejkowe dla węzłów sieci pakietowych, Pracownia Kompute- rowa Jacka Skalmierskiego, Gliwice 2013.

[2] Tikhonenko O., Kempa W.M., Queue-size distribution in M/G/1-type system with bounded capacity and packet dropping, Communications in Computer and Information Science 2013, 356, 177-186.