Algorithms for a continuous-review (s, S) inventory system

1981 ◽  
Vol 18 (2) ◽  
pp. 461-472 ◽  
Author(s):  
V. Ramaswami
Keyword(s):  
Steady State ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Continuous Review ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Phase Type ◽  
Markov Modulated ◽  
Stimulation Of

The steady-state distribution of the inventory position for a continuous-review (s, S) inventory system is derived in a computationally tractable form. Demands for items in inventory are assumed to form an N-process which is the ‘versatile Markovian point process' introduced by Neuts (1979). The N-process includes the phase-type renewal process, Markov-modulated Poisson process etc., as special cases and is especially useful in modelling a wide variety of qualitative phenomena such as peaked arrivals, interruptions, inhibition or stimulation of arrivals by certain events etc.

Algorithms for a continuous-review (s, S) inventory system

1981 ◽  
Vol 18 (02) ◽  
pp. 461-472
Author(s):  
V. Ramaswami
Keyword(s):  
Steady State ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Continuous Review ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Phase Type ◽  
Markov Modulated ◽  
Stimulation Of

The steady-state distribution of the inventory position for a continuous-review (s, S) inventory system is derived in a computationally tractable form. Demands for items in inventory are assumed to form an N-process which is the ‘versatile Markovian point process' introduced by Neuts (1979). The N-process includes the phase-type renewal process, Markov-modulated Poisson process etc., as special cases and is especially useful in modelling a wide variety of qualitative phenomena such as peaked arrivals, interruptions, inhibition or stimulation of arrivals by certain events etc.

On Growth-Collapse Processes with Stationary Structure and Their Shot-Noise Counterparts

2009 ◽  
Vol 46 (02) ◽  
pp. 363-371 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Steady State ◽  
Shot Noise ◽  
Growth Process ◽  
Initial Conditions ◽  
Steady State Distribution ◽  
State Distribution ◽  
Stationary Structure ◽  
Stationary Version ◽  
Special Cases ◽  
The Times

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes. We begin with a stationary setup with some relatively general growth process and observe that, under certain expected conditions, point- and time-stationary versions of the processes exist as well as a limiting distribution for these processes which is independent of initial conditions and necessarily has the marginal distribution of the stationary version. We then specialize to the cases where an independent and identically distributed (i.i.d.) structure holds and where the growth process is a nondecreasing Lévy process, and in particular linear, and the times between collapses form an i.i.d. sequence. Known results can be seen as special cases, for example, when the inter-collapse times form a Poisson process or when the collapse ratio is deterministic. Finally, we comment on the relation between these processes and shot-noise type processes, and observe that, under certain conditions, the steady-state distribution of one may be directly inferred from the other.

scholarly journals A TWO-ECHELON SPARE PARTS NETWORK WITH LATERAL AND EMERGENCY SHIPMENTS: A PRODUCT-FORM APPROXIMATION

2017 ◽  
Vol 32 (4) ◽  
pp. 536-555 ◽  
Author(s):  
Richard J. Boucherie ◽  
Geert-Jan van Houtum ◽  
Judith Timmer ◽  
Jan-Kees van Ommeren
Keyword(s):  
Steady State ◽  
Spare Parts ◽  
Inventory System ◽  
Product Form ◽  
Steady State Distribution ◽  
State Distribution ◽  
Poisson Demand ◽  
Lateral Transshipment ◽  
Repair Facility ◽  
Erlang Loss

We consider a single-item, two-echelon spare parts inventory model for repairable parts for capital goods with high downtime costs. The inventory system consists of multiple local warehouses, a central warehouse, and a central repair facility. When a part at a customer fails, if possible his request for a ready-for-use part is fulfilled by his local warehouse. Also, the failed part is sent to the central repair facility for repair. If the local warehouse is out of stock, then, via an emergency shipment, a ready-for-use part is sent from the central warehouse if it has a part in stock. Otherwise, it is sent via a lateral transshipment from another local warehouse, or via an emergency shipment from the external supplier. We assume Poisson demand processes, generally distributed leadtimes for replenishments, repairs, and emergency shipments, and a basestock policy for the inventory control.Our inventory system is too complex to solve for a steady-state distribution in closed form. We approximate it by a network of Erlang loss queues with hierarchical jump-over blocking. We show that this network has a product-form steady-state distribution. This enables an efficient heuristic for the optimization of basestock levels, resulting in good approximations of the optimal costs.

scholarly journals On some tractable growth-collapse processes with renewal collapse epochs

2011 ◽  
Vol 48 (A) ◽  
pp. 217-234 ◽  
Author(s):  
Onno Boxma ◽  
Offer Kella ◽  
David Perry
Keyword(s):  
Steady State ◽  
Stationary Distribution ◽  
Unit Interval ◽  
General Distribution ◽  
Phase Type Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
Positive Real ◽  
Ratio Distribution ◽  
Phase Type

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes with independent exponential intercollapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general distribution concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase-type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov-modulated shot noise process from which the steady-state distribution for the related growth-collapse model can be inferred via level crossing arguments.

On Growth-Collapse Processes with Stationary Structure and Their Shot-Noise Counterparts

2009 ◽  
Vol 46 (2) ◽  
pp. 363-371 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Steady State ◽  
Shot Noise ◽  
Growth Process ◽  
Initial Conditions ◽  
Steady State Distribution ◽  
State Distribution ◽  
Stationary Structure ◽  
Stationary Version ◽  
Special Cases ◽  
The Times

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes. We begin with a stationary setup with some relatively general growth process and observe that, under certain expected conditions, point- and time-stationary versions of the processes exist as well as a limiting distribution for these processes which is independent of initial conditions and necessarily has the marginal distribution of the stationary version. We then specialize to the cases where an independent and identically distributed (i.i.d.) structure holds and where the growth process is a nondecreasing Lévy process, and in particular linear, and the times between collapses form an i.i.d. sequence. Known results can be seen as special cases, for example, when the inter-collapse times form a Poisson process or when the collapse ratio is deterministic. Finally, we comment on the relation between these processes and shot-noise type processes, and observe that, under certain conditions, the steady-state distribution of one may be directly inferred from the other.

A continuous review (s, S) inventory system in a random environment

1978 ◽  
Vol 15 (03) ◽  
pp. 654-659 ◽  
Author(s):  
Richard M. Feldman
Keyword(s):  
Poisson Process ◽  
External Environment ◽  
Compound Poisson Process ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Compound Poisson ◽  
Continuous Review ◽  
Environmental Process ◽  
State Of The Environment

The steady-state distribution of the inventory position for a continuous review (s, S) inventory system is derived. The demand for items in inventory is dependent on an external environment. During an interval of time in which the environment is in a fixed state, the demand follows a discrete-valued compound Poisson process. The parameters of the compound Poisson process depend completely on the state of the environment. The environmental process is modeled as a continuous-time Markov process.

scholarly journals Tail Asymptotics for a Random Sign Lindley Recursion

2010 ◽  
Vol 47 (1) ◽  
pp. 72-83 ◽  
Author(s):  
Maria Vlasiou ◽  
Zbigniew Palmowski
Keyword(s):  
Steady State ◽  
Storage Systems ◽  
Queueing Systems ◽  
Service Systems ◽  
Tail Asymptotics ◽  
Steady State Distribution ◽  
State Distribution ◽  
Random Sign ◽  
Special Cases ◽  
Tail Behaviour

We investigate the tail behaviour of the steady-state distribution of a stochastic recursion that generalises Lindley's recursion. This recursion arises in queueing systems with dependent interarrival and service times, and includes alternating service systems and carousel storage systems as special cases. We obtain precise tail asymptotics in three qualitatively different cases, and compare these with existing results for Lindley's recursion and for alternating service systems.

scholarly journals On some tractable growth-collapse processes with renewal collapse epochs

2011 ◽  
Vol 48 (A) ◽  
pp. 217-234 ◽  
Author(s):  
Onno Boxma ◽  
Offer Kella ◽  
David Perry
Keyword(s):  
Steady State ◽  
Stationary Distribution ◽  
Unit Interval ◽  
General Distribution ◽  
Phase Type Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
Positive Real ◽  
Ratio Distribution ◽  
Phase Type

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes with independent exponential intercollapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general distribution concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase-type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov-modulated shot noise process from which the steady-state distribution for the related growth-collapse model can be inferred via level crossing arguments.

A continuous review (s, S) inventory system in a random environment

1978 ◽  
Vol 15 (3) ◽  
pp. 654-659 ◽  
Author(s):  
Richard M. Feldman
Keyword(s):  
Poisson Process ◽  
External Environment ◽  
Compound Poisson Process ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Compound Poisson ◽  
Continuous Review ◽  
Environmental Process ◽  
State Of The Environment

The steady-state distribution of the inventory position for a continuous review (s, S) inventory system is derived. The demand for items in inventory is dependent on an external environment. During an interval of time in which the environment is in a fixed state, the demand follows a discrete-valued compound Poisson process. The parameters of the compound Poisson process depend completely on the state of the environment. The environmental process is modeled as a continuous-time Markov process.

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